Electrical and electronic architectures for computing

Abstract

Aspects of this technical solution are directed to electrical and electronic systems for computing. In an aspect, this technical solution can operate according to a quantum computing environment without decoherence at a macro scale, using electronic computing systems configured to operate according to various probabilistic operations corresponding to quantum computing behavior.

Description

[0001] Atty. Dkt. 135589-0103

[0002] ELECTRICAL AND ELECTRONIC ARCHITECTURES FOR COMPUTING

[0003] TECHNICAL FIELD

[0004] [1] The present implementations relate generally to computing, including but not limited to electrical and electronic architectures that include quantum behaviors.

[0005] INTRODUCTION

[0006] [2] One of the most promising technologies of our era is quantum computing. Quantum computers can be astronomically more powerful than conventional computing devices and systems. However, present conventional quantum computing systems are susceptible to failure by exposure to the ambient environment through decoherence, and must be rigorously shielded from the ambient environment to perform computation benefitting from quantum characteristics including quantum entanglement and interference. Decoherence occurs as a result of interaction between the quantum system and its environment. This is due to the extreme sensitivity of bits capable of operating in quantum states, or qubits, where even slight external impact can cause them to lose their specific quantum characteristics through decoherence. Without interference from the environment surrounding a quantum system, quantum entanglement can occur, which can drive quantum computation. However, interaction with the environment of the quantum system changes or destroys quantum entanglement and quantum interference and eliminates functionality of a quantum system for computation, resulting in decoherence of the quantum system and loss of entanglement in the quantum system and corresponding loss of quantum computation.

[0007] [3] Conventional quantum systems require at least significant shielding from the ambient environment or significant redundancy to partially mitigate effects of decoherence. For example, vibrations, magnetic and electric fields, light, temperatures, and many other external influences cause errors and can destroy the computing process of a conventional quantum system. To keep the qubits stable, they must be well shielded. However, it must also be possible to manipulate qubits to be able to perform a calculation at all. To achieve this, superconducting qubits, for example, are cooled to temperatures close to absolute zero and shielded from the ambient environment. With such complex and expensive infrastructure, these conventional superconducting qubits can only remain stable, with quantum characteristics, for fractions of a second. This lack of temporal stability significantly hinders and eliminates the ability to perform calculations with conventional quantum systems. Other qubit technologies such as ions, can retain their properties over minutes or even hours, but take much longer for a single computing step. Further, a relatively large number of qubits are needed in conventional quantum systems,

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[0009] such as for error correction, to partially mitigate the effects of decoherence by an ambient environment. The number of qubits needed for error correction in conventional quantum systems can far exceed the number of qubits otherwise required for a computation by orders of magnitude.

[0010] SUMMARY

[0011] [4] The present technical solution is directed at least to implementing quantum-scale behaviors at macro scale, to achieve quantum computing behavior and operation via classical computer hardware (e.g., hardware of a type used in classical computing), classical computer software, or any combination thereof. In general, macro scale can be greater than an atomic or quantum scale at which quantum behavior affects outcomes relevant to objects at that physical size. For example, classical computers correspond to a macro scale, because quantum behavior does not affect outcomes, including calculations, of classical computers. In contrast, atomic or quantum scale can be at or below a size at which quantum behavior affects outcomes relevant to objects at that physical size. This technical solution is thus directed to encoding and executing at least models, devices, systems, or any combination thereof, that perform at a macro scale the types of operations generally understood to occur at an atomic or quantum scale, including via one or more probabilistic operations by electrical or electronic systems that can replicate quantum computing behaviors including quantum entanglement, quantum uncertainty, quantum interference, and collapse of quantum states upon measurement.

[0012] [5] Thus, a technical solution for a classical computing architecture for quantum computing is provided. This technical solution of quantum computing environment without decoherence at a macro scale can correspondingly provide at least a technical improvement of eliminating complex and costly shielding between a quantum computing system and its ambient environment. This technical solution of providing a quantum computing environment without decoherence at a macro scale can correspondingly provide at least a technical improvement of eliminating complex and costly redundant qubits from a quantum computing system for error correction caused by decoherence.

[0013] [6] These concepts can be used to create or otherwise implement a system. Such as system can include one or more processors (or processing circuitry), coupled with memory (e.g., to perform operations, such as the operations that follow). The system can obtain one or more first values (e.g., scalar values) indicative of a first direction corresponding to a quantum state. The quantum state can be a pure quantum state or a mixed quantum state. The system can obtain one or more second values (e.g., measurement scalar values) indicative of a second direction corresponding to a measurement direction for the quantum state. The system can generate, based on the one or more first values and the one or more second values, a probability that the first direction corresponds to the second direction, the probability can include a value (e.g., a scalar

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[0015] value). The system can provide an output determinative of a collapse of the quantum state, the output indicative of the value of the probability.

[0016] [7] At least one aspect is directed to a device corresponding to and / or operating as a quantum gate. The apparatus can include a circuit to output, based on one or more input values (e.g., scalar input values, input coordinates) indicative of a first quantum state (e.g., represented as a first qubit vector, an input vector) and according to a transform operation associated with a type of quantum gate, one or more output values (e.g., scalar output values, output coordinates) indicative of a second quantum state (e.g., represented as a second qubit vector, an output vector) corresponding to output of the quantum gate.

[0017] [8] At least one aspect is directed to a device corresponding to and / or operating as a quantum gate. The apparatus can include a first circuit to output, based on one or more first input coordinates of a first quantum state and according to a first transform operation associated with a type of quantum gate, a first output coordinate indicative of a second quantum state corresponding to output of the quantum gate. The apparatus can include a second circuit to output, based on one or more second input coordinates of the first quantum state and according to a second transform operation associated with the type of quantum gate, a second output coordinate indicative of the second quantum state.

[0018] [9] At least one aspect is directed to a device corresponding to a quantum gate. The apparatus can include a memory and one or more processors. The apparatus can determine a transform operation associated with a type of quantum gate. The apparatus can include generate, based on one or more input coordinates indicative of a first quantum state and according to the transform operation, one or more output coordinates indicative of a second quantum state corresponding to output of the quantum gate.

[0019]

[0010] At least one aspect is directed to a non-transitory computer readable medium can include one or more instructions stored thereon and executable by a processor. The processor can determine a transform operation associated with a type of quantum gate. The processor can generate, based on one or more input coordinates indicative of a first quantum state and according to the transform operation, one or more output coordinates indicative of a second quantum state corresponding to output of the quantum gate.

[0020]

[0011] At least one aspect is directed to a method. The method can include obtaining, by a processor, one or more first scalar values indicative of a first direction corresponding to a quantum state. The method can include obtaining, by the processor, one or more second scalar values indicative of a second direction corresponding to a measurement direction for the quantum state. The method can include generating, by the processor and based on the one or more first scalar values and the one or more second scalar values, a probability that the first direction

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[0022] corresponds to the second direction, the probability can include a scalar value. The method can include providing, by the processor, an output determinative of a collapse of the quantum state, the output indicative of the scalar value of the probability.

[0023]

[0012] At least one aspect is directed to a method. The method can include outputting or otherwise providing, based on one or more first input coordinates of a first quantum state and according to a first transform operation associated with a type of quantum gate, a first output coordinate indicative of a second quantum state corresponding to output of the quantum gate. The method can include outputting, based on one or more second input coordinates of the first quantum state and according to a second transform operation associated with the type of quantum gate, a second output coordinate indicative of the second quantum state.

[0024]

[0013] At least one aspect is directed to a method. The method can include determining a transform operation associated with a type of quantum gate. The method can include generating, based on one or more input coordinates indicative of a first quantum state and according to the transform operation, one or more output coordinates indicative of a second quantum state corresponding to output of the quantum gate.

[0025]

[0014] At least one aspect is directed to a system. The system can include one or more processors, coupled with memory. The system can identify, based on a plurality of scalar values associated with a first qubit (e.g., a control qubit) in a coordinate space and a second qubit (e.g., a target qubit) in the coordinate space, one or more samples of an ensemble indicative of the first qubit and the second qubit. The system can determine, in accordance with the sample(s) of the ensemble, one or more probabilities of the first qubit and the second qubit aligning. The system can determine, for each of the one or more probabilities, a density (e.g., of the ensemble) indicative of a likelihood of respective states in the coordinate space. The system can transform the density into a transformed density corresponding to a probability of a state of the first qubit and the second qubit in the coordinate space. The system can generate, based on the transformed density, a second plurality of scalar values in the coordinate space indicative of a second state of the first qubit and a second state of the second qubit.

[0026]

[0015] At least one aspect is directed to a method. The method can include identifying, based on a plurality of scalar values associated with a control qubit in a coordinate space and a target qubit in the coordinate space, a sample of an ensemble indicative of the control qubit and the target qubit. The method can include determining, in accordance with the sample of the ensemble, one or more probabilities of alignment of the control qubit and the target qubit. The method can include determining, for each of the one or more probabilities, a first density (e.g., of the ensemble) indicative of likelihood of respective states in the coordinate space. The method can include transforming the first density into a transformed density corresponding to a probability of

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[0028] a state of the control qubit and the target qubit in the coordinate space. The method can include generating, based on the transformed density, a second plurality of scalar values in the coordinate space indicative of a second state of the control qubit and a second state of the target qubit.

[0029]

[0016] At least one aspect is directed to a non-transitory computer readable medium that can include one or more instructions stored thereon and executable by a processor to perform operations (e.g., the operations that follow). A processor executing the instructions can identify, based on a plurality of scalar values associated with a control qubit in a coordinate space and a target qubit in the coordinate space, a sample of an ensemble indicative of the control qubit and the target qubit. A processor executing the instructions can determine, based on the sample of the ensemble, one or more probabilities of alignment of the control qubit and the target qubit. A processor executing the instructions can determine, for each of the one or more probabilities, a first density (e.g., of the ensemble) indicative of likelihood of respective states in the coordinate space. A processor executing the instructions can transform the first density into a transformed density corresponding to a probability of a state of the control qubit and the target qubit in the coordinate space. A processor executing the instructions can generate, based on the transformed density, a second plurality of scalar values in the coordinate space indicative of a second state of the control qubit and a second state of the target qubit.

[0030] BRIEF DESCRIPTION OF THE FIGURES

[0031]

[0017] These and other aspects and features of the present implementations are depicted by way of example in the figures discussed herein. Present implementations can be directed to, but are not limited to, examples depicted in the figures discussed herein.

[0032]

[0018] FIG. 1 depicts a diagrammatic view of an example system, according to one embodiment of the present disclosure.

[0033]

[0019] FIG. 2 depicts a diagrammatic view of an example qubit cell, according to one embodiment of the present disclosure.

[0034]

[0020] FIG. 3 depicts a diagrammatic view of an example quantum state measurement device, according to one embodiment of the present disclosure.

[0035]

[0021] FIG. 4 depicts a diagrammatic view of an example single-qubit quantum gate architecture, according to one embodiment of the present disclosure.

[0036]

[0022] FIG. 5 depicts a diagrammatic view of another example single-qubit quantum gate architecture, according to another embodiment of the present disclosure.

[0037]

[0023] FIG. 6 depicts a diagrammatic view of an example quantum gate interconnect architecture, according to one embodiment of the present disclosure.

[0038]

[0024] FIG. 7 depicts a diagrammatic view of an example first block diagram of a CNOT device architecture, according to one embodiment of the present disclosure.

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[0040]

[0025] FIG. 8 depicts a diagrammatic view of an example second block diagram of a CNOT device architecture, according to one embodiment of the present disclosure.

[0041]

[0026] FIG. 9 depicts a diagrammatic view of an example third block diagram of a CNOT device architecture, according to one embodiment of the present disclosure.

[0042]

[0027] FIG. 10 depicts a diagrammatic view of an example of a multiple qubit (e.g., two qubit) measurement device architecture, according to one embodiment of the present disclosure.

[0043]

[0028] FIG. 11 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0044]

[0029] FIG. 12 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0045]

[0030] FIG. 13 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0046]

[0031] FIG. 14 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0047]

[0032] FIG. 15 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0048]

[0033] FIG. 16 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0049]

[0034] FIG. 17 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0050]

[0035] FIG. 18 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0051]

[0036] FIG. 19 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

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[0053]

[0037] FIG. 20 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure.

[0054] DETAILED DESCRIPTION

[0055]

[0038] Electric or electronic architectures for accurately synthesizing quantum computing gates, circuit, and measurement devices are discussed herein. This technical solution includes, for example, electric or electronic implementations of single-input quantum gates (e.g., X or R quantum gates), measurement devices, and a multiple-input quantum gate (e.g., a controlled NOT (or “CNOT”) quantum gate). Each of the gates and quantum circuits at least as discussed herein, and any quantum circuit that can be constructed including any of the gates or measurement devices as discussed herein, can be implemented in digital electronics, electrical circuits, software technologies, or any combination thereof, to achieve a technical improvement of accurate and scalable quantum computational behavior with classical computing technology (e.g., electronics, semiconductor processors, high-level computing languages).

[0056]

[0039] To achieve the same computational power as quantum computing, this technical solution includes various components to perform probabilistic operations (e.g., computation) via one or more of the gates, measurement devices, or any component thereof. For example, probabilistic states can be represented with respect to multiple dimensions as a vector (e.g., an “arrow”) having an origin at an origin of a multidimensional (e.g., three-dimensional (3D) or higher-dimensional) coordinate space and a point in the multidimensional coordinate space. For example, the point of the vector can be placed at a predetermined distance from the origin to constitute a unit vector indicative of a probabilistic state. For example, the probabilistic state can represent a likelihood of collapse of the state into a binary one (“|1)’) or binary zero (“|0)’) upon measurement. In an aspect, the multidimensional coordinate space can be a Euclidean coordinate space that corresponds to or is based on an abstract or non-Euclidean space (e.g., a Hilbert space). For example, gates according to this disclosure can perform operations on corresponding arrows of each of the gates, so that the number of operations would be O(N), where N is the number of arrows. In an aspect, modifying an arrow corresponds to and conveys the same quantum state information as modifying the corresponding ket representation. Thus, by performing probabilistic operations on arrows associated with probabilistic quantum states of each gate, this technical solution can provide a technical improvement to change a ket representation of a quantum gate using classical computing technology with significantly reduced computational resources and power. In an aspect, single arrow gates can transform a single-arrow state with a certain polarization, into another single-arrow state with a different polarization. In an aspect, an angle of an arrow with respect to a predetermined polarization direction is a reference in a coordinate

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[0058] space for a statistical ensemble (e.g., probabilistic state) of a single-arrow state. Thus, based on a determination of how a given gate rotates the polarization direction, and a transformation via a probabilistic operation on every arrow for the gate, the gate generates an output probabilistic state corresponding to a quantum state modified by the gate, using, for example, a matrix representation in the basis {| 0), |1)}.

[0059]

[0040] This technical solution includes various electric or electronic devices structured to execute probabilistic operations providing or otherwise associated with quantum mechanical behavior, according to probabilities with respect to linear spaces. This technical solution includes various models constructed to represent discrete operational implementations under quantum mechanics as a subclass of corresponding discrete operational implementations under probability theory. For example, models are discussed herein that can individually or collectively describe probabilistic experiments whose outcomes are defined in relation to a measurement device, where outcomes are created through the act of measurement itself and cannot be defined prior to it. Thus, this technical solution provides at least a technical improvement to replicate quantum-scale (e.g., microscopic) phenomena macroscopically. For example, quantum interference is apparent through probability amplitudes, and is a result of transforming between different bases in Hilbert space. Those bases can represent non-compatible experiments (measuring along a first polarization along the z-axis and measuring along a second polarization vector n), which are represented by different bases. Thus, this technical solution can replicate quantum interference as a result of the uncertainty principle, on a macroscopic scale. For example, Equation (1) and Equation (2) illustrate that probabilities themselves can behave like fields, with the two fields coupled by the constraint of Equation (3).

[0060] P₀: S² → [0,1]: n̂ ↦ P₀(n̂) = ψ₀*ψ₀ Eqn. (1)P₁: S² → [0,1]: n̂ ↦ P₁(n̂) = ψ₁*ψ₁ Eqn. (2)P₀ + P₁ = 1 Eqn. (3)

[0061]

[0041] Thus, the interference effect can arise from the behavior of probabilities given by a certain quantum state as continuous fields. As discussed herein by way of example, the constraint of Equation (3) can represent a simplified form of entanglement, but is not limited thereto by this technical solution. For example, in a two qubit-state case, given any two-qubit quantum state, probabilities of the two-qubit quantum state also behave as continuous fields, as illustrated in Equations (4)-(7), with the four fields coupled by the constraint of Equation (8).

[0062] P₀₀: S² × S² → [0,1]: (n̂,m̂) ↦ P₀₀(n̂,m̂) = ψ*₀₀ψ₀₀ Eqn. (4)P₀₁: S² × S² → [0,1]: (n̂,m̂) ↦ P₀₁(n̂,m̂) = ψ*₀₁ψ₀₁ Eqn. (5)P₁₀: S² × S² → [0,1]: (n̂,m̂) ↦ P₁₀(n̂,m̂) = ψ*₁₀ψ₁₀ Eqn. (6)P₁₁: S² × S² → [0,1]: (n̂,m̂) ↦ P₁₁(n̂,m̂) = ψ*₁₁ψ₁₁ Eqn. (7)

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[0064] P₀₀ + P₀₁ + P₁₀ + P₁₁ = 1 Eqn. (8)

[0065]

[0042] For example, a measurement along some (ii, in) results in a value indictive of a collapsed quantum state for P00(_n, in) = 1 for these two directions. From the constraint, the probabilities of the other results will change conditionally, and a correlation based on the conditional probabilities will appear between those probability distributions. This conditional probability results in probabilistic behavior associated with nonlocal correlations of quantum entanglement. Thus, this technical solution can provide a technical improvement to achieve behavior corresponding to quantum entanglement using probabilistic experiments done on macro-systems. In an aspect, this technical solution can result in behavior corresponding to quantum entanglement, including in that manipulating one qubit can change 2nnumbers that represent the quantum state of n qubits in macro-systems. For example, in a system of n fair coins, the probability, if thrown, that all are heads, is —. Now, if one coin is fixed to be heads, then the

[0066]

[0067] probability will become Thus, by manipulating one coin, all other probabilities for the

[0068]

[0069] ensemble of coins will change, which contain 2nnumbers. Despite that, no effect traveled between the coins, which explains the no-signaling principle. Thus, probability distributions for qubit vectors as discussed herein demonstrate properties associated with quantum entanglement.

[0070]

[0043] Aspects of this technical solution are described herein with reference to the figures, which are illustrative examples of this technical solution. The figures and examples below are not meant to limit the scope of this technical solution to the present implementations or to a single implementation, and other implementations in accordance with present implementations are possible, for example, by way of interchange of some or all of the described or illustrated elements. Where certain elements of the present implementations can be partially or fully implemented using known components, only those portions of such known components that are necessary for an understanding of the present implementations are described, and detailed descriptions of other portions of such known components are omitted to not obscure the present implementations. Terms in the specification and claims are to be ascribed no uncommon or special meaning unless explicitly set forth herein. Further, this technical solution and the present implementations encompass present and future known equivalents to the known components referred to herein by way of description, illustration, or example.

[0071]

[0044] FIG. 1 is a diagram of an example computing system 100, according to one embodiment of the present disclosure. The system 100 can be a quantum computing system, or otherwise provide quantum computing (e.g., exhibiting or otherwise including quantum behavior), in a macroscopic computing environment 102. As illustrated by way of example in FIG. 1, the system 100 can include a bus 104, a qubit cell array 110, a quantum gate array 120, one or more

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[0073] system processors 130, one or more quantum probability processors 140, and a communication interface 150.

[0074]

[0045] The macroscopic computing environment 102 can correspond to and / or encompass an ambient environment of the system 100, and especially the qubit cell array 110. For example, the ambient environment can correspond to an indoor environment having an ambient temperature corresponding to room temperature. For example, room temperature can include temperatures ranging from 60 degrees Fahrenheit (°F) to 80 °F. The ambient environment is not limited to the range of temperatures discussed herein by way of example. For example, the ambient environment of the system 100 and the qubit cell array 110 can correspond to an environment having a temperature, a latent energy, or the like, exceeding that of a superconducting ambient environment. For example, a superconducting ambient environment can correspond to an environment having a superconducting temperature. For example, a superconducting temperature can be within the range of 0 Kelvin (K) to 150 K. Thus, this technical solution, and the system 100 and the qubit cell array 110, can provide the technical improvement of operating with quantum characteristics in a macroscopic computing environment even at temperatures exceeding superconducting temperatures.

[0075]

[0046] The bus 104 can couple a system processor 130 (and / or a corresponding quantum probability processor 140) with the qubit cell array 110 and / or the quantum gate array 120. For example, the bus 104 can be a data bus. In another example, the bus 104 can be a link. The bus 104 (or a portion of a bus) can couple the qubit cell array 110 and the quantum gate array 120. The bus 104 can be a data bus to communicate data between the qubit cell array 110, the quantum gate array 120, and the system processor 130. In an aspect, the bus 104 can be a data bus to further communicate data to the quantum probability processor 140. In another example, the bus 104 can be a communication bus and can include one or more digital, analog, or like channels, lines, traces, pipelines, or the like. In another example, the bus 104 can be a memory bus. In another example, the bus 104 can be a virtual bus (e.g., a simulated bus, such as in software), such as may be implemented in software, a software pipeline, as a data structure, in system memory, or the like.

[0076]

[0047] The qubit cell array 110 can be any appropriate memory or storage to store one or more quantum states each corresponding to a particular qubit. For example, the qubit cell array 110 can store states of an arbitrary number of qubits, which can thereby implement a quantum computing architecture with an arbitrary number of qubits. In an aspect, the qubit cell array 110 can include one or more qubit cells 112, each storing the state of a single qubit. The qubit cell array 110 can be structured as a system memory. For example, a system memory can store data associated with the qubit cell array 110 and one or more of the qubit cells 112. The system memory can include

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[0078] one or more hardware memory devices to store binary data, digital data, or the like. The system memory can include one or more electrical components, electronic components, programmable electronic components, reprogrammable electronic components, integrated circuits, semiconductor devices, flip flops, arithmetic units, or the like. The memory can include dynamic random-access memory (DRAM), static random access memory (SRAM), other random access memory and / or volatile memory. The system memory can include at least one of a non-volatile memory device, a solid-state memory device, a flash memory device, or a NAND memory device. The system memory can include one or more addressable memory regions disposed on one or more physical memory arrays. A physical memory array can include a NAND gate array disposed on, for example, at least one of a particular semiconductor device, an integrated circuit device, and a printed circuit board device.

[0079]

[0048] The qubit cells 112 can each store a quantum state of a qubit in a macroscopic environment. For example, a qubit cell 112 can include a memory or a portion of the system memory, and can store a quantum state according to, for example a Bloch sphere (or Bloch ball). The qubit cells 112 can include one or more metrics, parameters, or the like to represent a quantum state according to, for example, a probability of having a binary [1] or binary [0] in accordance with a Bloch sphere. In an implementation, the qubit cells 112 can each store a quantum state according to, for example, a qubit vector.

[0080]

[0049] At least one aspect is directed to determining and modifying pure states of a single qubit, which is stored in a qubit cell 112. As discussed herein, systems of qubits can include a plurality of arrows. A probability of finding a given arrow after measurement along n, given that the arrow was already along in, is cos2(θ / 2), where θ is the angle between n̂ and m̂. For example, if an arrow is along some direction n, it is considered to be in the state 10), while if the arrow is along — n, it is considered to be in the state |1). For example, if the arrow is along a different direction in, it is considered to be in the state |0) along with other arrows in the ensemble that have the same direction, while if the arrow is along — in, it is considered to be in the state | i ) along with other arrows in the ensemble that have the same direction. For example, a pure state |0) can correspond to an ensemble (e.g., a collection of arrows each representing probabilities) having all arrows oriented along and in a same direction as n. For example, a pure state |1) can correspond to an ensemble having one or more arrows oriented along and in a same direction as —n. For example, a pure state |0) can correspond to an ensemble having all arrows oriented along and in a same direction as in. For example, a pure state | i) can correspond to an ensemble having arrows oriented along and in a same direction as — in. In an aspect, a result of a measurement along some direction is defined after the act of measurement, which means that the very act of

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[0082] measurement creates the result. Thus, probabilistic operation according to this technical solution can violate Bell’s inequalities and demonstrate accurate entanglement behavior. In an aspect, measurements along two different directions will be incompatible, which ensures that the model respects the uncertainty principle in order for it to be able to give us interference.

[0083]

[0050] At least one aspect is directed to determining and modifying mixed states of a single qubit. For example, Equations (9), (10), and (11), can represent a mixed state of a single qubit according to the probabilistic property of Equation (12).

[0084] P = p|0)(0| + (1 — p)|l)(l|: p G [0,1] Eqn. (9) |0) = a|6) +?|i) Eqn. (10) |1) = y|0) + 5|1) Eqn. (11) p = p(a 10) + / ? | l»(<z*<01 + / ?*<! I) + (1 - p)Cr 10> + 5|l))(y*(0|+5*(l|) = [p\a\2+ (1 - p)|y|2]|0)(0| + [p\ / 3\2+ (1 - p)|5|2]|i)(i| Eqn. (12)

[0085]

[0086] + [pa(3* + (1 - p)yd*]|0)(l| + [p(3a* + (1 - p)5y*]|l)(0|

[0087]

[0051] In an aspect, Equation (13) can describe the probability of a result of |0) after measuring along in, and Equation (14) can describe the probability of a result of |1) after measuring along in.

[0088] <0|p|0) = p\a\2+ (1 - p)|y|2= pcos2( j + (1 - p)cos2(— — j / (n,m)\ / (-n,m)\

[0089] = pcos I — - — I + (1 — p)cos I - - - I

[0090] Eqn. (13) <1 |p| 1> = p\p\2+ (1 - p) |512= psin2Q) + (1 - p)sin2

[0091] = psin2+ (1 - p)sin

[0092]

[0093] 2

[0094] Eqn. (14)

[0095]

[0052] Thus, in an example, a mixed state p can include an ensemble having arrows oriented along and in a same direction as n, and other arrows oriented along and in an opposite direction from n, where the ratio of the arrows aligned with n to the total number is p, and the ratio of the arrows anti-aligned with n is 1 — p. In an aspect, an ensemble of arrows as discussed herein can be structured or represented according to a corresponding density matrix. For example, Equations (15), (16) or (17) can represent a density matrix of a single qubit, and can represent an equal mixture of arrows which are a mixture of the form v 11 ft and v 41 n. For example, Equation (17) can represent an equal mixture of vectors which satisfy v 11 m and v 41 in.

[0096] p = | |0)(0| + | |l)(l| Eqn. (15) p = | |0)(0| + | |1)(1| = l(|0><0| + |1><1|) = | Eqn. (16)

[0097]

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[0099] p = l = | (10)(0| + 11><11) = 110><0| + 11!><! I Eqn. (17)

[0100]

[0101]

[0053] Thus, a given density matrix can represent a given probabilistic state, including at least a probabilistic state corresponding to a quantum state of a single qubit system. For example, any measurement of an ensemble, along an arbitrary axis or vector in a coordinate space can result in identical probability distributions, according to, for example, Equation (18). For example, measuring any qubit where p = | along a given measurement direction corresponds with an equal probability of a result of a collapsed state corresponding to 0 or a collapsed state corresponding to 1 along the given measurement direction.

[0102] P(ut) = (ut\p\ut) = (uLHI Ut) = | Eqn. (18)

[0103]

[0104]

[0054] In an aspect, a pure state of a system of a plurality of qubits (e.g., two qubits) can be represented with a density matrix. In an aspect, the density matrix of the system of the plurality of qubits can be indicative of non-local correlations of the pure state of the system of the plurality of qubits.

[0105]

[0055] In an example, Equation (19) describes a given two-qubit state, Equation (20) describes a total density matrix corresponding to the given two-qubit state, and Equation (21) describes a density matrix in the basis {| 00), 101), 110), 111)} of the given two-qubit state.

[0106] |< Z>+> = ^=| 00> + ^=|11> Eqn. (19)

[0107] 1

[0108] P = |0+)(0+| = - (|00> +|11»«OO|+<11|)

[0109] 1

[0110] = - (|00)(00| + |11)(11| + |00)(ll| + |ll)(00|)

[0111] Eqn. (20)

[0112]

[0113] 4869-4255-3171.8 13Atty. Dkt. 135589-0103

[0114] In this example, Equation (22) can describe a reduced density matrix of the first qubit of the given two-qubit state, and Equation (23) can describe a reduced density matrix of the second qubit of the given two-qubit state.

[0115] PA = TrB(p) =B< 0|p|0)B+B< l|p|l)B

[0116] = ^|0>^j4< 0|) + (— |lh)(— ^< 1|) ^

[0117] 1 1

[0118] AE4 = 2 l°>-4<°l + 2| 1>-4<11 =21 / 4

[0119] Eqn. (22) 1 1

[0120]

[0121] PB = 2 I°> B<°I + 2| 1>«<11 =2lfi

[0122] Eqn. (23)

[0123] Thus, in an aspect, any measurement on A or on B, results in 0 or 1 with equal probability. Moreover, due to the no signaling principle, neither pAnor pBchange whether we made a measurement on the other qubit or not. However, non-local correlations can nonetheless be represented in the density matrix or total density matrix (e.g., “density”).

[0124]

[0056] For example, pAcontains only information that describes what (4) alone sees, without any reference to (B), and pBcontains only information that describes what (B) alone sees, without any reference to Q4). Here, neither pAnor pBcontains information regarding comparison of pAwith pB, where the comparison describes the whole system (e.g., the given two-qubit state). Equations (24), (25) and (26) illustrate non-local correlations in the total density matrix. Thus, a total density matrix that corresponds to the given two-qubit state as a whole, can capture nonlocal correlations. For example, p contains the same information as |+) because p corresponds to a pure state. Thus, p indeed has the needed correlations.

[0125] 1 1 1 1

[0126] p = — 100)(00| + -|11)(11| + - |00)(ll| + -|ll)(00|

[0127] Eqn. (24)

[0128] 1 1

[0129] p(00) = <00|p|00) = -,p(ll) = (lllplll) = -

[0130]

[0131] Eqn. (25)

[0132] p(01) = <01|p|01) = 0,p(10) = <10|p|10) = 0

[0133] Eqn. (26)

[0134]

[0057] The quantum gate array 120 can store one or more quantum transformations according to various quantum gate operations. For example, the quantum gate array 120 can perform one or

[0135] 4869-4255-3171.8 14Atty. Dkt. 135589-0103

[0136] more quantum gate operations on individual qubits, or a plurality of qubits in an entangled state or unentangled state with each other. A quantum gate operation can correspond to, for example, a modification of one or more parameters representing a Bloch sphere or representing a qubit cell 112. The quantum gate array 120 can be structured as a system memory, and can correspond to a portion of a system memory including the qubit cell array 110. The quantum gate array 120 can include one or more quantum gates 122.

[0137]

[0058] In an example, each of the one or more quantum gates 122 can store, associate with, or otherwise correspond to an operator corresponding to the quantum gate. For example, the quantum gates 122 may include or otherwise access memory that stores instructions of a quantum operation. In an aspect, the one or more quantum gates 122 can each include one or more logic circuits or register devices to store a matrix having particular values at particular elements, indices, or cells thereof, or particular coefficients thereof. For example, logic circuits can correspond to AND, OR, NOT, NAND, NOR, or XOR gates. The one or more quantum gates 122 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like. It is to be understood that any electrical, electronic, or like devices, or components associated with the of the one or more quantum gates 122 can also be associated with, integrated with, integrable with, replaced by, supplemented by, complemented by, or the like, the qubit cell array 110 or any component thereof.

[0138]

[0059] The quantum gates 122 can each store and / or implement a transformation to modify a quantum state of one or more of the qubit cells 112. For example, a quantum gate 122 can be linked with a qubit at an input or an output of the quantum gate 122, in accordance with a quantum circuit structure. For example, a quantum gate 122 can have a structure and / or operation that corresponds to a Hadamard (H) gate, and can be coupled with a qubit cell 112 at an input of the H gate. Thus, the qubit cell 112 can have a quantum state corresponding to the transformation of the quantum state of the qubit cell by the quantum gate according to the H gate. For example, one or more of the quantum gates 122 can correspond to a distinct quantum gate, including but not limited to an X gate, a Y gate, a Z gate, a T gate, and a controlled NOT (CNOT) gate.

[0139]

[0060] In one embodiment, the quantum gate array 120 includes a set of quantum gates including multiple quantum gates 122, each of which may provide an individual gate instance. Each quantum gate 122 of the set of quantum gates provides a transformation according to a quantum algorithm. The set of quantum gates configured according to the quantum algorithm can perform complex calculations. The complexity of these calculations for meaningful computation increases dramatically as each gate is added to the set, so manual processes would not be able to achieve this result.

[0140] 4869-4255-3171.8 15Atty. Dkt. 135589-0103

[0141]

[0061] In one embodiment, the system 100 can include one or more non-transitory memory devices having a digital memory architecture and configured to store one or more qubits in a digital format, each corresponding to one or more qubit cells 112. An example is shown and discussed more fully in FIG. 2.

[0142]

[0062] The system processor 130 can execute one or more instructions associated with the system 100. The system processor 130 can include an electronic processor, an integrated circuit, or the like including one or more of digital logic, analog logic, digital sensors, analog sensors, communication buses, volatile memory, nonvolatile memory, and the like. The system processor 130 can include, but is not limited to, at least one microcontroller unit (MCU), microprocessor unit (MPU), central processing unit (CPU), graphics processing unit (GPU), physics processing unit (PPU), embedded controller (EC), or the like. The system processor 130 can include a memory storing or operable to store one or more instructions for operating components of the system processor 130 and operating components operably coupled to the system processor 130. For example, the one or more instructions can include one or more of firmware, software, hardware, operating systems, embedded operating systems. The system processor 130 or the system 100 generally can include one or more communication bus controllers to effect communication between the system processor 130 and the other elements of the system 100.

[0143]

[0063] The quantum probability processor(s) 140 can execute one or more transformations according to one or more of the quantum gates 122 on one or more of the qubit cells 112. For example, a quantum probability processor 140 can include one or more circuits to execute one or more matrix operations of the quantum gates 122. A quantum probability processor 140 can generate or update one or more quantum states based on one or more probability parameters encoded at the quantum probability processor 140. For example, a quantum probability processor 140 can include one or more processor cores each configured or fabricated to execute a particular matrix operation corresponding to one or more of the quantum gates 122. For example, a quantum probability processor 140 can include one or more probabilistic models, or circuits corresponding to probabilistic models, to operate the system 100 in accordance with probabilistic behavior of a quantum computing system. For example, the quantum probability processor 140 can update a state of the system 100 including one or more of the qubit cells 112 of the qubit cell array 110. For example, the quantum probability processor 140 can update the state of one or more of the qubit cells 112 of the qubit cell array 110 in accordance with a system refresh period. For example, the system refresh frequency can correspond to a period of a particular number of nanoseconds or even picoseconds. Thus, the quantum probability processor 140 can provide a technical improvement of rapidly and efficiently executing instructions in a macroscopic or classical computing environment corresponding to physical quantum states of one or more qubit

[0144] 4869-4255-3171.8 16Atty. Dkt. 135589-0103

[0145] cells 112. Quantum entanglement or a similar quantum behavior or effect can manifest as an overlap of the probability states in qubits (and / or ensembles as will be discussed more fully below).

[0146]

[0064] The communication interface 150 can communicatively couple the system 100 with an external device. An external device can include, but is not limited to, a smartphone, mobile device, wearable mobile device, tablet computer, desktop computer, laptop computer, cloud server, local server, and the like. The communication interface 150 can communicate one or more instructions, signals, conditions, states, or the like between one or more of the system processor 130 and components, devices, blocks operatively coupled or couplable therewith. The communication interface 150 can include one or more digital, analog, or like communication channels, lines, traces, or the like. As one example, the communication interface 150 can include at least one serial or parallel communication line among multiple communication lines of a communication interface. The communication interface 150 can include one or more wireless communication devices, systems, protocols, interfaces, or the like. The communication interface 150 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like. The communication interface 150 can include one or more telecommunication devices including but not limited to antennas, transceivers, packetizers, and wired interface ports. Any electrical, electronic, or like devices, or components associated with the communication interface 150 can also be associated with, integrated with, integrable with, replaced by, supplemented by, complemented by, or the like, the system processor 130 or any component thereof.

[0147]

[0065] FIG. 2 depicts an example qubit cell 200, according to one embodiment of the present disclosure. As illustrated by way of example in FIG. 2, an example qubit cell 200 can include a qubit quantum state memory 220. Each of the qubit cells 112 of FIG. 1 can correspond at least partially in one or more of structure and operation to the qubit cell 200, and can each have parameters, operators, or characteristics differing to achieve various differing quantum states, including disentangled (not entangled) and entangled quantum states.

[0148]

[0066] The qubit quantum state memory 220 can store one or more state parameters 232 corresponding to a quantum state of the qubit cell 200. Each of the one or more state parameters 232 can correspond to or represent a single qubit. In an aspect, the qubit quantum state memory 220 can include or correspond to a memory register, a solid-state memory device, or a portion of the system memory. A state parameter 232 (e.g., qubit representation) can include or comprise a qubit vector 242 (e.g. qubit coordinates). In an aspect, the qubit vector 242 may comprise or include one or more scalar values 244x, 244y, 244z, 244n (generally or collectively scalar value(s) 244). For example, the state parameter 232 can correspond to a scalar value, vector

[0149] 4869-4255-3171.8 17Atty. Dkt. 135589-0103

[0150] value (e.g. coordinate), or object indicating a quantum state of the qubit cell 200. For example, the state parameter 232 can indicate a probability of a state of the qubit cell 200 having a value of [1] or [0] according to a Bloch sphere. Each qubit cell 200 can be one of the qubit cells 112 of FIG. 1 and can store corresponding state parameters 232 for corresponding qubits of the system 100. For example, the quantum probability processor 140 can set or modify a state parameter 232 of the qubit cell 200.

[0151]

[0067] In the embodiment of FIG. 2, the qubit cell 200 includes a qubit vector 242 that includes three scalar values 244x, 244y, 244z, each corresponding to or providing a coordinate value of a vector in three-space. In other embodiments, the qubit vector 242 may include any number of scalar values 244n, such as may correspond to an n-dimensional space. For example, the qubit vector 242 can include more than three scalar values 244. As another example, the qubit vector 242 can include fewer than three scalar values 244.

[0152]

[0068] As will be described, in one embodiment the state parameter 232, and more specifically the qubit vector 242, and still more specifically the scalar values 244, are accessed, sampled, or otherwise provided for use by quantum state measurement devices and quantum gates to perform quantum operations and / or to accomplish quantum computing algorithms.

[0153]

[0069] FIG. 3 depicts an example quantum state measurement device 300, according to one embodiment of the present disclosure.

[0154]

[0070] A measurement device is something from which can be read the result of some measurement. A measurement can be a direct interaction (e.g., of a qubit, or qubit cell), which is a measurement of a value that is already present - i.e., the value that is measured is present at the measurement. A measurement can be an indirect interaction, which may be a value that results from the measurement - i.e., the value that is measured is generated or created as a result of the measurement.

[0155]

[0071] There are at least six scenarios during a measurement that may be of interest and / or pertinence.

[0156]

[0072] In a first scenario, the measurement result is already defined before measurement, and in the case of composite systems the mapping is defined in the source, and is not changed as a result of the measurement. This can be like classical ensembles, which do not violate Bell’s inequalities.

[0157]

[0073] In a second scenario, the measurement result is already defined before measurement, and in the case of composite systems the mapping is defined in the source, but is changed during a measurement. In this scenario, since the mapping depends on the measurement, it is not clear whether or not the factoring can be done that is needed to prove Bell’s theorem, which means that in the cases where the needed factoring cannot be done, Bell’s inequalities may be violated.

[0158] 4869-4255-3171.8 18Atty. Dkt. 135589-0103

[0159]

[0074] In a third scenario, the measurement result is already defined before measurement, and in the case of composite systems the mapping is not defined at the source, meaning, before the measurement takes place, and it is only defined as a result to the act of measurement. In this case, since the mapping depends on the measurement, it is not clear whether or not the factoring can be done that is needed to prove Bell’s theorem, which means that in the cases where the needed factoring cannot be done, Bell’s inequalities may be violated.

[0160]

[0075] In a fourth scenario, the measurement result is created as a result of the act of measurement, and in the case of composite systems the mapping is defined in the source, and is not changed as a result of the measurement. In this scenario, the result of measurement might be contextual, which makes it not clear that Bell’s theorem can always be proven, which means that in the cases where that cannot be done, Bell’s inequalities may be violated.

[0161]

[0076] In a fifth scenario, the measurement result is created as a result of the act of measurement, and in the case of composite systems the mapping is defined in the source, but it gets changed during a measurement. In this scenario, since the mapping depends on the measurement, it’s not clear whether the factoring can be done that is needed to prove Bell’s theorem, which means that in the cases where the needed factoring cannot be done, Bell’s inequalities may be violated, and that the results of measurement might be contextual.

[0162]

[0077] In a sixth scenario, the measurement result is created as a result of the act of measurement, and in the case of composite systems the mapping is not defined at the source, meaning, before the measurement takes place, and it is only defined as a result to the act of measurement. In this scenario, since the mapping depends on the measurement, it is not clear whether or not we can make the factoring needed to prove Bell’s theorem, which means that in the cases where the needed factoring cannot be done, Bell’s inequalities may be violated, and that the results of measurement might be contextual.

[0163]

[0078] As is apparent from the foregoing, the inability to violate Bell’s inequalities is the exception not the norm, and through taking the considerations above into account, probabilistic models can be designed which are local, and at the same time violate Bell’s inequalities.

[0164]

[0079] A sidenote: if we have an ensemble of composite systems such that all the results of measurements of individual systems are only defined after the act of measurement, then the individual systems themselves lose their individual identities and they appear as if they were one entity. However, that does not mean they do not still have their own hidden variables that preserve their intrinsic identities. This might be used to explain the physics of some entangled states like the singlet state, for example.

[0165]

[0080] We stress here that the foregoing makes the assumption that Bell made to deduce his inequalities to separate the probability into a product to reflect locality. (See e.g., David J.

[0166] 4869-4255-3171.8 19Atty. Dkt. 135589-0103

[0167] Griffiths, Introduction to Quantum Mechanics, p. 378 (Prentice Hall, Inc. 1995). This may be a questionable assumption, since in no way enters to it the methods by which we should identify subsystems together, which is something we can do experimentally, according to Raedt. (See e.g., Hans De Raedt, et al., “Einstein-Podolsky-Rosen-Bohm experiments: A discrete data driven approach,” Annals of Physics v. 453, June 2023). From Raedt we see that we can multiply by a unit step function Θ(W − |t1− t2|), where W is some fixed time window, and t1and t2are the times of arrivals of the particles to the two detectors. Notice that if the differences between times are not constant, we cannot take the step function out of the integral, and the separation of probabilities is not possible. However, we see that what Bell did implicitly is that he treated the two particles as two classical particles moving with the same speed, so they get to the detectors at the same time. Hence the step function is always 1 and we can make such factorization of the probability. (Even if the particles traveled different distances, with the complete knowledge of their velocities we can account for the time difference and make the step function equal to 1).

[0168]

[0081] However, we have seen that the identification of subsystems is far from being an experimental artifact, and it enters into the definition of the state itself, meaning that the state itself carries in it a way that tells us which subsystems must be identified together, which translates into the way in which we should build our detectors. This is why the measurement device designed for a single qubit, is not guaranteed to work in a multiple qubit case, since it has a fixed design that doesn’t depend on how to couple subsystems. But at the same time properly designed single qubit measurement devices may be viable for any number of qubits, provided that they include a multiple qubit identification mechanism.

[0169]

[0082] As illustrated by way of example in FIG. 3, a quantum state measurement device 300 can include at least a quantum reliability processor 310, a qubit bus 320, a measurement vector bus 330, a random number generator 340, a polarization processor 350, and an output measurement direction vector memory 360. The quantum state measurement device 300 can measure a quantum state of a qubit cell, such as a qubit cell 200 of FIG. 2, to extract quantum information.

[0170]

[0083] For example, Equations (27), (28) and (29) describe properties of a state vector corresponding to a probability space (

[0171]

[0172] Ωρ(Ω), P), where Ω = {u1... uN}. Here, the state vector can contain more information than either of Q or P alone. Specifically, the state vector can contain information about all probabilistic experiments of the class CN. Thus, when we write that a state vector of a system according to Equation (30) can model knowledge of results of each experiment of the class CN, together with the probability distribution of the results.

[0173] 4869-4255-3171.8 20Atty. Dkt. 135589-0103

[0174] N IV>) = Ci l«i>

[0175] i=l

[0176] Eqn. (27)

[0177] = 1 Eqn. (28)

[0178] (uju7) = 8ij Eqn. (29)

[0179] N IV>) = Ct \ut)

[0180]

[0181] i=l

[0182] Eqn. (30)

[0183]

[0084] In an aspect, a single qubit pure state can be characterized by its polarization direction, which in turn can be described in 3-space using a unit vector n = (n1, n2, n3). Accordingly, the quantum state parameter in the quantum state memory 220 can be representative of a unit vector n = (n1, n2, n3), and therefore representative of a quantum state of a qubit cell to be measured. For example, the scalar values n1, n2, n3can be indicative of a first direction corresponding to a quantum state of a qubit cell to be measured by the quantum state measurement device 300. The measurement device for a single qubit can be described along unit vector m = (m1,m2,m3). The unit vector m can be indicative of a second direction that corresponds to a measurement direction for the quantum state of a qubit cell to be measured. Stated otherwise, the scalar values m1,m2,m3can be indicative of a second direction corresponding to a measurement direction for the quantum state of the qubit cell to be measured by the quantum state measurement device 300.

[0184]

[0085] An example quantum state can be described according to Equation (31). For example, a measurement of a first qubit results in the state |n = 0), the probability of finding the first qubit in the state \m = 0) is described by Equation (32), and the probability of finding the first qubit in the state \m = 1) is described by Equation (33), for any direction m. Accordingly, Equation (34) can describe the state of the second qubit, in this example, after measuring and finding the first qubit to be in the state |n = 0)).

[0185] \i ) = ^Poo\n= 0)|m = 0) + >oi|n = 0)|m = 1) + >10|n = l)|m = 0) + ^1±\n = l)|m = 1)

[0186] Eqn. (31)

[0187] 4869-4255-3171.8 21Atty. Dkt. 135589-0103

[0188] p((n = 0) A (m = 0))

[0189] p(m = 0|n = 0)

[0190] p(n = 0)

[0191] iV'ool2V>oo

[0192] I V>00 I2+ I V>0112Vl^ool2+ lV>oi l2

[0193] Eqn. (32), -< 1 P^n= 0~) A (m = 1))

[0194] p(m = 1 n = 0) = - - - - -1 7p(n = 0)

[0195] _ I V^oi 12_ _ ^01 _

[0196]

[0197] I V>00 I2+ I V>0112Vl^ool2+ IV>0112

[0198] Eqn. (33)

[0199]

[0086] Accordingly, Equation (34) can describe the state of the second qubit, in this example, after measuring and finding the first qubit to be in the state |n = 0)). For example, measuring the first qubit as |n = 1), the state of the second qubit can be described according to Equation (35). Thus, Equation (36) can describe a state before measurement. In an aspect, a measurement on a system may result in {|n = 0), |i >2)} with a probability | ip0012+

[0200]

[0201] | 12or {| n = 1), |2)} with a probability IV^IO I2+ lV>nl2- _ V^oo _ i / > m = 0) + - \m = 1) Vl^ool2+ iV'oil2Vl^ool2+ lV>oi I2

[0202] Eqn. (34)

[0203] i / >2) = ^10m = 0) + ^1: L\m = 1) Vl^iol2+ lV>nl2Vl^iol2+ lV>nl2

[0204] Eqn. (35) V>00 IVO = VlV>ool2+ lV>oil2l™ = 0) 0. — - — m = 0) + ^01m j = 1)

[0205] WlV>ool2+ lV>oil2Vl^ool2+ IV'OI I2

[0206] , ^11 Wl^iol2+ lV>nl2l™ = 1) 0 (., ^10= m = 0) + —, m = 1)

[0207] WlV>iol + IV>nl Vi^ioi2+ iv>n i2

[0208]

[0209] = Vl^ool2+ IV>01 |2|™ = 0) 0I^2> + 71^10 I2+ IV>11|2|™ = 1) 0IV>2>

[0210] Eqn. (36)

[0211]

[0087] In an example, the quantum state measurement device 300 can receive a pair of triplets of coordinates. A first triplet of coordinates (e.g., first scalar values), n1,n2,n3, can provide a quantum state of a qubit cell to be measured. A second triplet of coordinates (e.g., second scalar values, measurement coordinates), m1,m2,m3, can define a second vector in the three-dimensional space (e.g., corresponding to the Hilbert space), which is a direction of

[0212] 4869-4255-3171.8 22Atty. Dkt. 135589-0103

[0213] measurement. In an aspect, the quantum state measurement device 300 can include a plurality of electrical or electronic devices structured or configured to generate metrics indicative of relative probabilistic states.

[0214]

[0088] The quantum state measurement device 300 can perform one or more operations, including for example product operation(s), summation operation(s), comparison operations(s), and assignment operation(s), to output a measurement of the qubit cell and / or output a new polarization direction of the qubit cell. More specifically, the quantum state measurement device 300 can obtain or otherwise receive a first triplet of coordinates (n1,n2,n3), for example from first state coordinate memory 322, second state coordinate memory 324, and third state coordinate memory 326 (Ni, N2, N3), and can obtain or otherwise receive a second triplet of coordinates (m1, m2, m3), for example from first measurement coordinate memory 332, second measurement coordinate memory 334, and third measurement coordinate memory 336 (Mi, M2, M3), and perform operations to output a measurement of the qubit cell along m = (m1, m2, m3) and / or a new polarization (c1, c2, c3) of the qubit cell. The quantum state measurement device 300 can include circuitry to perform these one or more operations.

[0215]

[0089] The circuitry can be implemented by and / or otherwise include one or more processors, processing units, processing devices, or the like, which can be coupled to memory (e.g., a non-transitory computer readable medium, digital memory) that includes executable instructions (e.g., compiled software).

[0216]

[0090] A processor, processing unit, processing device, or the like can include one or more general purpose devices, such as an Intel®, AMD®, NVIDIA® or other central processing unit (CPU) or similar microprocessor.

[0217]

[0091] A processor, processing unit, processing device, or the like can include a special purpose processing device, such as a graphics processing unit (GPU), a physics processing unit (PPU), an embedded controller (EC), complementary metal-oxide semiconductor (CMOS) technology, a gate array, a programmable gate array (PGA), a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), a system on a chip (SoC), a system in a package (SiP), a programmable array logic (PAL), a programmable logic array (PLA), a field programmable logic array (FPLA), a programmable logic device (PLD), other customized, customizable or programmable device, a dedicated pipeline, and / or a sequencer.

[0218]

[0092] A processor, processing unit, processing device can, or the like include analog logic, analog components, including one or more opamps, sensors, and other analog voltage control components.

[0219]

[0093] A processor, processing unit, processing device, or the like can be an optical processing device that includes optical components, such as optical gates, optical switches, optical flip-flops,

[0220] 4869-4255-3171.8 23Atty. Dkt. 135589-0103

[0221] resonators, filters, modulators, etc., to perform digital computations using photons or otherwise use light as a primary means for carrying out calculations, reasoning, artificial intelligence, etc.

[0222]

[0094] A processor, processing unit, processing device can be implemented using semiconductor wafer technology, integrated circuit technology, flip-chip technology, and / or other appropriate semiconductor devices and / or technology.

[0223]

[0095] A processor, processing unit, processing device can include distributed (e.g., parallel) processing to execute or otherwise implement functionalities of the present embodiments. A processor, processing unit, processing device can execute or otherwise include an operating system, such as a standard operating system and perform standard operating system functions. It is recognized that any standard operating systems may be used, such as, for example, Microsoft® Windows®, Apple® MacOS®, Disk Operating System (DOS), UNIX, IRIX, Solaris, SunOS, FreeBSD®, Linux®, FreeRTOS™, IBM® OS / 2® operating systems, and so forth.

[0224]

[0096] The memory may include static RAM, dynamic RAM, flash memory, one or more flipflops, ROM, or any other computer-readable storage medium. The memory may include optical media. The memory may include magnetic media.

[0225]

[0097] The quantum reliability processor 310 of the quantum state measurement device 300 can perform one or more of product operations and / or summation operations on and / or using the first and second triplets of coordinates. As an example, the quantum reliability processor 310 can perform operations on and / or using (n1, n2, n3) and (m1;m2, m3) to determine a probability p (e.g., a scaled probability), according to Equation (37).

[0226] p = 50(1 + n1m1+ n2m2+ n3m3) Eqn. (37)

[0227]

[0098] For example, the quantum reliability processor 310 can provide circuitry to generate, based on one or more first scalar values (e.g., n1, n2, n3) and the one or more second scalar values (e.g., m1,m2, m3), a probability that a first direction (indicated by the first scalar values) corresponds to the second direction (indicated by the second scalar values). The probability can include a scalar value.

[0228]

[0099] The quantum reliability processor 310 can include a coordinate product circuit 312, and a coordinate summation circuit 314. The coordinate product circuit 312 and the coordinate summation circuit 314 can implement or otherwise enable operations to be performed on or using the first and second triplets.

[0229]

[0100] The coordinate product circuit 312 can perform product operations, such as multiplying coordinate scalar values, for example n1m1, n2m2, and n3m3. The coordinate product circuit 312 can also perform such operations as multiplying a scalar value by another value, such as a constant (e.g., 50). In an aspect, the coordinate product circuit 312 can include a multiply-accumulate (MAC) or multiply-add (MAD) operation that computes the product of two numbers

[0230] 4869-4255-3171.8 24Atty. Dkt. 135589-0103

[0231] and adds that product to an accumulator. In an aspect a MAC of the coordinate product circuit 312 can be implemented by a MAC unit.

[0232]

[0101] The coordinate summation circuit 314 can perform summation operations such as summing (e.g., adding) coordinate scalar values, for example n1m1+ n2m2+ n3m3. The coordinate summation circuit 314 can also perform such operations as adding a scalar value (e.g., a summation) to another value, such as a constant (e.g., 1). The coordinate summation circuit 314 can perform comparison operations such as comparing a scalar value to another.

[0233]

[0102] The qubit bus 320 can obtain the one or more first scalar values and store them in memory or otherwise provide them to or for the quantum state measurement device 300. The qubit bus 320 can include or access a first coordinate memory 322 (or Ni), a second coordinate memory 324 (or N2), and a third coordinate memory 326 (or N3).

[0234]

[0103] In an aspect, the one or more first scalar values can correspond to one or more first coordinates defining a first vector in a three-dimensional space corresponding to a Hilbert space, the first vector having the first direction. The one or more first coordinates (e.g., first scalar values) can represent or otherwise provide a quantum state of a qubit cell to be measured. In an aspect, the qubit bus 320 can provide one or more first scalar values that correspond to one or more first coordinates defining a first vector in a three-dimensional space corresponding to a Hilbert space, the first vector having the first direction. As an example, the first scalar values may be coordinates of the unit vector n = (n1,n2,n3'). In FIG. 3, the quantum state memory 220 of FIG. 2 is depicted, which stores the state of a qubit cell (e.g., a qubit cell 112 of FIG. 1). The qubit bus 320 can obtain the first scalar values through accessing the quantum state memory 220.

[0235]

[0104] In an aspect, the system can obtain the one or more first scalar values as output of a quantum gate (e.g., a quantum gate 122 of FIG. 1) configured to transform, according to a type of the quantum gate, one or more input scalar values into the one or more first scalar values. For example, the qubit bus 320 can obtain the one or more first scalar values as output of a quantum gate configured to transform, according to a type of the quantum gate processor, one or more input scalar values into the one or more first scalar values. In FIG. 3, the quantum state memory 220 of FIG. 2 is depicted, which can store the output of a quantum gate. The qubit bus 320 can obtain the first scalar values through accessing quantum state memory 220. In another aspect, the system can obtain the one or more first scalar values from a source for qubits (e.g., a qubit cell array 110 of FIG. 1).

[0236]

[0105] In an aspect, the one or more first scalar values correspond to a first triplet of coordinates, and the one or more second scalar values correspond to a second triplet of coordinates. For example, the qubit bus 320 can store or otherwise provide one or more first scalar values that correspond to a first triplet of coordinates. The qubit bus 320 can provide the first triplet of

[0237] 4869-4255-3171.8 25Atty. Dkt. 135589-0103

[0238] coordinates, for example, to first state coordinate memory 322, second state coordinate memory 324, and third state coordinate memory 326.

[0239]

[0106] The first state coordinate memory 322 (Ni), the second state coordinate memory 324 (N2), and the third state coordinate memory 326 (N3) can each store a scalar value (e.g., n1, n2, n3, respectively) of a first triplet of coordinates (e.g., n1, n2, n3of the unit vector n in a three-dimensional space corresponding to a Hilbert space). The first state coordinate memory 322 (Ni), the second state coordinate memory 324 (N2), and the third state coordinate memory 326 (N3) (generally and collectively “state coordinate memory”) may temporarily store the first scalar values of the first triplet of coordinates during or for use in the measurement process of the quantum state measurement device 300. The state coordinate memory 322, 324, 326 can be implemented as or otherwise include static RAM, dynamic RAM, flash memory, one or more flip-flops, ROM, or any other appropriate computer-readable storage medium.

[0240]

[0107] The measurement vector bus 330 can obtain one or more second scalar values and store them in memory or otherwise provide them for them to or for the quantum state measurement device 300. The measurement vector bus 330 can include or access a first measurement coordinate memory 332 (or Mi), a second measurement coordinate memory 334 (or M2), and a third measurement coordinate memory 336 (or M3).

[0241]

[0108] In an aspect, the one or more second scalar values corresponding to one or more second coordinates defining a second vector in the three-dimensional space corresponding to a Hilbert space, the second vector having the second direction. The one or more second coordinates (e.g., second scalar values) can provide a measurement vector (e.g., the second vector) for use in measuring a quantum state of a qubit cell. In an aspect, the measurement vector bus 330 can provide one or more second scalar values that correspond to one or more second coordinates (e.g., measurement coordinates) defining a second vector in the three-dimensional space corresponding to the Hilbert space, the second vector having the second direction. The one or more second coordinates can provide a measurement device for a single qubit, which can be described along the second vector having the second direction. As an example, the second scalar values may be coordinates of the unit vector m = (m1, m2, m3). In FIG. 3, a measurement vector memory 303 is depicted, which can store the second scalar values (e.g., second coordinates, measurement coordinates) for a measurement vector. The measurement vector memory 303 can include or correspond to a memory register, a solid state memory device, or a portion of the system memory. The measurement vector memory 303 may store or otherwise be populated with second scalar values (e.g., second coordinates, a measurement vector). The scalar values may be generated, specified, or otherwise provided by a separate process, such as by or from a control

[0242] 4869-4255-3171.8 26Atty. Dkt. 135589-0103

[0243] processor. The measurement vector bus 330 can obtain the second scalar values through accessing the measurement vector memory 303.

[0244]

[0109] In an aspect, the one or more first scalar values correspond to a first triplet of coordinates, and the one or more second scalar values correspond to a second triplet of coordinates. For example, the measurement vector bus 330 can store or otherwise provide one or more second scalar values that correspond to a second triplet of coordinates. The measurement vector bus 330 can provide the second triplet of coordinates, for example, to first measurement coordinate memory 332, second measurement coordinate memory 334, and third measurement coordinate memory 336.

[0245] [HO] The first measurement coordinate memory 332 (Mi), the second measurement coordinate memory 334 (M2), and the third measurement coordinate memory 336 (M3) can each store a scalar value (e.g., m1,m2,m3, respectively) of a second triplet of coordinates (e.g., m1, m2, m3of the arbitrary unit vector m in a three-dimensional space corresponding to a Hilbert space). The first measurement coordinate memory 332 (Mi), the second measurement coordinate memory 334 (M2), and the third measurement coordinate memory 336 (M3) (generally and collectively “measurement coordinate memory”) may temporarily store the second scalar values of the second triplet of coordinates during or for use in the measurement process of the quantum state measurement device 300. The measurement coordinate memory 332, 334, 336 can be implemented as or otherwise include static RAM, dynamic RAM, flash memory, one or more flip-flops, ROM, or any other appropriate computer-readable storage medium.

[0246] [Hl] The random number generator 340 can generate a random number to be provided to the polarization processor 350. The random number generator 340 can generate a random number or pseudo-random number as output. For example, the output of the random number generator 340 can correspond to quantum fluctuation, quantum uncertainty, or the like. The random number generator 340 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like. It is to be understood that any electrical, electronic, or like devices, or components associated with the random number generator 340 can also be associated with, integrated with, integrable with, replaced by, supplemented by, complemented by, or the like, the polarization processor 350 or any component thereof.

[0247]

[0112] The polarization processor 350 can generate third scalar values (e.g., measurement values, or polarization values), which are representative of or indicative of an output qubit, and which are based at least partially on a measurement direction for the quantum state that is being measured. For example, the third scalar values can represent an output qubit with a direction of either 1 or -1. The polarization processor 350 can, for example, determine whether two arrows

[0248] 4869-4255-3171.8 27Atty. Dkt. 135589-0103

[0249] (e.g., each pointing in a direction) in a multidimensional (e.g., 3D) space are aligned, and based on that determination can generate the third scalar values, which are representative of the output qubit and / or state thereof.

[0250]

[0113] In an aspect, the polarization processor 350 can perform operations to determine whether an arrow (e.g., pointing in a direction) providing a qubit state, and which may be characterized as a unit vector n = (n1, n2, n3) is aligned with a measurement direction (e.g., a predetermined direction), or arrow pointing in that measurement direction, which may be characterized as a unit vector m = (m1, m2, m3).

[0251]

[0114] In an example, the polarization processor 350 can obtain or otherwise receive a random number q from the random number generator 340 and can receive a probability p (e.g., see Eqn. (37) above) from the quantum reliability processor and can generate the third scalar values (e.g., Ci, c2, c3) based at least partially on a comparison of the probability p and the random number q. In an aspect, if random number q is less than or equal to probability p, then a third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to the second triplet of coordinates (e.g., the second scalar values m1, m2, m3). And, otherwise (i.e., if random number q is not less than or equal to probability p) the third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to an inverse of the second triplet of coordinates (e.g., — m1, — m2, — m3).

[0252]

[0115] In an aspect, a system according to the present disclosure can aggregate, into one or more aggregated coordinates, first coordinates of the one or more first scalar values with second coordinates of the one or more second scalar values. For example, the polarization processor 350 can aggregate, into one or more aggregated coordinates, first coordinates of the one or more first scalar values with second coordinates of the one or more second scalar values. In an aspect, the system can combine the aggregated coordinates into the probability. For example, the polarization processor 350 can combine the aggregated coordinates into the probability. In an aspect, the system can generate, according to the probability satisfying a threshold, one or more third scalar values based at least partially on the measurement direction for the quantum state. For example, the polarization processor 350 can generate, according to the probability satisfying a threshold, one or more third scalar values based at least partially on the measurement direction for the quantum state. The polarization processor 350 can include a threshold comparator 352, and a coordinate assignment circuit 354.

[0253]

[0116] The threshold comparator 352 can compare a probability (that a first direction / arrow for a qubit aligns with a second direction / arrow for a measurement) to a threshold. In an aspect, the threshold is based on at least one of a random value, a pseudorandom value, or a quasi-random value. In an aspect, the threshold can be, can include, or can be generated based on an output of a

[0254] 4869-4255-3171.8 28Atty. Dkt. 135589-0103

[0255] random number generator. For example, a threshold can be provided by the random number generator as a random number q.

[0256]

[0117] In an example, the threshold comparator 352 can compare a probability p to a threshold q. For example, the comparison may include comparing whether the threshold q is less than the probability p. For example, the comparison may include comparing whether the threshold q is less than or equal to the probability p. For example, the comparison may include comparing whether the threshold q is equal to the probability p. For example, the comparison may include comparing whether the threshold q is greater than or equal to the probability p. For example, the comparison may include comparing whether the threshold q is greater than the probability p.

[0257]

[0118] The threshold comparator 352 may provide a determination, result, value, or other indication of the result of the comparison. In an example, the comparison is fixed (e.g., is q < p) and the threshold comparator 352 simply provides a result of the fixed comparison, such as in a form of a binary (0, 1; yes / no), a Boolean, etc. In an example, the threshold comparator 352 simply provides the comparison in addition to the result of the comparing. In an aspect, the comparison may be configurable or otherwise changeable, such that a meaning of the result of the comparison is informed by the comparison performed. For example, a result “Yes” may be informed by and have different meaning depending on whether the comparison is defined as “q < p” or defined as “q > p”.

[0258]

[0119] The coordinate assignment circuit 354 can assign or otherwise designate a polarization of a qubit being measured, based on a determination, result, or other indication of the threshold comparator 352. In an aspect the threshold comparator 352 may generate or otherwise provide an indication of a comparison (e.g., of a probability and a threshold) to the coordinate assignment circuit 354. The coordinate assignment circuit 354 can then perform operations to assign or otherwise designate third scalar values (e.g., measurement values, or polarization values), which are representative of or indicative of an output qubit and which are based on the comparison. For example, the coordinate assignment circuit 354 may assign or otherwise designate third scalar values as being positive in the case the comparison is positive and may assign or otherwise designate the third scalar values as being negative in the case the comparison is negative. As another example, the coordinate assignment circuit 354 may assign or otherwise designate third scalar values as having the same sign (e.g., direction) as the measurement direction in the case the comparison is positive or otherwise successful and may assign or otherwise designate the third scalar values as having the opposite sign (e.g., direction) as the measurement direction in the case the comparison is negative, fails, or is otherwise unsuccessful.

[0259]

[0120] In an example, a qubit state being measured may be characterized as an arrow or direction defined by or indicated by a unit vector n = (n1, n2, n3) and a measurement direction (e.g., a

[0260] 4869-4255-3171.8 29Atty. Dkt. 135589-0103

[0261] predetermined direction), or arrow pointing in that measurement direction may be characterized as a unit vector m = (m1, m2, m3). A probability p that unit vector n is aligned with unit vector m may be characterized by Equation (37) above. And a threshold q may be a random number generated by the random number generator 340. The threshold comparator 352 may perform a comparison “c / < / ?”, and if the result is, for example, “True” (or “T”, “Yes”, “Y”, “1”, etc.) then a third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to the second triplet of coordinates (e.g., the second scalar values m1, m2, m3). However, if the result of the comparison “c / < p” is, for example, “False” (or “F”, “No”, “N”, “0”, etc.) then a third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to an inverse of the second triplet of coordinates (e.g., — m1, — m2, — m3).

[0262]

[0121] The coordinate assignment circuit 354 may transform the state of a qubit being measured, according to the third scalar values (e.g., a third triplet of coordinates, c1, c2, c3). For example, the coordinate assignment circuit 345 may assign, set, otherwise specify the third scalar values, c1, c2, c3to designate a vector with a direction of either 1 or -1, in accordance with the second scalar values (e.g., second triplet of coordinates m1, m2, m3).

[0263]

[0122] In an aspect, the system can generate the one or more third scalar values each equal in magnitude to respective values of the one or more second scalar values. For example, the coordinate assignment circuit 354 can generate the one or more third scalar values each equal in magnitude to respective values of the one or more second scalar values.

[0264]

[0123] In an aspect, the system can generate the one or more third scalar values each having a sign equal to (e.g., equal in direction to) the respective values of the one or more second scalar values, in response to a determination that the probability satisfies the threshold. For example, the coordinate assignment circuit 354 can generate the one or more third scalar values each having a sign equal to (e.g., equal in direction to) the respective values of the one or more second scalar values, in response to a determination that the probability satisfies the threshold.

[0265]

[0124] In an aspect, the system can generate the one or more third scalar values each having a sign opposite to (e.g., opposite in direction to) the respective values of the one or more first scalar values, in response to a determination that the probability does not satisfy the threshold. For example, the coordinate assignment circuit 354 can generate the one or more third scalar values each having a sign opposite to (e.g., opposite in direction to) the respective values of the one or more first scalar values, in response to a determination that the probability does not satisfy the threshold.

[0266]

[0125] The output measurement direction vector memory 360 can store third scalar values (e.g., a third triplet of coordinates, c1, c2, c3). In an aspect, the system can include the one or more third scalar values corresponding to one or more third coordinates defining a third vector in the three-

[0267] 4869-4255-3171.8 30Atty. Dkt. 135589-0103

[0268] dimensional space corresponding to the Hilbert space, the third vector having the third direction. For example, the output measurement direction vector memory 360 can provide the one or more third scalar values corresponding to one or more third coordinates defining a third vector in the three-dimensional space corresponding to the Hilbert space, the third vector having the third direction.

[0269]

[0126] In an aspect, the one or more third scalar values are indicative of a third direction, and the third direction corresponds to the collapse of the quantum state. For example, the one or more third scalar values of the output measurement direction vector memory 360 are indicative of a third direction, and the third direction corresponds to the collapse of the quantum state. The output measurement direction vector memory 360 can include a first output coordinate memory 362, a second output coordinate memory 364, and a third output coordinate memory 366.

[0270]

[0127] The output measurement direction vector memory 360 can include a first output coordinate memory 362 (Ci), a second output coordinate memory 364 (C2), and a third output coordinate memory 366 (C3), which can each store a scalar value (e.g., c1, c2, c3, respectively) of a third triplet of coordinates (e. g., of a unit vector c in a three-dimensional space). The first output coordinate memory 362 (Ci), the second output coordinate memory 364 (C2), and the third output coordinate memory 366 (C3), may temporarily store the third scalar values of the third triplet of coordinates during, or at completion of, the measurement process of the quantum state measurement device 300.

[0271]

[0128] In an aspect, the output coordinate memory 362, 364, 366 can be implemented as or otherwise include static RAM, dynamic RAM, flash memory, one or more flip-flops, ROM, registers, or any other appropriate computer-readable storage medium.

[0272]

[0129] In an aspect, the output coordinate memory 362, 364, 366 can be variables, such as implemented in software.

[0273]

[0130] In an aspect, the output coordinate memory 362, 364, 366 is separate (distinct) from the qubit quantum state memory 220, and can be temporary storage for then specifying values to the qubit quantum state memory 220 to thereby transform the qubit being measured and / or otherwise generate an output qubit. A connection between the output coordinate memory 362, 364, 366 and the qubit quantum state memory 220 can be a direct connection (e.g., hardwired, traces).

[0274]

[0131] In an aspect, the output coordinate memory 362, 364, 366 is contiguous with, integrated with, or otherwise directly associated with the qubit quantum state memory 220. Although depicted in FIG. 3 as separate or distinct, in an aspect the output coordinate memory 362, 364, 366 and the qubit quantum state memory 220, can be the same, such that storing to the output coordinate memory 362, 364, 366 can be storing to the quantum state memory 220 and thereby transforming or setting the state of the target qubit cell (i.e., qubit cell that is measured).

[0275] 4869-4255-3171.8 31Atty. Dkt. 135589-0103

[0276]

[0132] Transforming or otherwise setting the qubit quantum state memory 220 for the target qubit cell, according to the third scalar values (e.g., a third triplet of coordinates, c1, c2, c3), thereby accomplishes the measurement process for measuring the state of the target qubit (e.g., first scalar values, first triplet of coordinates, n1, n2, n3). In an embodiment, the third scalar values can be output coordinates defining an output vector (e.g., in a multidimensional space) having a third direction.

[0277]

[0133] FIGs. 4-5 are directed to example implementations of single-input quantum gates with electrical or electronic architectures. FIG. 4 depicts an implementation of a single input quantum gate having logic, functionality, and / or operations that can be implemented in data structures, objects, class structures, modules, engines, or the like. FIG. 5 depicts an implementation of a single input quantum gate having logic, functionality, and / or operations implemented as a device and having logic, functionality, and / or operations that can be implemented as circuitry and / or firmware. While FIGs. 4 and 5 are depicted and described as alternatives, or alternate implementations, with FIG. 4 implemented as instructions to accomplish the logic, operations and functionality and FIG. 5 implemented as circuitry and / or firmware, a person having ordinary skill will readily understand that one or more elements, components, logic, operations, functionality, and the like of FIG. 4 can be implemented in hardware (e.g., as circuitry, firmware), and similarly, a person having ordinary skill will readily understand that one or more elements, components, logic, operations of FIG. 5 can be implemented in modules, engines, software and the like, including for example structures, objects, class structures, modules, engines. In an aspect, FIG. 4 depicts an implementation of a single input quantum gate having a pipeline architecture and / or configuration. In an aspect, FIG. 5 depicts an implementation of a single input quantum gate having a sequenced architecture and / or configuration.

[0278]

[0134] As discussed herein, the embodiments in each of FIGs. 4 and 5 can each operate according to an X gate, a Z gate, a T gate, an H gate, an R gate, and P gate, but are not limited thereto. For example, an embodiment according to FIG. 4 can implement one of, or selectably implement any or all of, an X gate, a Z gate, a T gate, an H gate, an R gate, and P gate. For example, a device according to Fig. 5 can implement one of, or selectably implement any or all of, an X gate, a Z gate, a T gate, an H gate, an R gate, and P gate. This technical solution is not limited to the quantum gates discussed herein by way of example. For example, all other single qubit gates can be implemented according to the probabilistic architecture discussed herein.

[0279]

[0135] In an aspect, an X gate can be implemented according to the probabilistic model architecture discussed herein, using an electric or electronic circuit. The X gate (sometimes referred to as a " Pauli-X gate") is a fundamental quantum logic gate performs a bit-flip operation on a single qubit, analogous to the classical NOT gate. It swaps the basis states |0) and |1).

[0280] 4869-4255-3171.8 32Atty. Dkt. 135589-0103

[0281] Applying the X gate to |0) yields |1). Applying the X gate to |1) yields |0). For example, Equations (38) and (39) describe linear transform properties associated with operation of an X gate according to the probabilistic model architecture discussed herein. Accordingly, Equation (40) describes input-output operation of an X gate implemented according to the probabilistic model architecture discussed herein.

[0282] fO 1\ > (a'\ (fix > (a'\ a' = p Eqn. (38) 11 0 / W - [p ') =* U ~ \P')[p' = a ni' = a’P’* + a’*P’ = Pa* + p*a = nxn2= i(a'P'* — a'*P') = i(J3a* — p*a) = —i(ap* — Pa*) = —n2Eqns. (39) n

[0283]

[0284] 3= a'*a' — P'*P' = p*p — a*a = —(a*a — P*P) = — n3X (n1, n2, n3) -> (n -n2, -n3) Eqn. (40)

[0285]

[0136] In an aspect, a Z gate can be implemented according to the probabilistic model architecture discussed herein, using an electric or electronic circuit. The Z gate (sometimes referred to as the " Pauli-Z gate") is a single-qubit quantum gate and another fundamental quantum logic gate that plays an important role in quantum operations by affecting the phase of a qubit's state. The Z gate introduces a phase shift of n (or 180 degrees) to the |1)| state of a qubit, while leaving the |0) state unchanged. For example, Equations (41) and (42) describe linear transform properties associated with operation of a Z gate according to the probabilistic model architecture discussed herein. Accordingly, Equation (43) describes input-output operation of a Z gate implemented according to the probabilistic model architecture discussed herein.

[0286] Z|i / >) = IV>') =>

[0287] / I 0 \ (a\ > (a'\ (a\ > (a'\{a' = a Eqn. (41)

[0288] VO -17 \P) ~ \P') y-PJ ~ \P'){p' = -P ni' = a’P’* + a’*P’ = a(—p*) + a*(—P) = — (aP* + a*P) = —n^ n2= i(a'P'* — a'*P') = i[a(— P)* — a*(~PJ] = —i(ap* — a*P) = —n2Eqns. (42) n

[0289]

[0290] 3= a'* a' — P’*P’ = a* a — {—P*){—P) = a* a — p*p = n3

[0291] Z: (n1, n2, n3) → (-n1, -n2, n3) Eqn. (43)

[0292]

[0137] In an aspect, a T gate can be implemented according to the probabilistic model architecture discussed herein, using an electric or electronic circuit. The T gate (also known as the n / 8 gate) is a single-qubit quantum gate that plays a significant role in manipulating the phase of a qubit's state. It can be considered a special case of a more general family phase gates. The T gate applies a phase shift of 7i / 4 radians (or 45 degrees) to the |1) state of a qubit while leaving the |0) state unchanged. For example, Equations (44) and (45) describe linear transform properties associated with operation of a T gate according to the probabilistic model architecture

[0293] 4869-4255-3171.8 33Atty. Dkt. 135589-0103

[0294] discussed herein. Accordingly, Equation (46) describes input-output operation of a T gate implemented according to the probabilistic model architecture discussed herein.

[0295] / I 0 \ za\ (a'\ (a\ (a'\ a' = a Eqn. (44) (o W “ G'H w) - W = a' (3'* + a'*(3' = ae~li(3* + a*eli(3 = e-i4 (a(3* + a*(3el2^ _iTt / 1 i \ = e i(a(3* + ia*(3*) = — —J (a / ?* + ia*(3*)

[0296] 1 1 = — (1—0(a)?* + ia*(3*) = —[=(0.(3* + ia*(3 — ia(3* + a*)?)

[0297] 1 „,, n-, - n?= — [(a(3* + a* (3) — i(a(3* — a* (3}] = — — — n2= i(a'(3'* — a'* (3') = i (ae~l4 / 3* — a*eli / 3^

[0298] .71 z.71 \.71 = ie~li \ a(3* — el^a*(3\ = ie~l^(a(3* — ia*(3) _. Tt_ 1 = el4(ia(3* + a*)?) = - / =(1—i)(ia(3* + a*j?)

[0299] 1 1 = — (ia(3* + a* (3 + a(3* — ia*[3) = -j= [(a(3* + a* / ?) + i(a(3* — a'O]=n±+ n2

[0300] V2 n3= a'* a' — (3'* (3' = a* a — e~l^(3rel^(3 = a* a — 13*13 = n3

[0301] Eqns. (45)z(nr- n2n±+ n2\ T: (n^n^n^ -> [ ’ ^2 V

[0302]

[0303] Eqn. (46)

[0304]

[0138] In an aspect, an H gate can be implemented according to the probabilistic model architecture discussed herein, using an electric or electronic circuit. The H gate (also known as the Hadamard gate) is a fundamental single-qubit gate that plays a crucial role in creating superpositions and in various quantum algorithms. The H gate creates a superposition of the |0) and |1) basis states. For example, Equations (47) and (48) describe linear transform properties associated with operation of an H gate according to the probabilistic model architecture discussed herein. Accordingly, Equation (49) describes input-output operation of an H gate implemented according to the probabilistic model architecture discussed herein.

[0305] 4869-4255-3171.8 34Atty. Dkt. 135589-0103

[0306] a

[0307] Eqn. (47)

[0308] | cr |2— ap* + Pa* — \p\2+ |a|2— a*p + p*a — \p\2

[0309] 2

[0310] . / |<z|2— ap* + a*p — | / ?|2|a|2— a*p + ap* — \p\2~12 2

[0311] = 2 Clal2 - a^*+ a*@ ~ l / ?l2 -lal2 + a*^ ~a@*+l^l2) = -(2a’P — 2aPJ = i(a*p — apj = —i(aft* — a* / T) = —n2

[0312] \a\2+ a*p + aft* + | / ?|2|<z|2— a*p — aft* + | / ?|2

[0313] 2 2

[0314] 1

[0315] = - (1 + a*p + aft* — 1 + a*p + apj

[0316] 1

[0317] = —(2ap* + 2a* p*) = ap* + a*p = n±

[0318]

[0319] Eqns. (48)

[0320] H: (n1, n2, n3) → (n3, -n2, n1) Eqns. (49)

[0321]

[0139] In an aspect, an Rk gate can be implemented according to the probabilistic model architecture discussed herein, using an electric or electronic circuit.

[0322]

[0140] The Rk gate applies a phase shift of to the |1) state of a qubit and it leaves |0)

[0323]

[0324] unchanged. For example, Equations (50), (51) and (52) describe linear transform properties associated with operation of an Rk gate according to the probabilistic model architecture discussed herein. Accordingly, Equation (53) describes input-output operation of an Rk gate implemented according to the probabilistic model architecture discussed herein. Here, because the T gate can correspond to the R3gate, the T gate can be derived from Equation (53), where k = 3.

[0325] 4869-4255-3171.8 35Atty. Dkt. 135589-0103

[0326] / i0\

[0327] Rk= ( 2m I

[0328] \0 e zk /

[0329] Eqn. (50)

[0330]

[0331] Eqns. (51)

[0332] n1f = aof a1f* + i a0f* axf = aQax*e — 2k+ i cz0*a1e —2 / j7c= (ZQO^ * / I cos-^ — • ■z / ti +i aQ *^ / I cos-^ + i • ■z / t1 = cos^Cao^ + a^) - ism^a^ - a^) = nrcos-^ - n2sin^ I T * ( 2n.. 2n\ * f 2n. 2n n2= i^aQ^ — a0a^J = 11 aoa^e 2k— 2k\ = I cos-^ — isin-^ I — aQ^ I cos-^ + isin-^- = i[cos|^ (<z0< Zi — «oai)—isin|^(<z0< Zi + «oai)] = cos^[i(cz0ai - «o«i)] + sm^(aoai + «o«i) = «2cos^ + Thsin^ n

[0333]

[0334] 3= < ZQ*< ZQ—«i*ai= aoao ~e zka^e2ka1= < ZQ< Z0— aiai= n3

[0335] Eqns. (52)

[0336] / 2?r 2?r 2?r 2?r

[0337] 7?fc: (ni,n2,n3) I ^cos— - n2sin —, ^sin— + n2cos —, n3

[0338]

[0339] Eqn. (53)

[0340]

[0141] In an aspect, a

[0341]

[0342] gate can be implemented according to the probabilistic model architecture discussed herein, using an electric or electronic circuit. The

[0343]

[0344] gate (also known as the Phase gate) is a single-qubit gate that introduces a phase shift of (p to the |1) state of a qubit and it leaves |0) unchanged. It is closely related to other phase gates and is commonly used to manipulate the phase of qubits. For example, Equations (54) and (55) describe linear transform properties associated with operation of an P^ gate according to the probabilistic model architecture discussed herein. Accordingly, Equation (56) describes input-output operation of an Pip gate implemented according to the probabilistic model architecture discussed herein.

[0345] Pv= A ko0AA = UA

[0346] a0 \= / «o\ ra0 = «0

[0347]

[0348] eirpaj ~

[0349] Eqn. (54)

[0350] 4869-4255-3171.8 36Atty. Dkt. 135589-0103

[0351] n-y' = ao'a^ +

[0352] = aoe~l,pa^ + alel<par

[0353] = a0a^(cos(p — isimp) + aoa1(cos<p + isimp) = cos(p(a0al + < ZQ< ZI) — ism(p(aoal — < Zoai) =*

[0354] ni=n^coscp — n2sm(p

[0355] n2= i(ao'ai' — a^a-J'

[0356] = i(aoe~l<pai — a^e^a-J

[0357] = i[a0ai(cos(p — isimp) — a^a1(cos(p + isimp)]

[0358] = i[cos<p(a0< Zi — — isin(p(aoa^ + < Zoai)]

[0359] = cos(p[i(a0ai — < Zoai)] + sin<p(a0< Zi + < Zoai) =*

[0360] n2= n2cos(p + H-L sirup

[0361] n3' = a0'*aQ — a^ao'

[0362] = a^a0— el<palel<pa1

[0363] — aoao

[0364] H3 = n3Eqns. (55)

[0365]

[0366] P(p-. (a, b, c) -> (acoscp — bsin(p, bcos<p + asm(p, c) Eqn. (56)

[0367]

[0142] FIG. 4 depicts an example single-qubit quantum gate 400, according to one embodiment of the present disclosure. The single-qubit quantum gate 400 can have a multifunction quantum gate architecture 401. As illustrated by way of example in FIG. 4, a multifunction quantum gate architecture 401 can include an input logic 410 to receive input values 402, a transform controller 420, a coordinate triplet transformer 430, and an output logic 440. In some implementations, the single-qubit quantum gate 400 and / or the multifunction quantum gate architecture 401 corresponds to, or performs similar functions to, the system 100 of FIG. 1. The multifunction quantum gate architecture 401 can receive input values 402 (e.g., input scalar values, input coordinates). In some embodiments, the input values can be received via a qubit input bus and provide output values 404 (e.g., out scalar values, output coordinates) via a qubit output bus.

[0368]

[0143] In some implementations, the single-qubit quantum gate 400, and multifunction quantum gate architecture 401 thereof, receives three input scalar values: n1, n2, n3, corresponding to coordinates in a Bloch sphere (or Bloch ball) for a quantum state. Groups of three input values can be referred to as coordinate triplets. The input values 402 can be received from a memory array, such as the qubit cell array 110 of FIG. 1. The input values 402 can be received, for example, as digital signals.

[0369]

[0144] The input logic 410 can receive the input values 402 and provide the input values 402 to the transform controller 420 and / or the coordinate triplet transformer 430. The input logic 410 may queue input values 402 as they are received for transformation. In some implementations,

[0370] 4869-4255-3171.8 37Atty. Dkt. 135589-0103

[0371] the input logic 410 receives hundreds, thousands, hundreds of thousands, or millions of input values 402 per minute, or per second, calling for rapid queuing and transmission of the input values 402 to the transform controller 420 and / or coordinate triplet transformer 430. In this way, the input logic 410 can receive input values 402 in a manner for transforming the input values 402 beyond practical performance via manual processes. The input logic 410 provides the input values 402 to the transform controller 420 and / or coordinate triplet transformer 430. In the embodiment of FIG. 4, the input logic 410 can be a data structure, an object, class structure, module, and / or engine, such as may be implemented in software. In another embodiment, the input logic 410 can be implemented in electronics (e.g., circuitry, hardware). In another embodiment, the input logic 410 can be implemented in a combination of electronics and software. In another embodiment, the input logic 410 can be implemented in a mechanical system.

[0372]

[0145] The transform controller 420 can control or otherwise determine a configuration and / or operation of the single-qubit quantum gate 400 having the multifunction quantum gate architecture 401. Stated otherwise, the transform controller 420 can include logic, functionality, instructions, or otherwise configure the single-qubit quantum gate 400 to implement a given type of quantum gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to a X gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to a Z gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to a T gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to an H gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to an R gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to and P gate. For example, the transform controller 420 can configure the multifunction quantum gate architecture 401 of the single-qubit quantum gate 400 to operate according to any appropriate quantum computing gate.

[0373]

[0146] The transform controller 420 can include a transform storage 422 and a transform selector 424. The transform storage 422 can include a memory, such as a non-transitory, computer-readable medium. The transform storage 422 can include multiple computer memories. The transform storage 422 can include transformations (e.g., algorithms, functions, etc.) for transforming the input values 402. The transformations stored in the transform storage 422 can

[0374] 4869-4255-3171.8 38Atty. Dkt. 135589-0103

[0375] correspond to transformations enacted by or implemented by quantum gates, such as the quantum gates 122 of FIG. 1. The transformations stored in the transform storage 422 can include transform instructions (e.g., computer-readable instructions, computer code) for executing transformations of the input values 402. The transform storage 422 may include a mapping of quantum gates to transform instructions such that transform instructions can be selected based on a type of quantum gate to be implemented.

[0376]

[0147] The transform selector 424 can select transform instructions from the transform storage 422 to be enacted and / or implemented. In some implementations, the transform selector 424 includes a circuit configured to obtain transform instructions from the transform storage 422. The transform selector 424 can receive instructions indicating which quantum gate is to be implemented and retrieve the corresponding transform instructions from the transform storage 422. In some implementations, the transform selector 424 selects transform instructions from the transform storage 422 based on instructions (e.g., from the transform controller 420 or another source) indicating which quantum gate or gates are to be implemented. The transform selector 424 may provide the selected transform instructions to the coordinate triplet transformer 430. In some implementations the transform selector 424 can select one or more of the input values 402 as input to a quantum gate (i.e., a transform operation and / or corresponding transform instructions). In some implementations, the transform instructions indicate which input values 402 are input to the transform instructions.

[0377]

[0148] In an aspect, the transform controller 420 can be implemented in electronics (e.g., circuitry, hardware). In another aspect, the transform controller 420 can be a data structure, an object, class structure, module, and / or engine, such as may be implemented in software.

[0378]

[0149] In another embodiment, the multifunction quantum gate architecture 401 can include a first logic (e.g., the transform selector 424) to obtain the one or more first transform instructions from the first memory (e.g., the transform storage 422). In an aspect, the first logic can select a transform operation that corresponds to the type of quantum gate. The first logic can obtain the transform operation (i.e., corresponding transform instructions) from a storage (e.g., the transform storage 422) in a memory. For example, the transform selector 424 can select the transform operation that corresponds to the type of quantum gate. The transform selector 424 can obtain the transform operation, and more specifically corresponding transform instructions, from the transform storage 422 of the memory. In an aspect, the transform storage 422 stores a plurality of transform operations, which can include the transform operation that is selected. For example, the transform storage 422 stores a plurality of transform operations that include the transform operation corresponding to a selected quantum gate. In an aspect, the plurality of transform operations correspond to a plurality of types of quantum gates that can include the type of

[0379] 4869-4255-3171.8 39Atty. Dkt. 135589-0103

[0380] quantum gate. In some implementations, the transform selector 424 selects one or more of the input values 402 as input to a quantum gate (i.e., a transform operation and / or corresponding transform instructions). In some implementations, the transform instructions indicate which input values 402 are input to the transform instructions. In an example, the transform instructions indicate that one, two, or three of the input values 402 are transformed using a quantum gate. In an example, transform instructions corresponding to a quantum Z gate indicate that three of the input values 402 are transformed using the quantum Z gate. The transform instructions can function to transform a magnitude and / or sign (e.g., direction) of the input values 402.

[0381]

[0150] In another embodiment, the transform controller 420 can include or otherwise be associated with circuitry (e.g., a processor). In an aspect, the multifunction quantum gate architecture 401 can include a first circuit (e.g., included in or associated with the transform selector 424) to obtain the one or more first transform instructions from the first memory (e.g., the transform storage 422). In an aspect, the first circuit can select a transform operation that corresponds to the type of quantum gate. The first circuit can obtain the transform operation (i.e., corresponding transform instructions) from the transform storage 422 of the memory. In an aspect, the multifunction quantum gate architecture 401 can include a first memory coupled with the first circuit and storing one or more first transform instructions corresponding to the first transform operation. For example, the device can include a transform storage 422 coupled with the first circuit and storing one or more first transform instructions corresponding to the first transform operation.

[0382]

[0151] The coordinate triplet transformer 430 can execute the transform instructions received from the transform selector 424 to implement the quantum gates. The coordinate triplet transformer 430 applies the transform instructions to the input values (e.g., input coordinate triplet) to transform the input values. The coordinate triplet transformer 430 can include a first coordinate transformer 432, a second coordinate transformer 434, and a third coordinate transformer 436. The coordinate triplet transformer 430 transforms the input values into output values 404 (e.g., output scalar values, output coordinate triplet). The output values 404 are indicative of quantum state corresponding to the quantum state indicated in the input values 402 as transformed by the selected transform instructions corresponding to the quantum gates. In this way, the multifunction quantum gate architecture 401 applies the quantum gates to a qubit (e.g., quantum state).

[0383]

[0152] The first coordinate transformer 432 can apply selected transform instructions to one or more of the input values to obtain an output value (e.g., output coordinate) of the output values 404(e.g., output coordinates). In some implementations, the first coordinate transformer 432 applies a first set of selected transform instructions from the transform selector 424 to one or

[0384] 4869-4255-3171.8 40Atty. Dkt. 135589-0103

[0385] more of the input values to obtain a first output value of the output values 404. The second coordinate transformer 434 can apply selected transform instructions to one or more of the input values to obtain an output value of the output values 404. In some implementations, the second coordinate transformer 434 applies a second set of selected transform instructions from the transform selector 424 to one or more of the input values to obtain a second output value of the output values 404. The third coordinate transformer 436 can apply selected transform instructions to one or more of the input values to obtain an output value of the output values 404. In some implementations, the third coordinate transformer 436 applies a third set of selected transform instructions from the transform selector 424 to one or more of the input values to obtain a third output value of the output values 404. The first coordinate transformer 432, the second coordinate transformer 434, and the third coordinate transformer 436 can execute the transform instructions provided by the transform selector 424. In this way, each of the first coordinate transformer 432, the second coordinate transformer 434, and the third coordinate transformer 436 can implement any quantum gate for which corresponding instructions are stored in the transform storage 422.

[0386]

[0153] In some implementations, the transform storage 422 includes multiple memories corresponding to the multiple coordinate transformers of the coordinate triplet transformer 430. In an example, the transform storage 422 includes a first memory coupled to the first coordinate transformer 432, a second memory coupled to the second coordinate transformer 434, and a third memory coupled to the third coordinate transformer. In some implementations, the transform selector 424 selects transform instructions for the coordinate transformers from within their respective memories within the transform storage 422. In some implementations, the transform selector 424 selects the transform instructions for the coordinate transformers by copying transform instructions from a shared memory in the transform storage 422 to the memories corresponding to the coordinate transformers. In this way, the coordinate transformers can retrieve the transform instructions from their respective memories. In an example, the transform storage 422 includes a first memory, a second memory, and a third memory, corresponding to the first coordinate transformer 432, the second coordinate transformer 434, and the third coordinate transformer 436, respectively. In this example, the transform selector 424 provides first transform instructions for the first coordinate transformer by copying the first transform instructions from a shared memory in the transform storage 422 to the first memory, from which the first coordinate transformer 432 retrieves the first instructions. In this example, the transform selector 424 similarly provides second and third transform instructions to the second and third memories, from which the second coordinate transformer 434 and the third coordinate transformer 436 retrieve the second and third transform instructions.

[0387] 4869-4255-3171.8 41Atty. Dkt. 135589-0103

[0388]

[0154] The coordinate triplet transformer 430 provides the output values to the output logic 440. The output logic 440 can provide the output values 404 to a memory and / or one or more processors for analysis. In some implementations, the output logic 440 can provide the output values 404 to additional systems implementing quantum gates. In this way, a computing device including multiple quantum gates can be implemented using the multifunction quantum gate architecture 401 and other similar systems.

[0389]

[0155] In the embodiment of FIG. 4, the coordinate triplet transformer 430 can be a data structure, an object, class structure, module, and / or engine, such as may be implemented in software. In another embodiment, the coordinate triplet transformer 430_can be implemented in electronics (e.g., circuitry, hardware). In another embodiment, the coordinate triplet transformer 430 can be implemented in a combination of electronics and software.

[0390]

[0156] In an aspect, the multifunction quantum gate architecture 401 can transform, according to a transform operation corresponding to the type of the quantum gate processor, at least one magnitude from at least one coordinate of one or more first scalar values of the input values 402, to a corresponding coordinate of one or more second scalar values of the output values 404. For example, the multifunction quantum gate architecture 401 can transform, according to a transform operation corresponding to the type of the quantum gate processor, at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values. In an aspect, the system can transform, according to the type of the quantum gate processor, at least one sign (e.g., direction) from at least one coordinate of the one or more first scalar values, to an opposite sign (e.g., opposite direction) of a corresponding coordinate of the one or more second scalar values. For example, the multifunction quantum gate architecture 401 can transform, according to the type of the quantum gate processor, at least one sign (e.g., direction) from at least one coordinate of the one or more first scalar values, to an opposite sign (e.g., direction) of a corresponding coordinate of the one or more second scalar values. In an aspect, the type of the quantum gate processor corresponds to a quantum X gate, a quantum Z gate, a quantum H gate, a quantum R gate, or a quantum P gate. In an aspect, the type of quantum gate processor corresponds to any appropriate quantum gate.

[0391]

[0157] FIG. 5 depicts an example single-qubit quantum gate 500, according to another embodiment of the present disclosure. The single-qubit quantum gate 500 can have a multifunction quantum gate architecture 501. As illustrated by way of example in FIG. 5, a multifunction quantum gate architecture 501 can include at least a qubit input bus 502, a qubit output bus 504, an input circuit 510, transform control circuitry 520, transform storage 522, a transform selector 524, one or more coordinate transformers 530, and an output circuit 540. In

[0392] 4869-4255-3171.8 42Atty. Dkt. 135589-0103

[0393] some implementations, the multifunction quantum gate architecture 501 corresponds to, or performs similar function(s) to, the system 100 of FIG. 1. In some implementations, the multifunction quantum gate architecture 501 corresponds to, or performs similar function(s) to, the single-qubit quantum gate 400 and / or the multifunction quantum gate architecture 401 of FIG. 4. The single-qubit quantum gate 500 and / or the multifunction quantum gate architecture 501 can receive input values (e.g., input scalar values, input coordinates) via the qubit input bus 502 and provide output values via the qubit output bus 504. In the embodiment of FIG. 5, a quantum state memory 505 can be accessed by, or otherwise provide the input values to, the qubit input bus 502. Also, in the embodiment of FIG. 5, the quantum state memory 505 can be accessed by, or otherwise receive output values from, the qubit output bus 504.

[0394]

[0158] The qubit input bus 502 can receive and / or provide input values (e.g., input scalar values, input coordinates) indicative of qubit vectors. In some implementations, the qubit input bus 502 receives three input values: m,, ns, corresponding to coordinates in a Bloch sphere (or Bloch ball) for a qubit vector. Groups of three input values can be referred to as coordinate triplets. In some implementations, the qubit input bus 502 can correspond to the bus 104 of FIG. 1. In some embodiments, the quantum state memory 505 can correspond to a quantum state memory 220 of the qubit cell 200 of FIG. 2. In some embodiments, the qubit input bus 502 can receive the input values from a memory array, such as the qubit cell array 110 of FIG. 1. The qubit input bus 502 can receive the input values as digital signals.

[0395]

[0159] The qubit input bus 502 can be a bus for carrying signals. In an example, the bus can be, for example, a physical bus including metal wires for carrying electrical signals. In another example, the bus can be a software bus, a simulated bus, a virtual bus, and / or other representation of a bus providing the functionalities, actions, and / or operations of a bus. In some implementations, the qubit input bus 502 includes multiple channels (e.g., multiple different wires) for carrying the different input values. In an example, the qubit input bus 502 includes three channels for receiving the input values. The qubit input bus 502 can include three channels for receiving coordinate triplets. The qubit input bus 502 provides the input values to the input circuit 510.

[0396]

[0160] The input circuit 510 can receive the input values and provide the input values to the transform control circuitry 520. The input circuit 510 may queue input values as they are received (e.g. for transformation according to an implementation of quantum gate). In some implementations, the input circuit 510 receives hundreds, thousands, hundreds of thousands, or millions of input values per minute, or per second, calling for rapid queuing and transmission of the input values to the transform control circuitry 520. In this manner, the input circuit 510 can receive input values in a manner for transforming the input values beyond practical performance

[0397] 4869-4255-3171.8 43Atty. Dkt. 135589-0103

[0398] via manual processes. The input circuit 510 provides the input values to the transform control circuitry 520. In the embodiment of FIG. 5, the input circuit 510 can be implemented in electronics (e.g., circuitry, hardware). In another embodiment, the input circuit 510 can be implemented in a combination of electronics and software. The input circuit 510 can determine which of the input values to provide to the transform control circuitry 520, based on types of quantum gates implemented by their respective coordinate transformers.

[0399]

[0161] The transform control circuitry 520 can control or otherwise determine a configuration and / or operation of the single-qubit quantum gate 500 having the multifunction quantum gate architecture 501. Stated otherwise, the transform control circuitry 520 can include logic, functionality, instructions, or otherwise configure the single-qubit quantum gate 500 to operate as, or otherwise provide functionality according to, a given type of quantum gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500 to operate according to a X gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the singlequbit quantum gate 500 to operate according to a Z gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500 to operate according to a T gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500 to operate according to an H gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500 to operate according to an R gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500 to operate according to a P gate. For example, the transform control circuitry 520 can configure the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500 to operate according to any appropriate quantum computing gate.

[0400]

[0162] The transform control circuitry 520 can include a transform storage 522, a transform selector 524, and one or more coordinate transformer(s).

[0401]

[0163] The transform storage 522 can include a memory, such as static RAM, dynamic RAM, flash memory, one or more flip-flops, ROM, magnetic media, optical media, or any other computer-readable storage medium. The transform storage 522 can include multiple computer memories. The transform storage 522 can include transformations (e.g., algorithms, functions, etc.) for transforming the input values. The transformations stored in the transform storage 522 can correspond to transformations enacted by or implemented by quantum gates, such as the quantum gates 122 of FIG. 1. The transformations stored in the transform storage 522 can include transform instructions (e.g., firmware, computer-readable instructions, computer code) for

[0402] 4869-4255-3171.8 44Atty. Dkt. 135589-0103

[0403] executing transformations of the input values. The transform storage 522 may include a mapping to transform instructions for various given quantum gates, such that transform instructions can be selected based on a type of quantum gate to be implemented by the multifunction quantum gate architecture 501 of the single-qubit quantum gate 500.

[0404]

[0164] The transform selector 524 can select transform instructions from the transform storage 522 to be enacted and / or implemented. In some implementations, the transform selector 524 includes circuitry configured to obtain transform instructions from the transform storage 522. The transform selector 524 can receive an indication of or instructions indicating which quantum gate is to be implemented and retrieve the corresponding transform instructions from the transform storage 522. In some implementations, the transform selector 524 selects transform instructions from the transform storage 522 based on instructions (e.g., from the transform control circuitry 520 or another source) indicating which quantum gate or gates are to be implemented. The transform selector 524 may provide the selected transform instructions to the one or more coordinate transformers 530 (e.g. triplet transformers).

[0405]

[0165] The coordinate transformer(s) 530 can execute or otherwise implement the transform instructions selected by the transform selector 524 and obtained or otherwise accessible from the transform storage 522. In the embodiment of FIG. 5, the coordinate transformer(s) can be implemented in electronics (e.g., circuitry, hardware). In another embodiment, the coordinate transformer(s) 530 can be implemented in a combination of electronics and software. The coordinate transformer(s) 530 can execute transform instructions to implement a given quantum gate. The coordinate transformer(s) 530 can apply the transform instructions to the input values to transform the input values.

[0406]

[0166] In an embodiment, a single coordinate transformer 530 includes circuitry to apply transform instructions to multiple input values and to transform the multiple input values to multiple output values. For example, a coordinate transformer 530 can be a circuit that transforms three input values (e.g., input coordinate triplet) into three output values (e.g., output coordinate triplet). In another example, a coordinate transformer can be a circuit that transforms n input values (e.g., input scalar values) into n output values (e.g., output scalar values). The output values are indicative of quantum state corresponding to the quantum state indicated in the input values as transformed by the selected transform instructions corresponding to the quantum gate. In this way, the multifunction quantum gate architecture 501 applies the quantum gates to a qubit quantum state. In an example, the output values are indicative of new quantum state corresponding to the input values (e.g., the quantum state indicated thereby) as transformed by the selected transform instructions corresponding to the quantum gate.

[0407] 4869-4255-3171.8 45Atty. Dkt. 135589-0103

[0408]

[0167] In another embodiment, the transform control circuitry 520 can include a plurality of the coordinate transformers 530. For example, the coordinate transformers 530 can include a first coordinate transformer, a second coordinate transformer, and a third coordinate transformer to apply the transform instructions to the input values (e.g., input coordinate triplet) to transform the input values. The first coordinate transformer 530 can apply selected transform instructions to one or more of the input values to obtain an output value of the output values. In some implementations, the first coordinate transformer applies a first set of selected transform instructions from or selected by the transform selector 524 to one or more of the input values to obtain a first output value (e.g., a first output coordinate) of the output values (e.g., output coordinates). In some implementations, the second coordinate transformer can apply selected transform instructions to one or more of the input values to obtain a second output value of the output values. In some implementations, the second coordinate transformer applies a second set of selected transform instructions from the transform selector 524 to one or more of the input values to obtain a second output value (e.g., a second output coordinate) of the output values. In some implementations, the third coordinate transformer can apply selected transform instructions to one or more of the input values to obtain an output value of the output values. In some implementations, the third coordinate transformer applies a third set of selected transform instructions from the transform selector 524 to one or more of the input values to obtain a third output value (e.g., a third output coordinate) of the output values. Stated otherwise, the first coordinate transformer, the second coordinate transformer, and the third coordinate transformer can execute the transform instructions provided or otherwise selected by the transform selector 524. In this way, each of the first coordinate transformer, the second coordinate transformer, and the third coordinate transformer can implement any quantum gate for which corresponding instructions are stored in the transform storage 522. In this way, the coordinate transformer(s) 530 transform input values (e.g., input scalar values, input coordinate triplet) into output values (e.g., output scalar values, output coordinate triplet). The output values are indicative of quantum state corresponding to the quantum state indicated in the input values as transformed by the selected transform instructions corresponding to the quantum gate. In this way, the multifunction quantum gate architecture 501 applies the quantum gates to a qubit (and / or its quantum state).

[0409]

[0168] In an aspect, the multifunction quantum gate architecture 501 can copy at least one magnitude from at least one input values of one or more input values (e.g., first scalar values, first input coordinates) to a corresponding output value of one or more output values (e.g., second scalar values, second output coordinates). In an aspect, the multifunction quantum gate architecture 501 can copy at least one magnitude from at least one coordinate of one or more first scalar values, to a corresponding coordinate of one or more second scalar values. For example,

[0410] 4869-4255-3171.8 46Atty. Dkt. 135589-0103

[0411] the coordinate transformer(s) 530 can copy at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values. In an aspect, the coordinate transformer(s) 530 can copy at least one sign (e.g., direction) from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values. For example, the coordinate transformer(s) 530 can copy at least one sign (e.g., direction) from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values.

[0412]

[0169] In an embodiment, the transform storage 522 can include multiple memories (or storage devices) corresponding to multiple coordinate transformers 530. In an example, the transform storage 522 includes a first memory coupled to a first coordinate transformer, a second memory coupled to a second coordinate transformer, and a third memory coupled to a third coordinate transformer. In some implementations, the transform selector 524 selects transform instructions for the coordinate transformers from within their respective memories within the transform storage 522. In some implementations, the transform selector 524 selects the transform instructions for the coordinate transformers by copying transform instructions from a shared memory in the transform storage 522 to the memories corresponding to the coordinate transformers. In this way, the coordinate transformers can retrieve the transform instructions from their respective memories. In an example, the transform storage 522 includes a first memory, a second memory, and a third memory, corresponding to a first coordinate transformer, a second coordinate transformer, and a third coordinate transformer, respectively. In this example, the transform selector 524 provides first transform instructions for the first coordinate transformer by copying the first transform instructions from a shared memory in the transform storage 522 to the first memory, from which the first coordinate transformer retrieves the first instructions. In this example, the transform selector 524 similarly provides second and third transform instructions to the second and third memories, from which the second coordinate transformer and the third coordinate transformer retrieve the second and third transform instructions.

[0413]

[0170] The coordinate transformer 530 provides the output values to the output circuit 540. The output circuit 540 can, using the qubit output bus 504, provide the output values to a memory (e.g., the quantum state memory 505) and / or one or more additional gates. In some implementations, the output circuit 540 can provide the output values to additional systems and / or devices implementing quantum gates. In this way, a computing device including multiple quantum gates can be implemented using the multifunction quantum gate architecture 501 and other similar systems.

[0414]

[0171] The qubit output bus 504 can provide the output values to a memory and / or one or more processors for analysis. In an example, the bus can be, for example, a physical bus including

[0415] 4869-4255-3171.8 47Atty. Dkt. 135589-0103

[0416] metal wires for carrying electrical signals. In another example, the bus can be a software bus, a simulated bus, a virtual bus, and / or other representation of a bus providing the functionalities, actions, and / or operations of a bus. In some implementations, the qubit output bus 504 can provide the output values to additional systems implementing quantum gates. In this way, a computing device including multiple quantum gates and transform operations can be implemented using the multifunction quantum gate architecture 501 and other similar systems.

[0417]

[0172] FIG. 6 depicts an example quantum gate interconnect architecture, according to the present disclosure. As illustrated by way of example in FIG. 6, a quantum gate interconnect architecture 600 can include at least a qubit input bus 602, a qubit output bus 604, a first quantum gate 610, a second quantum gate 620, and a third quantum gate 630.

[0418]

[0173] At least one aspect of the architecture 600 is directed to single qubit interference for qubits, implemented as discussed herein. Because the correct probability distributions of single qubits are encoded in the polarization direction, then all we need to do is to show that our gates produce the correct polarization direction.

[0419]

[0174] For example, a plurality of quantum gates can be structured according to the architecture 600, where the first quantum gate 610 can correspond to a first H gate, the second quantum gate 620 can correspond to a P(pgate, and the third quantum gate 630 can correspond to a second H gate. For example, the first quantum gate 610 can receive an input of |0) and provide an output of |i >out). Thus, Equation (57) can describe output of the circuit according to the architecture 600, and Equation (58) can describe a polarization direction of \ipout in athree-dimensional (e.g., Euclidean) space. Thus, Equation (59) describes output of the circuit according to architecture 600, based on the input of 10), where (n1, n2, n3) = (0,0,1).

[0420] [pout) = cos^| 0) - isin^ |1) Eqn. (57)

[0421]

[0422] nx= 0, n2= — simp, n3= costp Eqn. (58)

[0423] (0,0,1) -> (1,0,0) -> (coscp, simp, 0) -> (0, — simp, cos<p) ®qn. (59)

[0424]

[0175] The qubit input bus 602 can receive input values (e.g., input scalar values, input coordinates) indicative of quantum state(s) (e.g., qubit vector(s)). In some implementations, the qubit input bus 602 receives three input values: m,, ns, corresponding to coordinates in a Bloch sphere (or Bloch ball) for a quantum state. A group of three input values can be referred to as a coordinate triplet. In an aspect, the qubit input bus 602 can receive one or more triplets. In some implementations, the qubit input bus 602 corresponds to the bus 104 of FIG. 1. The qubit input bus 602 can receive the input values from a memory array, such as the qubit cell array 110 of

[0425] 4869-4255-3171.8 48Atty. Dkt. 135589-0103

[0426] FIG. 1. In some implementations, the qubit input bus 602 corresponds to the qubit input bus 502 of FIG. 5. The qubit input bus 602 can receive the input values as digital signals.

[0427]

[0176] The qubit input bus 602 can be a bus for carrying electrical signals, such as a bus including metal wires for carrying electrical signals. In some implementations, the qubit input bus 602 includes multiple channels (e.g., multiple different wires) for carrying the different input values. In an example, the qubit input bus 602 includes three channels for receiving the input values. The qubit input bus 602 can include three channels for receiving coordinate triplets. The qubit input bus 602 provides the input values to the first quantum gate 610.

[0428]

[0177] The first quantum gate 610 can receive the input values from the qubit input bus 602. The first quantum gate 610 can implement a quantum gate to transform the input values into first intermediate values. The first intermediate values are indicative of a quantum state corresponding to the quantum state indicated in the input values as transformed by the quantum gate implemented by the first quantum gate 610. In this way, the first quantum gate 610 applies the quantum gate to the quantum state. In an aspect, the first quantum gate 610 applies the quantum gate to manipulate the quantum state. The first quantum gate 610 can correspond to the singlequbit gate 500 of FIG. 5, the single-qubit quantum gate 400 of FIG. 4, and / or the quantum gate(s) 122 of FIG. 1. The first quantum gate 610 can be a multi-function quantum gate that is configurable to implement any quantum gate, including any quantum gate discussed herein. In some implementations, the first quantum gate 610 can be configured to implement different quantum gates (e.g., different types of quantum gates) at different times. In an example, the first quantum gate 610 implements a first type of quantum gate to transform a first set of input values by retrieving transform instructions corresponding to the first type of quantum gate from memory and implements a second type of quantum gate to transform a second set of input values by retrieving transform instructions corresponding to the second type of quantum gate from a memory. The first quantum gate 610 can transform the input values from the qubit input bus 602 into the first intermediate values. The first quantum gate 610 can represent a first quantum gate in a series of quantum gates implemented in the architecture 600.

[0429]

[0178] The architecture 600 and / or the first quantum gate 610 can include a gate output bus 612. The gate output bus 612 can provide the first intermediate values to the second quantum gate 620.

[0430]

[0179] The second quantum gate 620 can receive the first intermediate values from the first quantum gate 610. The second quantum gate 620 can implement a quantum gate to transform the first intermediate values into second intermediate values. The second intermediate values are indicative of a quantum state corresponding to the quantum state indicated in the first intermediate values as transformed by the quantum gate implemented by the second quantum gate 620. In this way, the second quantum gate 620 applies the quantum gate to the quantum state. The

[0431] 4869-4255-3171.8 49Atty. Dkt. 135589-0103

[0432] second quantum gate 620 can correspond to the single-qubit quantum gate 500 of FIG. 5, the single-qubit quantum gate 400 of FIG. 4, and / or the quantum gate(s) 122 of FIG. 1. The second quantum gate 620 can be a multifunction quantum gate that is configurable to implement any quantum gate, including any quantum gate discussed herein. In an example, the second quantum gate 620 can be configured to implement different quantum gates (e.g., different types of quantum gates) at different times. In an example, the second quantum gate 620 implements a first type of quantum gate to transform a first set of input values by retrieving transform instructions corresponding to the first type of quantum gate from memory and implements a second type of quantum gate to transform a second set of input values by retrieving transform instructions corresponding to the second type of quantum gate from memory. The second quantum gate 620 can transform the first intermediate values from the first quantum gate 610 into the second intermediate values. The second quantum gate 620 can represent a second quantum gate in a series of quantum gates implemented in the architecture 600.

[0433]

[0180] The architecture 600 and / or the second quantum gate 620 can include a gate output bus 622. The gate output bus 622 can provide the second intermediate values to the third quantum gate 630.

[0434]

[0181] The third quantum gate 630 can receive the second intermediate values from the second quantum gate 620. The third quantum gate 630 can implement a quantum gate to transform the second intermediate values into output values. The output values are indicative of a qubit vector corresponding to the quantum state indicated in the second intermediate values as transformed by the quantum gate implemented by the third quantum gate 630. In this way, the third quantum gate 630 applies the quantum gate to the quantum state. The third quantum gate 630 can correspond to the single-qubit quantum gate 500 of FIG. 5, the single-qubit quantum gate 400 of FIG. 4, and / or the quantum gate(s) 122 of FIG. 1. The third quantum gate 630 can be a multifunction quantum gate that is configurable to implement any quantum gate, including any quantum gate discussed herein. In an example, the third quantum gate 630 can be configured to implement different quantum gates (e.g., different types of quantum gates) at different times. In an example, the third quantum gate 630 implements a first type of quantum gate to transform a set of input values by retrieving transform instructions corresponding to the first type of quantum gate from memory and implements a second type of quantum gate to transform a second set of input values by retrieving transform instructions corresponding to the second type quantum gate from memory. The third quantum gate 630 can transform the second intermediate values from the second quantum gate 620 into the output values. The third quantum gate 630 can represent a third quantum gate in a series of quantum gates implemented in the architecture 600.

[0435] 4869-4255-3171.8 50Atty. Dkt. 135589-0103

[0436]

[0182] The architecture 600 and / or the third quantum gate 630 can provide the output values to the qubit output bus 604, which can provide the output values to a memory and / or one or more processors for output and / or analysis. In some implementations, the qubit output bus 604 can provide the output values to one or more additional quantum gates.

[0437]

[0183] In an aspect, the type of quantum gate implemented by the first quantum gate 610, the second quantum gate 620, and / or the third quantum gate 630 is a Z gate, the one or more first input values include a first coordinate of a qubit vector (e.g., a quantum state), the one or more second input values include a second coordinate of the qubit vector, and the one or more third input values include a third coordinate of the qubit vector. In an aspect, the first transform operation corresponds to generating the first output having a magnitude equal to a magnitude of the first coordinate, and having a sign (e.g., direction) opposite to a sign (e.g., direction) of the first coordinate. In an aspect, the second transform operation corresponds to generating the second output having a magnitude equal to a magnitude of the second coordinate, and having a sign (e.g., direction) opposite to a sign (e.g., direction) of the second coordinate. In an aspect, the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the third coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate.

[0438]

[0184] In an aspect, the type of quantum gate implemented by the first quantum gate 610, the second quantum gate 620, and / or the third quantum gate 630 is a T gate, the one or more first input values include a first coordinate of a qubit vector (e.g., a quantum state), the one or more second input values include a second coordinate of the qubit vector, and the one or more third input values include a third coordinate of the qubit vector. In an aspect, the first transform operation corresponds to generating the first output as an arithmetic product of a constant with a difference between the first coordinate and the second coordinate. In an aspect, the second transform operation corresponds to generating the second output as an arithmetic product of a constant with a sum between the first coordinate and the second coordinate. In an aspect, the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the second coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate.

[0439]

[0185] In an aspect, the type of quantum gate implemented by the first quantum gate 610, the second quantum gate 620, and / or the third quantum gate 630 is an H gate, the one or more first input values include a first coordinate of the qubit vector (e.g., a quantum state), the one or more second input values include a second coordinate of the qubit vector, and the one or more third input values include a third coordinate of the qubit vector. In an aspect, the first transform operation corresponds to generating the first output having a magnitude equal to a magnitude of

[0440] 4869-4255-3171.8 51Atty. Dkt. 135589-0103

[0441] the third coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate. In an aspect, the second transform operation corresponds to generating the second output having a magnitude equal to a magnitude of the second coordinate, and having a sign (e.g., direction) opposite to a sign (e.g., direction) of the second coordinate. In an aspect, the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the first coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the first coordinate.

[0442]

[0186] In an aspect, the type of quantum gate is a P gate or an R gate, the one or more first input values include a first coordinate of the qubit vector (e.g., a quantum state), the one or more second input values include a second coordinate of the qubit vector, and the one or more third input values include a third coordinate of the qubit vector. In an aspect, the first transform operation corresponds to generating the first output as a difference between a first trigonometric operation and a second trigonometric operation, the first trigonometric operation based on the first coordinate and the second trigonometric operation based on the second coordinate. In an aspect, the type of quantum gate is the R gate, the first trigonometric operation is based on a constant, the second trigonometric operation is based on the constant, and the constant is based on the R gate. In an aspect, the second transform operation corresponds to generating the second output as a sum of a first trigonometric operation and a second trigonometric operation, the first trigonometric operation based on the first coordinate and the second trigonometric operation based on the second coordinate. In an aspect, the type of quantum gate is the R gate, the first trigonometric operation is based on a constant, the second trigonometric operation is based on the constant, and the constant is based on the R gate. In an aspect, the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the third coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate.

[0443]

[0187] In an aspect, the one or more input values correspond to a first triplet of coordinates in a three-dimensional space corresponding to a Hilbert space, and the one or more output values correspond to a second triplet of coordinates in the three-dimensional space corresponding to the Hilbert space. In an aspect, the non-transitory computer readable medium can include one or more instructions executable by the processor. The processor can select the transform operation that corresponds to the type of quantum gate. The processor can obtain the transform operation from a transform storage of the memory.

[0444]

[0188] FIGs. 7-9 are directed to an example implementation of a device corresponding to a two-qubit gate using an electrical or electronic architecture, according to an embodiment of the present disclosure. A two-qubit gate provides additional complexity in functionality and / or

[0445] 4869-4255-3171.8 52Atty. Dkt. 135589-0103

[0446] operation than a single qubit gate (such as the single qubit gate of FIG. 4, the single qubit gate of FIG. 5, and the like) to perform computing operations of greater complexity. The present disclosure illustrates examples of one-qubit gates and measurement devices and two-qubit gates and measurement devices, and multi-qubit gates and measurement devices performing operations on or using more than two qubits can be implemented according to the concepts, techniques, and / or principles herein. For example, a three-qubit gate may be implemented by a combination of a one-qubit gate and a two-qubit gate. For example, a three-qubit gate may be implemented in a manner similar to the two-qubit gate(s) described herein, with enhanced complexity. In an aspect, a three-qubit measurement device may be implemented by a combination of a one-qubit measurement device and a two-qubit measurement device. For example, a three-qubit measurement device be implemented in a manner similar to the two-qubit measurement device(s) described herein, with enhanced complexity. For example, a four-qubit gate may be implemented by a combination of a pair of two-qubit gates. For example, a four-qubit gate may be implemented in a manner similar to the two-qubit gate(s) described herein, with enhanced complexity. In an aspect, a four-qubit measurement device may be implemented by a combination of a pair of two-qubit measurement devices. For example, a four-qubit measurement device can be implemented in a manner similar to the two-qubit measurement device(s) described herein, with enhanced complexity.

[0447]

[0189] The two-qubit gate of FIGs. 7-9 can be, or otherwise implement quantum operations of, a controlled NOT (CNOT) quantum gate. Other two-qubit gates (e.g., controlled phase gate, etc.) can be implemented in a similar manner. As illustrated by way of example herein, a CNOT gate according to FIGs. 7-9 can provide a technical solution to operate an electric or electronic device according to a CNOT gate, to achieve a technical improvement to achieve quantum gate behavior of a CNOT gate, including behavior according to entanglement of multiple qubits. For example, a CNOT gate according to this disclosure can be implemented in at least one of a semiconductor device (e.g., die, wafer), an electronic circuit (e.g., integrated circuit, printed circuit board), a computing system (e.g., compiled high-level language stored in a computer-readable medium), or any combination thereof, but is not limited thereto. As another example, a CNOT gate according to this disclosure can be implemented in a mechanical system. Thus, the CNOT gate of FIGs. 7-9 can provide a technical improvement at least to achieve computational behavior corresponding to a CNOT gate using digital electronics, analog electrical circuits, or any combination thereof. In this example implementation, a CNOT gate can include multiple stages (e.g., seven stages), as illustrated by way of example in FIGs. 7-9.

[0448]

[0190] For example, a CNOT gate according to this disclosure can be implemented according to probabilistic computational processes. Probabilistic output of the CNOT gate, corresponding to

[0449] 4869-4255-3171.8 53Atty. Dkt. 135589-0103

[0450] results of measurement of quantum states, can be correlated according to identification parameters, such as time tags that indicate time stamps of probabilistic outputs of the CNOT gate. For example, the number of identification parameters is equal to the number of qubits, which allows correlation of outputs of a CNOT gate to accurately model probabilistic computational behavior corresponding to quantum entanglement. In some implementations, identification parameters (e.g., time tags) can be used to correlate outputs of a CNOT gate based on windowing. In an example, two qubits can be considered part of the same composite system if their identification parameters are within an identification window, where the window represents, for example, a maximum allowable difference between the identification parameters.

[0451]

[0191] FIG. 7 depicts an example first block diagram of a CNOT device architecture 700, according to this disclosure. As illustrated by way of example in FIG. 7, a first block diagram of a CNOT device architecture 700 can include at least a first stage 701. For example, the first block diagram can correspond to a portion of a CNOT device architecture that includes two inputs, where the two inputs each respectively correspond to inputs for distinct qubits. The first stage 701 can generate a probabilistic representation of relative directions between two input qubits. For example, the first stage can provide technical improvement to achieve behavior corresponding to quantum entanglement via a technical solution to generate a plurality of probabilities according to arrows for each input qubit. In an aspect, the first stage can include a plurality of electrical or electronic devices structured or configured to generate metrics indicative of relative probabilistic states. Thus, the first stage 701 can achieve “mixing” between probabilistic states of two input probabilistic states to probabilistically model a quantum entanglement state over the stages of FIGs. 7-9 as discussed herein. The first stage 701 can include a quantum tomography circuit 702.

[0452]

[0192] Quantum tomography can be understood as reconstructing quantum state(s) by repeatedly performing many measurements. One example implementation of quantum tomography implementation is described with reference to FIG. 7. However, persons of ordinary skill may recognize that other quantum tomograph implementations are available and may be suitable to accomplish similar operations and / or functionality. Other example implementations of quantum tomography or types of quantum tomography that may be suitable include, but are not limited to: linear inversion, maximum likelihood estimation, Bayesian methods, quantum measurement tomography, and the like.

[0453]

[0193] The quantum tomography circuit 702 can also be referred to as a multi-qubit axis comparator, as the quantum tomography circuit 702 can be used to measure the combined state of a control qubit and a target qubit. The quantum tomography circuit 702 can also be referred to as a multi-qubit measurement device, as the quantum tomography circuit 702 can be used to measure the combined state of a control qubit and a target qubit. The quantum tomography circuit

[0454] 4869-4255-3171.8 54Atty. Dkt. 135589-0103

[0455] 702 can also be used to measure the reduced state of a control qubit and a target qubit. The quantum tomography circuit 702 can include an arbitrary vector processor 720, a first axis comparator 731, a second axis comparator 733, a third axis comparator 735, a fourth axis comparator 737, a first time tag generator 732, a second time tag generator 734, a third time tag generator 736, a fourth time tag generator 738, an X-axis probability processor 740, a Y-axis probability processor 742, a Z-axis probability processor 744, and an arbitrary-axis probability processor 746. The quantum tomography circuit 702 can include and / or receive as input a control qubit input 710 and a target qubit input 712. The quantum tomography circuit 702 can include and / or provide as output a first probability object output 750, a second probability object output 752, a third probability object output 754, and a fourth probability object output 756.

[0456]

[0194] The control qubit input 710 can correspond at least partially in one or more of structure and operation to a first coordinate triplet as discussed herein. For example, the control qubit input 710 can correspond to output of a quantum gate according to FIGs. 4 or 5, or an output of a sequence of quantum gates according to FIG. 6 as discussed herein, but is not limited thereto. For example, the control qubit input 710 can correspond to a first coordinate triplet for any quantum gate, sequence of quantum gates, or qubit states at least as discussed herein.

[0457]

[0195] The target qubit input 712 can correspond at least partially in one or more of structure and operation to the control qubit input 710. For example, the target qubit input 712 can correspond to a second coordinate triplet distinct from the first coordinate triplet of the control qubit input 710. For example, the target qubit input 712 can be output of a quantum gate, or a sequence of quantum gates, distinct from output of a quantum gate or sequence of quantum gates corresponding to the control qubit input 710. For example, the control qubit input 710 can correspond to a first unit vector in a 3D coordinate space, and the target qubit input 712 can correspond to a second unit vector in the 3D coordinate space that can be distinct from the first unit vector.

[0458]

[0196] The arbitrary vector processor 720 can generate a coordinate triplet in an arbitrary direction in a 3D coordinate space. For example, the arbitrary vector processor 720 can generate three scalar values, with each respectively corresponding to a distance from an origin along a corresponding axis (e.g., X-axis, Y-axis, Z-axis) in the 3D coordinate space. For example, the coordinate triplet in the arbitrary direction can correspond to an arrow or arbitrary direction in the 3D coordinate space corresponding to a unit vector. In an aspect, the arbitrary vector processor 720 can generate the arbitrary direction to improve computation of the system that can solve a linear system of multiple equations. For example, the CNOT gate of FIGs. 7-9 can operate according to a linear system of 16 equations with 16 unknowns. Outputs of the probability processors 740, 742, 744, 746 each output 4 values to provide 16 values that are used to solve the

[0459] 4869-4255-3171.8 55Atty. Dkt. 135589-0103

[0460] 16 equations. The arbitrary vector processor 720 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0461]

[0197] In some implementations, the quantum tomography circuit 702 does not include the arbitrary vector processor 720, the fourth axis comparator 737, the fourth time tag generator 738, and the arbitrary axis probability processor 746. Instead, only the first axis comparator 731, the second axis comparator 733, and the third axis comparator 735 output probability elements, such that the x-axis probability processor 740, the y-axis probability processor 742, and the z-axis probability processor 744 each output 4 values to provide 12 values that are used to solve the 12 equations. In some implementations, only 10 values are used to solve 10 equations. For example, the CNOT gate of FIGs. 7-9 can operate according to a linear system of 10 equations with 10 unknowns.

[0462]

[0198] The output of the quantum tomography circuit 702 (i.e., the outputs of the probability processors 740, 742, 744, 746) reflect quantum entanglement according to probabilistic operations beyond the capability of manual processes to achieve. While FIG. 7 illustrates a simplified implementation with a lower number of qubits, computational complexity increases dramatically for any number of qubits involved in a practical application, beyond the capability of manual processes to achieve.

[0463]

[0199] In some implementations, the arbitrary vector processor 720 can provide the target qubit input 712 to the first axis comparator 731, the second axis comparator 733, the third axis comparator 735, the fourth axis comparator 737, the first time tag generator 732, the second time tag generator 734, the third time tag generator 736, and the fourth time tag generator 738. In some implementations, the target qubit input 712 is provided directly to the first axis comparator 731, the second axis comparator 733, the third axis comparator 735, the fourth axis comparator 737, the first time tag generator 732, the second time tag generator 734, the third time tag generator 736, and the fourth time tag generator 738.

[0464]

[0200] The first axis comparator 731 can measure a quantum state of one or more qubit cells to extract quantum information. The first axis comparator 731 can measure the quantum state of the control qubit (represented by the control qubit input 710) and the target qubit (represented by the target qubit input 712), otherwise referred to as the “combined state.” The combined state can include either a mixed state or a pure state. In some implementations, the first axis comparator 731 measures the unit vector of the control qubit with the unit vector of the target qubit along the X-axis. The first axis comparator 731 can take a dot product of the X-axis vector of the control qubit input 710 (dot product with the X-axis unit vector) and the dot product of the X-axis vector

[0465] 4869-4255-3171.8 56Atty. Dkt. 135589-0103

[0466] of the target qubit input 712 (dot product with the X-axis unit vector) and take the signs of the dot product results to determine the probability element output by the first axis comparator 731.

[0467]

[0201] The first time tag generator 732 can correlate inputs and / or outputs of the first axis comparator 731. The first axis comparator 731 can generate multiple measurements over time. Outputs (i.e., measurements) can be assigned time tags to identify pairings of outputs. An output corresponding to an input may be generated within a predetermined identification window (e,g., time period) after the input is received, allowing for correlation of outputs based on the time tags. The time tags generated by the first time tag generator 732 can be combined with the outputs of the first axis comparator 731.

[0468]

[0202] The first axis comparator 731 outputs probability elements corresponding to the quantum state of the control qubit and the target qubit. The first axis comparator 731 provides a probability element (i.e., one or more scalar values) to the X-axis probability processor 740. The probability element corresponds to a sample of a probability space for the quantum state of the control qubit and the target qubit. The first axis comparator 731 may provide a plurality of probability elements corresponding to the quantum state of a plurality of control qubits and target qubits. The X-axis probability processor 740 accumulates an ensemble (i.e., set, collection, plurality) of the probability elements (e.g., samples) to map the probability space and selects values, based on the ensemble (e.g., a distribution or density of the ensemble), to provide as the first probability object output 750. In this way, the probability elements are each a sample of the ensemble accumulated by the X-axis probability processor 740, where the ensemble maps the probability space for the quantum state of the control qubit and the target qubit. In this way, the ensemble is indicative of the quantum state of the control qubit and the target qubit. In some implementations, the X-axis probability processor 740 accumulates a global ensemble representing a mapping of the probability space for the quantum state of the control qubit and target qubit.

[0469]

[0203] A qubit can correspond to or otherwise be associated with various ensembles, each of which have different grouping criteria. A combined state can correspond to an ensemble of pairs of qubits. A reduced state (single control qubit or single target qubit) can correspond to an ensemble of single qubits. An ensemble can be a way of grouping portions of the state by a shared condition, description, or other commonality.

[0470]

[0204] The second axis comparator 733 can measure a quantum state of one or more qubit cells to extract quantum information. The second axis comparator 733 can measure the quantum state of the control qubit (represented by the control qubit input 710) and the target qubit (represented by the target qubit input 712), otherwise referred to as the “combined state.” In some implementations, the second axis comparator 733 measures the unit vector of the control qubit with the unit vector of the target qubit along the Y-axis. The second axis comparator 733 can take

[0471] 4869-4255-3171.8 57Atty. Dkt. 135589-0103

[0472] a dot product of the Y-axis vector of the control qubit input 710 (dot product with the Y-axis unit vector) and the dot product of the Y-axis vector of the target qubit input 712 (dot product with the Y-axis unit vector) and take the signs of the dot product results to determine the probability element output by the second axis comparator 733.

[0473]

[0205] The second time tag generator 734 can correlate inputs and / or outputs of the second axis comparator 733. The second axis comparator 733 can generate multiple measurements over time. Outputs (i.e., measurements) can be assigned time tags to identify pairings of outputs. An output corresponding to an input may be generated within a predetermined identification window (e,g., time period) after the input is received, allowing for correlation of outputs based on the time tags. The time tags generated by the second time tag generator 734 can be combined with the outputs of the second axis comparator 733.

[0474]

[0206] The second axis comparator 733 outputs probability elements corresponding to the quantum state of the control qubit and the target qubit. The second axis comparator 733 provides a probability element (i.e., one or more scalar values) to the Y-axis probability processor 742. The probability element corresponds to a sample of a probability space for the quantum state of the control qubit and the target qubit. The second axis comparator 733 may provide a plurality of probability elements corresponding to the quantum state of a plurality of control qubits and / or target qubits. The Y-axis probability processor 742 accumulates an ensemble (i.e., set, collection, plurality) of the probability elements to map the probability space and selects values, based on the ensemble (e.g., a distribution or density of the ensemble), to provide as the second probability object output 752. In this way, the probability elements are each a sample of the ensemble accumulated by the Y-axis probability processor 742, where the ensemble maps the probability space for the quantum state of the control qubit and the target qubit. In this way, the ensemble is indicative of the quantum state of the control qubit and the target qubit. In some implementations, the Y-axis probability processor 742 accumulates a global ensemble representing a mapping of the probability space for the quantum state of the control qubit and target qubit.

[0475]

[0207] The third axis comparator 735 can measure a quantum state of a qubit cell to extract quantum information. The third axis comparator 735 can measure the quantum state of the control qubit (represented by the control qubit input 710) and the target qubit (represented by the target qubit input 712), otherwise referred to as the “combined state.” In some implementations, the third axis comparator 735 measures the unit vector of the control qubit with the unit vector of the target qubit along the Z-axis. The third axis comparator 735 can take a dot product of the Z-axis vector of the control qubit input 710 (dot product with the Z-axis unit vector) and the dot product of the Z-axis vector of the target qubit input 712 (dot product with the Z-axis unit vector) and

[0476] 4869-4255-3171.8 58Atty. Dkt. 135589-0103

[0477] take the signs of the dot product results to determine the probability element output by the third axis comparator 735.

[0478]

[0208] The third time tag generator 736 can correlate inputs and / or outputs of the third axis comparator 735. The third axis comparator 735 can generate multiple measurements over time. Outputs (i.e., measurements) can be assigned time tags to identify pairings of outputs. An output corresponding to an input may be generated within a predetermined identification window (e,g., time period) after the input is received, allowing for correlation of outputs based on the time tags. The time tags generated by the third time tag generator 736 can be combined with the outputs of the third axis comparator 735.

[0479]

[0209] The third axis comparator 735 outputs probability elements corresponding to the quantum state of the control qubit and the target qubit. The third axis comparator 735 provides a probability element (i.e., one or more scalar values) to the Z-axis probability processor 744. The probability element corresponds to a sample of a probability space for the quantum state of the control qubit and the target qubit. The third axis comparator 735 may provide a plurality of probability elements corresponding to the quantum state of a plurality of control qubits and / or target qubits. The Z-axis probability processor 744 accumulates an ensemble (i.e., set, collection, plurality) of the probability elements to map the probability space and selects values, based on the ensemble (e.g., a distribution or density of the ensemble), to provide as the third probability object output 754. In this way, the probability elements are each a sample of the ensemble accumulated by the Z-axis probability processor 744, where the ensemble maps the probability space for the quantum state of the control qubit and the target qubit. In this way, the ensemble is indicative of the quantum state of the control qubit and the target qubit. In some implementations, the Z-axis probability processor 744 accumulates a global ensemble representing a mapping of the probability space for the quantum state of the control qubit and target qubit.

[0480]

[0210] The fourth axis comparator 737 can measure a quantum state of a qubit cell to extract quantum information. The fourth axis comparator 737 can measure the quantum state of the control qubit (represented by the control qubit input 710) and the target qubit (represented by the target qubit input 712). In some implementations, the fourth axis comparator 737 measures the unit vector of the control qubit with the unit vector of the target qubit along the arbitrary axis. The fourth axis comparator 737 can take a dot product of the control qubit input 710 (dot product with the arbitrary-axis unit vector) and the dot product of the target qubit input 712 (dot product with the arbitrary axis unit vector) and take the signs of the dot product results to determine the probability element output by the fourth axis comparator 737.

[0481]

[0211] The fourth time tag generator 738 can correlate inputs and / or outputs of the fourth axis comparator 737. The fourth axis comparator 737 can generate multiple measurements over time.

[0482] 4869-4255-3171.8 59Atty. Dkt. 135589-0103

[0483] Outputs (i.e., measurements) can be assigned time tags to identify pairings of outputs. An output corresponding to an input may be generated within a predetermined identification window (e.g., time period) after the input is received, allowing for correlation of outputs based on the time tags. The time tags generated by the fourth time tag generator 738 can be combined with the outputs of the fourth axis comparator 737.

[0484]

[0212] The fourth axis comparator 737 outputs probability elements corresponding to the quantum state of the control qubit and the target qubit. The fourth axis comparator 737 provides a probability element (i.e., one or more scalar values) to the arbitrary axis probability processor 746. The probability element corresponds to a sample of a probability space for the quantum state of the control qubit and the target qubit. The fourth axis comparator 737 may provide a plurality of probability elements corresponding to the quantum state of a plurality of control qubits and target qubits. The arbitrary axis probability processor 746 accumulates an ensemble (i.e., set, collection, plurality) of the probability elements to map the probability space and selects values, based on the ensemble (e.g., a distribution or density of the ensemble), to provide as the fourth probability object output 756. In this way, the probability elements are each a sample of the ensemble accumulated by the arbitrary axis probability processor 746, where the ensemble maps the probability space for the quantum state of the control qubit and the target qubit. In this way, the ensemble is indicative of the quantum state of the control qubit and the target qubit. In some implementations, the arbitrary axis probability processor 746 accumulates a global ensemble representing a mapping of the probability space for the quantum state of the control qubit and target qubit.

[0485]

[0213] The probability processors 740, 742, 744, 746 can generate a plurality of probabilities each corresponding to an axis or direction provided by the axis comparators 731, 733, 735, 737. For example, the probability processors 740, 742, 744, 746 can generate probabilities based on probability elements generated by the axis comparators 731, 733, 735, 737. In an aspect, the probability processors 740, 742, 744, 746 can generate a probability with respect to each axis or direction of the axis comparators 731, 733, 735, 737. Thus, the probability processors 740, 742, 744, 746 can generate a plurality of probabilities, each indicative of a corresponding probability that a combined mixed state of the control qubit and the target qubit is along a given axis or direction. In an example, each of the probability processors 740, 742, 744, 746 has four inputs corresponding to the four components output by the axis comparators 731, 733, 735, 737 and the time tag generators 732, 734, 736, 738. In an aspect, the four inputs (corresponding to the four components output by the axis comparators 731, 733, 735, 737) can include probability elements and identification parameters (e.g., time tags). In an example, each of the probability processors 740, 742, 744, 746 has four inputs corresponding to the four components output by the axis

[0486] 4869-4255-3171.8 60Atty. Dkt. 135589-0103

[0487] comparators 731, 733, 735, 737 and four inputs corresponding to the time tag generators 732, 734, 736, 738.

[0488]

[0214] The X-axis probability processor 740 can generate a first probability for a direction of a combined state of the control qubit input 710 and the target qubit input 712, with respect to the X-axis. For example, the X-axis probability processor 740 can generate a probability based on the ensemble output by the first axis comparator 731. In some implementations, the X-axis probability processor 740 includes a qubit time window processor to correlate the outputs of the first axis comparator 731 using the time tags generated by the first time tag generator 732. Thus, the X-axis probability processor 740 can generate a probability corresponding to a combined probabilistic state for a plurality of qubits, beyond the capability of manual processes to achieve. The X-axis probability processor 740 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software. The first probability object output 750 can correspond to a linear system that is relative to the X axis. The first probability object output 750 can correspond to a probability of the control qubit and the target qubit aligning with the X axis. For example, the first probability object output 750 can correspond to a first set of four equations that indicate probabilities of each of the relative orientations of the first axis comparator 731 with respect to the X axis (e.g., probabilities of each of the relative orientations aligning with the X axis).

[0489]

[0215] The Y-axis probability processor 742 can correspond at least partially in one or more of structure and operation to the X-axis probability processor 740, and can generate a second probability for a direction of a combined state of the control qubit input 710 and the target qubit input 712, with respect to the Y-axis. For example, the Y-axis probability processor 742 can generate a probability with respect to each relative orientation of the control qubit and the target qubit as output by the second axis comparator 733. In some implementations, the Y-axis probability processor 742 includes a qubit time window processor to correlate the outputs of the second axis comparator 733 using the time tags generated by the second time tag generator 734. Thus, the Y-axis probability processor 742 can generate a probability corresponding to a combined probabilistic state for a plurality of qubits, beyond the capability of manual processes to achieve. The Y-axis probability processor 742 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software. The second probability object output 752 can correspond to a linear system that is relative to the Y axis. The second probability object output 752 can correspond to a probability of the control qubit and the target qubit aligning with the Y axis. For example, the second probability object

[0490] 4869-4255-3171.8 61Atty. Dkt. 135589-0103

[0491] output 752 can correspond to a second set of four equations that indicate probabilities of each of the relative orientations of the second axis comparator 733 with respect to the Y axis (e.g., probabilities of each of the relative orientations aligning with the Y axis).

[0492]

[0216] The Z-axis probability processor 744 can correspond at least partially in one or more of structure and operation to the X-axis probability processor 740, and can generate a third probability for a direction of a combined state of the control qubit input 710 and the target qubit input 712, with respect to the Z-axis. For example, the Z-axis probability processor 744 can generate a probability with respect to each relative orientation of the control qubit and the target qubit as output by the third axis comparator 735. In some implementations, the Z-axis probability processor 744 includes a qubit time window processor to correlate the outputs of the third axis comparator 735 using the time tags generated by the third time tag generator 736. Thus, the Z-axis probability processor 744 can generate a probability corresponding to a combined probabilistic state for a plurality of qubits, beyond the capability of manual processes to achieve. The Z-axis probability processor 744 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software. The third probability object output 754 can correspond to a linear system that is relative to the Z axis. The third probability object output 754 can correspond to a probability of the control qubit and the target qubit aligning with the Z axis. For example, the third probability object output 754 can correspond to a third set of four equations that indicate probabilities of each of the relative orientations of the third axis comparator 735 with respect to the Z axis (e.g., probabilities of each of the relative orientations aligning with the Z axis).

[0493]

[0217] The arbitrary-axis probability processor 746 can correspond at least partially in one or more of structure and operation to the X-axis probability processor 740, and can generate a fourth probability for a direction of a combined state of the control qubit input 710 and the target qubit input 712, with respect to the arbitrary direction generated by the arbitrary vector processor 720. For example, the arbitrary-axis probability processor 746 can generate a probability with respect to each relative orientation of the control qubit and the target qubit as output by the fourth axis comparator 737. In some implementations, the arbitrary-axis probability processor 740 includes a qubit time window processor to correlate the outputs of the fourth axis comparator 737 using the time tags generated by the fourth time tag generator 738. Thus, the arbitrary-axis probability processor 746 can generate a probability corresponding to a combined probabilistic state for a plurality of qubits, beyond the capability of manual processes to achieve. The arbitrary-axis probability processor 746 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and

[0494] 4869-4255-3171.8 62Atty. Dkt. 135589-0103

[0495] the like and / or can be implemented at least in part in software. The fourth probability object output 756 can correspond to a linear system that is relative to the arbitrary direction. The fourth probability object output 756 can correspond to a probability of the control qubit and the target qubit aligning with the arbitrary axis. For example, the fourth probability object output 756 can correspond to a fourth set of four equations that indicate probabilities of each of the relative orientations of the fourth axis comparator 737 with respect to the arbitrary direction (e.g., probabilities of each of the relative orientations aligning with the arbitrary axis).

[0496]

[0218] FIG. 8 depicts an example second block diagram of a CNOT device architecture, according to this disclosure. As illustrated by way of example in FIG. 8, a second block diagram of a CNOT device architecture 800 can include at least a second stage 802, a third stage 803, a fourth stage 804, and a fifth stage 805. The probability object outputs 750, 752, 754 and 756 of the architecture 700 of FIG. 7 can couple with the architecture 800 as discussed herein. The architecture 700 of FIG. 7 and the architecture 800 can be part of a CNOT gate, as discussed herein.

[0497]

[0219] The second stage 802 can generate a density matrix (e.g., a “density” as discussed herein) that is indicative of a combined state of the control qubit and the target qubit. As discussed herein, the second stage 802 can generate a linear system indicative of a likelihood of a combined state having one or more, or all potential states corresponding to a Bloch sphere or Bloch ball. Thus, the second stage 802 can generate a representation of a quantum-entangled state of the control qubit input 710 and the target qubit input 712 of the architecture 700. Thus, a density as discussed herein can correspond to a linear system (e.g., a matrix representation) that stores a likelihood that a combined state corresponds to any given probabilistic state. The second stage 802 can include a density processor 810.

[0498]

[0220] The density processor 810 can generate a first density for a combined state of the control qubit input 710 and the target qubit input 712. The first density can be indicative of a likelihood of states (e.g., probabilistic states) of the control qubit input 710 and the target qubit input 712. In an aspect, the density processor 810 can generate a linear system of sixteen equations that correspond to a likelihood that the combined state corresponds to and / or aligns with the X-axis, Y-axis, Z-axis, and the arbitrary axis discussed in FIG. 7. In an aspect, the linear system can be used to generate the density matrix. In an aspect, the linear system or density can be represented as one or more matrices. The density processor 810 can generate a linear system of four equations that correspond to a likelihood that the combined state corresponds to and / or aligns with the X-axis. The density processor 810 can generate a linear system of four equations that correspond to a likelihood that the combined state corresponds to and / or aligns with the Y-axis. The density processor 810 can generate a linear system of four equations that correspond to a likelihood that

[0499] 4869-4255-3171.8 63Atty. Dkt. 135589-0103

[0500] the combined state corresponds to and / or aligns with the Z-axis. The density processor 810 can generate a linear system of four equations that correspond to a likelihood that the combined state corresponds to and / or aligns with the arbitrary axis. The density processor 810 can thus generate a linear system having 16 equations and 16 variables. The density processor 810 can generate, as discussed below, a density corresponding to the density for the combined state of the control qubit input 710 and the target qubit input 712. The density may be represented, as shown below, as a density matrix. The density processor 810 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0501]

[0221] For example, a reduced density matrix for a collection of electrons is not affected by measurement, which reflects the no-signaling principle which in turn manifests in the nonuniqueness of purification. For example, correlations do not appear unless the results of both measurements are compared together (this happens sub-luminally), and even then, correlation does not imply causation. Thus, the correlation can be traced back to the source when the electrons interacted in the past, corresponding to the time tag correlation operations discussed herein. In an aspect, a density matrix can describe a given ket with a much simpler computational resource requirement, including for densities that represent combined states. For example, Equation (60) can represent a pure state of three qubits, according to a summation convention in which an index repeated only twice is summed over, one downstairs and one upstairs, from zero to one. Based on Equation (60), Equations (61) can generate a density operator that represents the first two qubits via taking a partial trace over the third qubit. In an aspect, for a density matrix according to Equation (62), then in the basis {lz^zv): z, v G {0,l}2(no sum)}, the density matrix can be represented according to Equation (63).

[0502] IV>) = C^va\n^mvlff} Eqn. (60) P12 = Tr3[p] = Tr3[(C^Vff|n^mv / ff))(C^Vff,(n^mv / / c7 / |)] = Tr3[C^Qvrar\n^mvlff)(n^mv,lff,\] = C^aC^ / arTr-i[\n^mvlr7)(nll'mv' = C^ffC^vV(|n^mv)(n^mv'|( / 0|r)( / c7 / | / 0) + |n^mv)(n^mv'|( / 1|r)( / c7 / | / 1))

[0503] = CgvoC^Vo|n^mv)(n^mv,| +

[0504]

[0505] = (C / zvo^Vo + B^^v>\n^mv)(n^' mv' \ Eqns. (61)

[0506] 4869-4255-3171.8 64Atty. Dkt. 135589-0103

[0507] P

[0508]

[0509] = CnVnrvr\z^zv){z^'zv'\ Eqn. (62)

[0510] EQOOO Qooi EQOIO Qon

[0511] Q1OO ioi Quo in

[0512] Eqn. (63)

[0513] Qooo Ciooi Qoio Cion

[0514]

[0515] Q1OO Qioi Quo Cun

[0516]

[0222] In an aspect, in Cμν μ′ν′, (μ ν) = 00, 01, 01, 11 numbers the rows, while (μ′ν′) = 00, 01, 01, 11 numbers the columns. Equation (64) can describe a pure state of two qubits provided as input to a CNOT-gate, and Equation (65) can provide an output.

[0517] Pin = \ipin)(Pin\ Eqn. (64)

[0518] Pout I ^P out).^P out I

[0519] = CNOT\ipin){ipin\CNOT^ =

[0520] Pout = CNOTpinCNOT

[0521] Eqn. (65)

[0522]

[0223] In an aspect, the density matrix can take the same form as in Equation (63). Thus, the effect of a CNOT would be consistent regardless of whether input qubits are entangled with other qubits. We will now deduce some very important equations that we will use in the CNOT implementation as discussed herein by way of example. In an aspect, a CNOT architecture according to FIGs. 7-9, including the density processor 810, can be based on Equation (66), and an input density matrix according to Equation (67). Equations (70) and (73) can correspond to probabilities with respect to the X axis, and Equations (74) and (78) can correspond to probabilities with respect to the Y axis for {|y°), ly1)}. Equations (69) can correspond to probabilities with respect to the Z axis..

[0523] |Z°):= |O)

[0524]

[0525] := |1>

[0526] |X°):= |+)

[0527] Ix1) ^!-)

[0528] |y°):= |y+)

[0529] 1 ):= ^-)

[0530] Eqns. (66)

[0531] P

[0532]

[0533] in C^v^v'\z^zv){z^'zv'\ Eqn. (67)

[0534] 4869-4255-3171.8 65Atty. Dkt. 135589-0103

[0535] P zffz ) = {zffz \pin\zaz'n)

[0536] = Cfivu'v'^z71\z^zv}{zIJ-'zv' \zffzV} = Ctlvtllvl(ztJ\z^(z^\zv)(z^’ \zff)(zv’ \z^)

[0537] = c ^(jriari Eqn. (68) P(z°'z,?) = CO-JJO-JJ, where a, p G {0, 1} Eqn. (69)

[0538] P(xffxv) = (xffxv\pin\xffx'n)

[0539] = Ctlv^vi(xaxri\zp-zv')(zp-' zv' \xaxri) = C^vyv'{xa\z^){x^ \zv)(z^' |xCT)(zv' lx17)

[0540] Eqn. (70) 1 1 \|x“> =v?|z”) +vi|zl>i (-ir lx1) = — |z°) - — Iz1)^

[0541] Eqn. (71)

[0542] (-1)^ V2 Eqn. (72) (-l)^ (-l)^ (-1 / * P(xffx^ = C^>v>—j= - — - - 2 V2 f—l')O'(zi+Zi')+J7(v+v') P(X-X^ = - - - C^v,

[0543] 4 Eqn. (73)

[0544] P(yffyv) = {yffyv\pin\yffyri)

[0545] = CflVfl'v'(yay,1\ztlzv}(ztl' zv' \yayri)

[0546]

[0547] =cuLvuL'v'<yffI^Xy77|zv)(z / / lyCT>(zv' \yn)

[0548] Eqns. (74)

[0549] 4869-4255-3171.8 66Atty. Dkt. 135589-0103

[0550] ly0> =^lz“> +^lzl)l 1 (-1)’ ly1) = -j=\z°) - —\zr) j

[0551] Eqn. (75)

[0552] [e^+1)pe^(2,+D V2 ~ V2 ~ V2

[0553] c -iyg(2a+l) (yff|z^)

[0554] V2 Eqn. (77)

[0555] ”1 77 ”1 TT ”1 7T, ”1 7T, P(yV’) = C^^, —e-t2^ + l)_e-i^ + l)_ei^ (2a+l) i^V (2a+l)

[0556] 1 7T P(v°vV~) -,,

[0557]

[0558] rv7 J J 4UjWVjU v Eqn. (78)

[0559]

[0224] Thus, Equation (79) can describe a Bloch sphere representation of a unit vector n in 3-space designated by the spherical coordinates

[0560]

[0561] Thus, the density processor 810 can generate, according to one or more of Equations (79)-(91), a portion (e.g., a first portion) of a density according to a linear system having four variables and four components (Equation (912)), as part of a linear system of the second stage having 16 variables and 16 components. In an aspect, the density generated by the second stage, including the portion of the density, corresponds as a whole to the density for the combined state of the control qubit input 710 and the target qubit input 712.

[0562] 9 0

[0563] |n°) = cos— |z°) + e1^ sin— Iz1)

[0564] Eqn. (79)

[0565] In1) = cos(^\z°) + ed< / >+^) sin^y^lz1) = sin^|z°) — el$ cos^lz1)

[0566]

[0567] Eqn. (80)

[0568] 4869-4255-3171.8 67Atty. Dkt. 135589-0103

[0569] 6 9 (z°|n°) = cos — <=> (n°|z°) = cos —

[0570] Eqn. (81)

[0571] 0 0 (z1^0) = el$ sin— <=> (n^z1) = e~l<^ sin — 2 2 Eqn. (82)

[0572] 0 0 (zQIn1) = sin— <=> (n^z0) = sin —

[0573] Eqn. (83)

[0574] 0 0 (z1^1) = —el$ cos— <=> (n1|z1) = —e~l$ cos — 2 2 Eqn. (84) (eI cos —, / z = 0n e. (n^|z°) = < = cos ( / z— — — ) I u \ 2 2 / (^sin —, / z = 1

[0575] Eqn. (85)

[0576] / n 9\ (n^z0) = cosl / z- - -I

[0577] Eqn. (86)

[0578] / n 9\ <^> (z0!^) = cos hi- - -)

[0579] Eqn. (87)

[0580] e‘^ sin —, / z = 0 / n 9\ (z1!^) = < = (-1)^6^ sin ( / z— + — j (no sum)

[0581] (-^ COS-, ^ 1

[0582]

[0583] Eqn. (88)

[0584] 4869-4255-3171.8 68Atty. Dkt. 135589-0103

[0585] ( TC 0\

[0586] / z — + — ] (no sum)

[0587] Eqn. (89)

[0588] ( 7T 0\

[0589] / z — + — ] (no sum)

[0590] Eqn. (90)

[0591] P(nc7n’7) = (n°'n’?|fyn|n°'n,?)

[0592] = CnVll!v!{nann\ztlzv){ztl' zv' \nanri')

[0593]

[0594] = \zv){z11' \na){zv' \n^)

[0595] Eqns. (91)

[0596]

[0225] The third stage 803 can modify a density that represents a combined state into an output combined state that corresponds to a transformation according to a CNOT gate. For example, the third stage 803 can include one or more linear transformations configured to receive a matrix from the second stage 802 and to generate an output combined state with a transformed density corresponding to a quantum state provided by a CNOT gate. In an example, the third stage 803 receives a density matrix from the second stage 802 and outputs a transformed density matrix. The third stage 803 can include a density transformer 820.

[0597]

[0226] The density transformer 820 can transform at least a portion of the density received from the density processor 810, and can generate a modified combined state (e.g., transformed density) that is representative of output by the CNOT gate, or a state of the control qubit and the target qubit. For example, the density transformer 820 can transform the density into the modified combined state (e.g., transformed density) according to Equation (92). The modified combined state, otherwise referred to as the transformed density, corresponds to a probability of a state (e.g., a probabilistic state) of the control qubit and the target qubit. The density transformer 820 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0598] 0000 0001 0010 0011 1 0 0 0 0000 0001 0010 oon\ / I 0 0 0 0100 0101 0110 0111 0 1 0 0 0100 0101 0110 oin 11 0 1 0 0 1000 1001 1010 1011 0 0 0 1 1000 1001 1010 ion / 1 0 0 0 1

[0599]

[0600] 1100 1101 1110 1111 0 0 1 0 1100 1101 1110 ini / \o 0 1 0

[0601] Eqn. (92)

[0602] 4869-4255-3171.8 69Atty. Dkt. 135589-0103

[0603]

[0227] The fourth stage 804 can reduce a linear system from a first number (e.g., 16 equations) to a second number (e.g., 12 equations) to describe a modified combined state according to a 3D coordinate space. For example, the fourth stage 804 can receive 16 inputs each corresponding to the modified combined state output by the third stage 803, and can convert the linear system of 16 equations into a linear system of 12 equations. The linear system of 12 equations can include multiple scalar values that, within the linear system of 12 equations, indicate a probability of a state (e.g., a probabilistic state) of the control qubit and the target qubit. The fourth stage 804 can include a quantum direction probability processor 830. The quantum direction probability processor 830 can reduce the linear system according to one or more of Equations (69), (73) (78), with C′μνμ′ν′in place of C. Here,

[0604]

[0605] can correspond to the modified combined state received from the third stage 803. Thus, the quantum direction probability processor 830 can provide a technical improvement to identify output combined states that have probabilities corresponding to the quantum states of qubits according to the control qubit input 710 and the target qubit input 712.

[0606]

[0228] The fifth stage 805 can identify an order for a plurality of probabilities (e.g., scaled probabilities) generated by the fourth stage 804. In an aspect, the fifth stage can order a plurality of probabilities (e.g., scaled probabilities) by a probability that the combined state corresponds to an output of the CNOT gate. For example, the fourth stage 804 can generate a density according to a combined state, and the fifth stage 805 can generate a plurality of arrows ranked by their frequency of presence in the modified combined state as reduced by the fourth stage 804. The fifth stage 805 can include a first order transform processor 840, a second order transform processor 842, and a third order transform processor 844, to output a first ranked direction probabilities output 850, a second ranked direction probabilities output 852, and a third ranked direction probabilities output 854, respectively.

[0607]

[0229] The first order transform processor 840 can generate a first ordering of arrows, including for example, an ordering or relative orientations of a first direction of a control qubit and a second direction of a target qubit after transformation according to the fourth stage 804. The first ranked direction probabilities output 850 can include the arrows and the ordering of the arrows generated by the first order transform processor 840. Thus, the first ranked direction probabilities output 850 can correspond to a first partial model output indicative of a plurality of probabilistically-ranked states of a CNOT gate output.

[0608]

[0230] The second order transform processor 842 can generate a second ordering of arrows, including for example, an ordering or relative orientations of a first direction of a control qubit and a second direction of a target qubit after transformation according to the fourth stage 804. The second ranked direction probabilities output 852 can include the arrows and the ordering of

[0609] 4869-4255-3171.8 70Atty. Dkt. 135589-0103

[0610] the arrows generated by the second order transform processor 842. Thus, the second ranked direction probabilities output 852 can correspond to a second partial model output indicative of a plurality of probabilistically-ranked states of a CNOT gate output.

[0611]

[0231] The third order transform processor 844 can generate a third ordering of arrows, including for example, an ordering or relative orientations of a first direction of a control qubit and a second direction of a target qubit after transformation according to the fourth stage 804. The third ranked direction probabilities output 854 can include the arrows and the ordering of the arrows generated by the third order transform processor 844. Thus, the third ranked direction probabilities output 854 can correspond to a third partial model output indicative of a plurality of probabilistically-ranked states of a CNOT gate output.

[0612]

[0232] Each of the partial model outputs corresponding to the first ranked direction probabilities output 850, the second ranked direction probabilities output 852, and the third ranked direction probabilities output 854 can collectively correspond to a set of probabilistically-ordered outputs of potential states of the CNOT gate of FIGs. 7-9.

[0613]

[0233] FIG. 9 depicts an example third block diagram of a CNOT device architecture, according to this disclosure. As illustrated by way of example in FIG. 9, a third block diagram of a CNOT device architecture 900 can include at least a sixth stage 906, and a seventh stage 907. For example, the ranked direction probabilities outputs 850, 852 and 854 of the architecture 800 can couple with the architecture 900 as discussed herein.

[0614]

[0234] The sixth stage 906 can segment one or more probabilities of the ranked direction probabilities outputs 850, 852 and 854 into a set that includes arrows whose likelihoods of occurrence correspond to the quantum probability of the CNOT gate. For example, the sixth stage 906 can output vectors representing samples of an ensemble corresponding to the state of the control qubit and the state of the target qubit based on the probabilities output by the fifth stage 805. The sixth stage 906 can include a first probability classification processor 910, a second probability classification processor 912, a third probability classification processor 914, a first randomizer circuit 905, a second randomizer circuit 908, a first coordinate selection processor 930, a second coordinate selection processor 932, and a third coordinate selection processor 934.

[0615]

[0235] The first probability classification processor 910 can classify the first ranked direction probabilities output 850 according to Equations (93). The classifications of the first ranked direction probabilities output 850 can correspond to signs (e.g., positive, negative) of the coordinate pairs of the first ranked direction probabilities output 850. In some implementations, the classifications of the first ranked direction probabilities output 850 can correspond to spin signs, or relative magnitudes of spins, of the coordinate pairs of the first ranked direction probabilities output 850. In some implementations, the classifications of the first ranked direction

[0616] 4869-4255-3171.8 71Atty. Dkt. 135589-0103

[0617] probabilities output 850 can correspond to directions of spin vectors. In some implementation, the classifications of the first ranked direction probabilities output 850 can correspond to other types of qubits (e.g., polarization of photons, energy levels of an atom, etc.). The first probability classification processor 910 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0618] n

[0619]

[0620] < 1

[0621] S < r < S + S2X-> 2

[0622] S + S2X< r < S* + S2X+ S3X3

[0623] otherwise -> 4

[0624] Eqns. (93)

[0625]

[0236] The second probability classification processor 912 can correspond at least partially in one or more of structure and operation to the first probability classification processor 910. The second probability classification processor 912 can classify the second ranked direction probabilities output 852 according to Equations (94). The classifications of the second ranked direction probabilities output 852 can correspond to signs (e.g., positive, negative) of the coordinate pairs of the second ranked direction probabilities output 852. In some implementations, the classifications of the second ranked direction probabilities output 852 can correspond to spin signs, or relative magnitudes of spins, of the coordinate pairs of the second ranked direction probabilities output 852. In some implementations, the classifications of the second ranked direction probabilities output 852 can correspond to directions of spin vectors. In some implementation, the classifications of the second ranked direction probabilities output 852 can correspond to other types of qubits (e.g., polarization of photons, energy levels of an atom, etc.). The second probability classification processor 912 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0626] r

[0627]

[0628] r1≤ S1y→ 1

[0629] S1y≤ r1≤ S1y+ S2y→ 2

[0630] S1y+ S2y≤ r1≤ S1y+ S2y+ S3y→ 3

[0631] otherwise -> 4

[0632] Eqns. (94)

[0633] 4869-4255-3171.8 72Atty. Dkt. 135589-0103

[0634]

[0237] The third probability classification processor 914 can correspond at least partially in one or more of structure and operation to the first probability classification processor 910. The third probability classification processor 914 can classify the third ranked direction probabilities output 854 according to Equations (95). The classifications of the third ranked direction probabilities output 854 can correspond to signs (e.g., positive, negative) of the coordinate pairs of the third ranked direction probabilities output 854. In some implementations, the classifications of the third ranked direction probabilities output 854 can correspond to spin signs, or relative magnitudes of spins, of the coordinate pairs of the third ranked direction probabilities output 854. In some implementations, the classifications of the third ranked direction probabilities output 854 can correspond to directions of spin vectors. In some implementation, the classifications of the third ranked direction probabilities output 854 can correspond to other types of qubits (e.g., polarization of photons, energy levels of an atom, etc.). The third probability classification processor 914 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0635] r1≤ S1z→ 1

[0636] S1z≤ r1≤ S1z+ S2z→ 2

[0637] S

[0638]

[0639] S1z+ S2z≤ r1≤ S1z+ S2z+ S3z→ 3

[0640] otherwise -> 4

[0641] Eqns. (95)

[0642]

[0238] Thus, the probability classification processors 910, 912 and 914 can classify the ranked direction probabilities outputs 850, 852, 854 and can provide a technical improvement to probabilistically determine the quantum state of the control qubit and the target qubit beyond the capability of manual processes to achieve.

[0643]

[0239] The first randomizer circuit 905 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software. The first and second randomizer circuits 905 and 908 can each include one or more independently-executable random number generators. For example, the first randomizer circuit 905 can generate one or more random numbers, pseudo-random numbers, or quasi random numbers, and is not limited to true random numbers. The output of the first randomizer circuit 905 can correspond to quantum fluctuation, quantum uncertainty, or the like. In an example, the first randomizer circuit 905 can generate a different random number between one and one hundred for each of the first probability

[0644] 4869-4255-3171.8 73Atty. Dkt. 135589-0103

[0645] classification processor 910, the second probability classification processor 912, and the third probability classification processor 914 based on a first random seed input. In some implementations, the random numbers generated by the first randomizer circuit 905 can be compared to combinations of the input probabilities to classify the sign of the ranked direction probabilities outputs 850, 852, 854.

[0646]

[0240] The second randomizer circuit 908 can correspond at least partially in one or more of structure and operation to the first randomizer circuit 905, and can generate one or more random numbers, pseudo-random numbers, or quasi random numbers based on a second random seed input distinct from the first random seed input. As noted above, the first and second randomizer circuits 905 and 908 can each include one or more independently-executable random number generators. The output of the second randomizer circuit 908 can correspond to quantum fluctuation, quantum uncertainty, or the like. One or more random numbers output by the second randomizer circuit 908 can be used to modify, or be combined with, the output of the probability classification processors 910, 912, 914. In some implementations, the one or more random numbers output by the second randomizer circuit 908 can be multiplied with the classifications, or signs corresponding to the classifications, output by the probability classification processors 910, 912, 914. In an example, the second randomizer circuit 908 generates four different random numbers for each of the classifications of each of the probability classification processors 910, 912, 914 according to Equations (95).

[0647]

[0241] The first coordinate selection processor 930 can apply the classification or indication generated by the first probability classification processor 910 to one or more random numbers generated by the second randomizer circuit 908. In an example, the first coordinate selection processor 930 assigns a sign corresponding to a classification generated by the first probability classification processor 910 to a random number generated by the second randomizer circuit. In some implementations, the first coordinate selection processor 930 assigns a sign to a random number generated by the second randomizer circuit 908 based on the classified probabilities of the first ranked direction probabilities output 850. In an example, the first coordinate selection processor 930 can assign to each random number generated by the second randomizer circuit 908

[0648]

[0649] a “++,” a a or a The first coordinate selection processor 930 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0650]

[0242] The second coordinate selection processor 932 can correspond at least partially in one or more of structure and operation to the first coordinate selection processor 930, and can apply the classification or indication generated by the second probability classification processor 912 to one

[0651] 4869-4255-3171.8 74Atty. Dkt. 135589-0103

[0652] or more random numbers generated by the second randomizer circuit 908. The third probability classification processor 914 can correspond at least partially in one or more of structure and operation to the first probability classification processor 910, and can apply the classification or indication generated by the third probability classification processor 914 to one or more random numbers generated by the second randomizer circuit 908. Thus, the probability classification processors 910, 912 and 914 can assign signs to random numbers generated by the second randomizer circuit 908 to generate coordinates that are indicative of probabilities of a quantum state for a CNOT gate, and can provide a technical improvement to generate the probabilities beyond the capability of manual processes to achieve.

[0653]

[0243] The seventh stage 907 can generate arrows (e.g., vectors) for the target qubit and the control qubit based on the coordinates generated by the sixth stage 906. In some implementations, each arrow includes a coordinate triplet (e.g., X, Y, Z). In an example each of the arrows for the target qubit and the control qubit can include one coordinate from each of the first coordinate selection processor 930, the second coordinate selection processor 932, and the third coordinate selection processor 934. The seventh stage 907 can include a first output unit vector generator 950 to output a first output unit vector 970 and a second output unit vector generator 952 to output a second output unit vector 972.

[0654]

[0244] The first output unit vector generator 950 can select an X-coordinate, a Y-coordinate, and a Z-coordinate from the first coordinate selection processor 930, the second coordinate selection processor 932, and the third coordinate selection processor 934, respectively. The first output unit vector generator 950 can select the coordinates. The first output unit vector generator 950 can normalize the coordinates to the unit sphere. The first output unit vector generator 950 can generate an arrow including the selected X-coordinate, Y-coordinate, and Z-coordinate. In some implementations, the first output unit vector generator 950 generates a unit vector based on the arrow according to Components (96).

[0655] >

[0656] n'l / J(nA)2+ Qr'D2+ (n'l)2

[0657] «'! / J(«l)2+ («'l)2+ («1)2

[0658] n'l / J(«l)2+ («'l)2+ (n'l)2

[0659]

[0660] Components (96)

[0661]

[0245] The second output unit vector generator 952 can select an X-coordinate, a Y-coordinate, and a Z-coordinate from the first coordinate selection processor 930, the second coordinate selection processor 932, and the third coordinate selection processor 934, respectively. The second output unit vector generator 952 can select the coordinates. The second output unit vector

[0662] 4869-4255-3171.8 75Atty. Dkt. 135589-0103

[0663] generator 952 can normalize the coordinates to the unit sphere. The second output unit vector generator 952 can generate an arrow including the selected X-coordinate, Y-coordinate, and Z-coordinate. In some implementations, the second output unit vector generator 952 generates a unit vector based on the arrow according to Components (97).

[0664] n'i / J(XB)2+ (n'b)2+ (n'|)2

[0665] n'l / J(«'B)2+ (n'D2+ (n'l)2

[0666] n'l / J(«'B)2+ (n'D2+ (n'D2

[0667]

[0668] Components (97)

[0669]

[0246] The first output unit vector 970 can be a unit vector output by the first output unit vector generator 950. The first output unit vector 970 can be an output coordinate triplet for the control qubit.

[0670]

[0247] The second output unit vector 972 can be a unit vector output by the second output unit vector generator 952. The second output unit vector 972 can be an output coordinate triplet for the target qubit.

[0671]

[0248] As indicated previously, the two-qubit gate of FIGs. 7-9 can be, or otherwise implement quantum operations of, a controlled NOT (CNOT) quantum gate. However, similar and / or analogous principles, concepts, structures, circuitry, and / or architecture may be used for implementing other two-qubit gates. Different types of two-qubit gates will have different transform matrices. For example, similar and / or analogous principles concepts, structures, circuitry, and / or architecture can be used for implementing a controlled phase gate. A controlled phase gate implemented according to similar and / or analogous principles concepts, structures, circuitry, and / or architecture of FIGs. 7-9 can provide a technical solution to operate an electric or electronic device according to a controlled phase gate, to achieve a technical improvement to achieve quantum gate behavior of a controlled phase gate, including behavior according to entanglement of multiple qubits. For example, a controlled phase gate according to this disclosure can be implemented in at least one of a semiconductor device (e.g., die, wafer), an electronic circuit (e.g., integrated circuit, printed circuit board), a computing system (e.g., compiled high-level language stored in a computer-readable medium), or any combination thereof, but is not limited thereto. Thus, a controlled phase gate implemented according to similar and / or analogous principles concepts, structures, circuitry, and / or architecture of FIGs. 7-9 can provide a technical improvement at least to achieve computational behavior corresponding to a controlled phase gate using digital electronics, analog electrical circuits, or any combination thereof. For example, a controlled phase gate according to this disclosure can be implemented according to probabilistic

[0672] 4869-4255-3171.8 76Atty. Dkt. 135589-0103

[0673] computational processes. Probabilistic output of a controlled phase gate, corresponding to results of measurement of quantum states, can be correlated according to time tags that indicate time stamps of probabilistic outputs of such controlled phase gate. For example, the number of time tags is equal to the number of qubits, which allows correlation of outputs of a controlled phase gate to accurately model probabilistic computational behavior corresponding to quantum entanglement. In some implementations, time tags can be used to correlate outputs of a controlled phase gate based on windowing. In an example, two qubits can be considered part of the same composite system if their time tags are within an identification window, where the window represents, for example, a maximum allowable difference between the time tags.

[0674]

[0249] FIG. 10 depicts an example two-qubit measurement device 1000, according to an embodiment of the present disclosure. The two-qubit measurement device 1000 can measure two qubits output by the CNOT device architecture according to this disclosure, a controlled phase device architecture, any other two-qubit device architecture, or otherwise provided. The two-qubit measurement device 1000 can measure any two qubits, regardless of whether they are outputted by a gate or provided from another source. The two-qubit measurement device 1000 can measure the output of a CNOT gate, such as the first output unit vector 970 and the second output unit vector 972 of the CNOT gate as illustrated in FIGS. 7-9. The two-qubit measurement device 1000 can receive the outputs of the seventh stage: the first output unit vector 970 and the second output unit vector 972. While the two-qubit measurement device 1000 is illustrated as measuring the state of two qubits, similar structures can be implemented to measure the state of any number of qubits, such as an n-qubit measurement device.

[0675]

[0250] The two-qubit measurement device 1000 can take as input the first output unit vector 970 and the second output unit vector 972 and output a first output qubit 1050 and a second output qubit 1052. The first output qubit 1050 can correspond to the control qubit, or state of the control qubit, as measured by the two-qubit measurement device 1000 and the second output qubit 1052 can correspond to the target qubit, or state of the target qubit, as measured by the two-qubit measurement device 1000.

[0676]

[0251] The two-qubit measurement device 1000 includes a first measurement device 1020, a second measurement device 1022, a first time tag generator 1030, a second time tag generator 1032, and a qubit time window processor 1040. The first measurement device 1020, the second measurement device 1022, the first time tag generator 1030, the second time tag generator 1032, and the qubit time window processor 1040 can include one or more logical or electronic devices including but not limited to integrated circuits, logic gates, flip flops, gate arrays, programmable gate arrays, and the like and / or can be implemented at least in part in software.

[0677] 4869-4255-3171.8 77Atty. Dkt. 135589-0103

[0678]

[0252] The first measurement device 1020 and the first time tag generator 1030 can receive the first output unit vector 970. The first measurement device 1020 can measure a quantum state of one or more qubit cells to extract quantum information. The first measurement device 1020 can measure the quantum state of the control qubit (represented by the first output unit vector 970) In some implementations, the first measurement device 1020 measures the unit vector of the control qubit along an axis (i.e., any arbitrary axis). For example, the first measurement device 1020 takes a dot product of the unit vector of the control qubit and a unit vector of an arbitrary axis and outputs the result to the qubit time window processor 1040.

[0679]

[0253] The first time tag generator 1030 can generate time tags (i.e., identification parameters) to correlate outputs of the first measurement device 1020 with outputs of the second measurement device 1022. The first measurement device 1020 can generate multiple measurements over an indication period (e.g., a time period). Outputs (i.e., measurements) can be assigned time tags to identify pairings of outputs. An output corresponding to an input may be generated within a predetermined indication window according to the input is received, allowing for correlation of outputs based on the time tags (e.g., correlation of outputs from the first measurement device 1020 and the second measurement device 1022). The time tags generated by the first time tag generator 1030 can be combined with the outputs of the first measurement device 1020. Outputs of the first measurement device 1020 and the second measurement device 1022 can be correlated by the qubit time window processor 1040 based on the outputs falling within a same indication window, as indicated by their respective indication parameters (e.g., time tags).

[0680]

[0254] The second measurement device 1022 and the second time tag generator 1032 can receive the second output unit vector 972. The second measurement device 1022 can measure a quantum state of one or more qubit cells to extract quantum information. The second measurement device 1022 can measure the quantum state of the target qubit (represented by the second output unit vector 972) In some implementations, the second measurement device 1022 measures the unit vector of the target qubit along an axis (i.e., any arbitrary axis). The axis used for measurement by the second measurement device 1022 can be different than the axis used for measurement by the first measurement device 1020. For example, the second measurement device 1022 takes a dot product of the unit vector of the target qubit and a unit vector of an arbitrary axis and outputs the result to the qubit time window processor 1040.

[0681]

[0255] The second time tag generator 1032 can generate time tags (i.e., indication parameters) to correlate outputs of the second measurement device 1022 with outputs of the first measurement device 1020. The second measurement device 1022 can generate multiple measurements over an indication period (e.g., a time perioed). Outputs (i.e., measurements) can be assigned time tags to identify pairings of outputs. An output corresponding to an input may be generated within a

[0682] 4869-4255-3171.8 78Atty. Dkt. 135589-0103

[0683] predetermined indication window according to the input is received, allowing for correlation of outputs based on the time tags (e.g., correlation of outputs from the first measurement device 1020 and the second measurement device 1022). The time tags generated by the second time tag generator 1032 can be combined with the outputs of the second measurement device 1022. Outputs of the second measurement device 1022 and the first measurement device 1020 can be correlated based on the outputs falling within a same indication window, as indicated by their respective indication parameters (e.g., time tags).

[0684]

[0256] The second measurement device 1022 and the second time tag generator 1032 can provide a plurality of measurements and associated time tags to the qubit time window processor 1040 for the qubit time window processor 1040 to associate (e.g., map, correlate) with the outputs of the first measurement device 1020 and the first time tag generator 1030. In some implementations, there is no association or correlation between an output of the first measurement device 1020 and an output of the second measurement device 1022. A number of the time tags is equal to the number of qubits, allowing for scalability when using hundreds or thousands of qubits.

[0685]

[0257] The qubit time window processor 1040 correlates the outputs of the first measurement device 1020 and the second measurement device 1022 based on the time tags (i.e., identification parameters) associated with the outputs of the first measurement device 1020 and the second measurement device 1022. The qubit time window processor 1040 associates (e.g., maps, correlates) the outputs of the first measurement device 1020 with the outputs of the second measurement device 1022 (e.g., as part of a same composite system). The qubit time window processor 1040 relies on an identification window (e.g., an interval, or aperture) for measurement, which allows the associations (e.g., mappings, correlations) to be made. The identification window may be referred to as a time window. In an example, two qubits can be considered part of the same composite system if their identification parameters (e.g., time tags) are within an identification window, where the window represents, for example, a maximum allowable difference between the identification parameters.

[0686]

[0258] The qubit time window processor 1040 may output the first output qubit 1050 and the second output qubit 1052 based on the first output qubit 1050 and the second output qubit 1052 having time tags (i.e., indication parameters) falling within a same time window (i.e., indication window). The correlated outputs of the first measurement device 1020 and the second measurement device 1022 can be output as the first output qubit 1050 and the second output qubit 1052. The first output qubit 1050 and the second output qubit 1052 may represent measurements of the control qubit input 710 and the target qubit input 712.

[0687] 4869-4255-3171.8 79Atty. Dkt. 135589-0103

[0688]

[0259] The foregoing description of FIG. 10 includes aspects of measuring states of qubits or quantum state. The two-qubit measurement device 1000 may effectuate a measurement that would be a collapsing of the quantum state. For example, a series of quantum gates, including single qubit gates (e.g., the single qubit gates of FIGs. 4 and 5, configured as any of a quantum X gate, a quantum Z gate, a quantum H gate, a quantum R gate, or a quantum P gate) and dual qubit gates (e.g., the CNOT gate of FIGs. 7-9), can be interconnected or otherwise arranged to implement a quantum algorithm. At the end of the series of quantum gates, which may implement a quantum algorithm, a measurement of the quantum state may be obtained using the two-qubit measurement device 1000. The two-qubit measurement device 1000 can obtain a measurement of the resultant quantum state, which may be a collapse of the quantum state, and that measurement of the quantum state may indicate a collapsed state. The two-qubit measurement device 1000 can receive as input a first output unit vector and a second output unit vector (e.g., the first output unit vector 970 and the second output unit vector 972, as illustrated). In an aspect, a first measurement device (e.g., see measurement device 1020) within the two-qubit measurement device 1000 can collapse the state of a first qubit (e.g., a control qubit) and a second measurement device (e.g., see measurement device 1022) within the two-qubit measurement device 1000 can collapse the state of a second qubit (e.g., a target qubit).

[0689]

[0260] The first output qubit 1050 and the second output qubit 1052 can be indicative of a collapsed quantum state, as measured by the two-qubit measurement device 1000. The first output qubit 1050 can be the control qubit, or state of the control qubit (e.g., a collapsed state), as measured by the two-qubit measurement device 1000 and the second output qubit 1052 can be the target qubit, or state of the target qubit (e.g., a collapsed state), as measured by the two-qubit measurement device 1000.

[0690]

[0261] FIG. 11 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1100. The method 1100 can be performed by other systems, components and / or combinations thereof.

[0691]

[0262] At 1110, the method 1100 can obtain one or more first scalar values indicative of a first direction. The first scalar values can be input values. The first scalar values can be a first triplet of coordinates. The first scalar values can correspond to coordinates in a Bloch sphere or Bloch ball. The first scalar values can be coordinates or other values for or indicating a qubit vector. At 1112, the method 1100 can obtain the first scalar values for a quantum state. In an aspect, the quantum state can be represented as a qubit vector. In an aspect, the first direction can be a direction of a quantum state. In an aspect, the first direction can be a direction of a qubit vector.

[0692] 4869-4255-3171.8 80Atty. Dkt. 135589-0103

[0693] In some embodiments, the quantum state can be represented in a data structure, such as a vector data structure. The first scalar values can indicate the quantum state. In an example, the first scalar values can be n1, n2, n3. At 1114, the method 1100 can obtain the first scalar values by a processor or other circuitry.

[0694]

[0263] At 1120, the method 1100 can obtain one or more second scalar values indicative of a second direction. The second scalar values can be input values. The second scalar values can be a second triplet of coordinates. The second scalar values can correspond to coordinates in a Bloch sphere or Bloch ball. At 1122, the method 1100 can obtain the second scalar values. At 1124, the method 1100 can obtain the second scalar values for a measurement direction. In an example, the first scalar values can be m1,m2,m3. At 1126, the method 1100 can obtain the second scalar values by a processor or other circuitry.

[0695]

[0264] FIG. 12 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1200. The method 1200 can be performed by other systems, components and / or combinations thereof.

[0696]

[0265] At 1210, the method 1200 can generate a probability that the first direction corresponds to the second direction. At 1212, the method 1200 can generate the probability based on the one or more first scalar values. At 1214, the method 1200 can generate the probability based on the one or more second scalar values. At 1216, the method 1200 can generate the probability including a scalar value. At 1218, the method 1200 can generate the probability by the processor or other circuitry. In an aspect, the method 1200 can generate the probability according to Equation (37).

[0697]

[0266] For example, first scalar values (e.g., n1, n2, n3) and second scalar values (e.g., m1,m2,m3), can be used to determine a probability that a first direction (indicated by the first scalar values) corresponds to the second direction (indicated by the second scalar values). The probability can include a scalar value. The method 1200 can include implementing or otherwise performing operations using the first and second triplets, such as multiplying coordinate scalar values (e.g., n1m1, n2m2, and n3m3) and / or summing (e.g., adding) coordinate scalar values (e.g., n1m1+ n2m2+ n3m3).

[0698]

[0267] At 1220, the method 1200 can provide an output determinative of the quantum state. At 1222, the method 1200 can provide the output indicative of the value of the probability. At 1224, the method 1200 can provide the output indicative of a collapse of the quantum state. At 1226, the method 1200 can provide the output by the processor or other circuitry.

[0699]

[0268] In an example, at 1228, the method 1200 can include comparing a probability value to another value (e.g., a q value) as part of providing 1220 output (e.g., one or more output

[0700] 4869-4255-3171.8 81Atty. Dkt. 135589-0103

[0701] coordinates) determinative of quantum state. The another value can be a q value that is a random value (e.g., a random number generated by a random number generator).

[0702]

[0269] In an aspect, the method 1200 can include comparing a probability (that a first direction (e.g., an arrow) for a qubit aligns with a second direction (e.g., an arrow) for a measurement) to a threshold. In an aspect, the threshold is based on at least one of a random value, a pseudorandom value, or a quasi-random value. In an aspect, the threshold can be, can include, or can be generated based on an output of a random number generator. For example, a threshold can be provided by the random number generator as a random number q.

[0703]

[0270] In an aspect, the method 1200 can generate three output coordinates (e.g., third triplet of coordinates, third scalar values (e.g., c1, c2, c3) based at least partially on a comparison of the probability p and the random number q. For example, a threshold comparator (e.g., threshold comparator 352 of FIG. 3) can compare the probability p (that a first direction (e.g., an arrow) for a qubit aligns with a second direction (e.g., an arrow) for a measurement) to the random number q. In an aspect, if the random number q is less than or equal to the probability p, then a third triplet of coordinates (e.g., the third scalar values,

[0704]

[0705] c2, c3) are generated equal to the second triplet of coordinates (e.g., the second scalar values m1, m2, m3). And, otherwise (i.e., if random number q is not less than or equal to probability p) the third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to an inverse of the second triplet of coordinates (e.g., — m1(— m2, — m3).

[0706]

[0271] In an aspect, a method 1200 can include aggregating, into one or more aggregated coordinates, first coordinates of the one or more first scalar values with second coordinates of the one or more second scalar values. For example, the method 1200 can aggregate, into one or more aggregated coordinates, first coordinates of the one or more first scalar values with second coordinates of the one or more second scalar values. In an aspect, the method 1200 can combine the aggregated coordinates into the probability. In an aspect, the method can generate, according to the probability satisfying a threshold, one or more third scalar values based at least partially on the measurement direction for the quantum state. For example, the method can generate, according to the probability satisfying a threshold, one or more third scalar values based at least partially on the measurement direction for the quantum state.

[0707]

[0272] In an aspect, the method 1200 can assign or otherwise designate a polarization of a qubit being measured, based on a determination, result, or other indication of a comparison. In an aspect, the method 1200 may generate or otherwise provide an indication of a comparison (e.g., of a probability and a threshold). In an aspect, the method 1200 can then perform operations to assign or otherwise designate third scalar values (e.g., measurement values, or polarization values), which are representative of or indicative of an output qubit and which are based on the

[0708] 4869-4255-3171.8 82Atty. Dkt. 135589-0103

[0709] comparison. For example, the method 1200 may assign or otherwise designate third scalar values as being positive in the case the comparison is positive or otherwise successful and may assign or otherwise designate the third scalar values as being negative in the case the comparison is negative or otherwise unsuccessful. As another example, the method 1200 may assign or otherwise designate third scalar values as having the same sign (e.g., direction) as the measurement direction in the case the comparison is positive or otherwise successful and may assign or otherwise designate the third scalar values as having the opposite sign (e.g., direction) as the measurement direction in the case the comparison is negative, fails, or is otherwise unsuccessful.

[0710]

[0273] In an example, a qubit state being measured may be characterized as an arrow or direction defined by or indicated by a unit vector n̂ = (n1, n2, n3) and a measurement direction (e.g., a predetermined direction), or arrow pointing in that measurement direction may be characterized as a unit vector m = (m1, m2, m3). A probability p that unit vector n is aligned with unit vector m may be characterized by Equation (37) above. And a threshold q may be a random number generated by the random number generator. The method 1200 may compare “c / < / ?”, and if the result is, for example, “True” (or “T”, “Yes”, “Y”, “1”, etc.) then a third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to the second triplet of coordinates (e.g., the second scalar values m1, m2, m3). However, if the result of the comparison “q < p” is, for example, “False” (or “F”, “No”, “N”, “0”, etc.) then a third triplet of coordinates (e.g., the third scalar values, c1, c2, c3) are generated equal to an inverse of the second triplet of coordinates (e.g., -m1(—m2, - m3).

[0711]

[0274] The method 1200 may transform the state of a qubit being measured, according to the third scalar values (e.g., a third triplet of coordinates, c1, c2, c3). For example, the method 1300 may include assigning, setting, or otherwise specifying the third scalar values, c1, c2, c3to designate a vector with a direction of either 1 or -1, in accordance with the second scalar values (e.g., second triplet of coordinates m1, m2, m3).

[0712]

[0275] In an aspect, the method 1200 can generate the one or more third scalar values each equal in magnitude to respective values of the one or more second scalar values. For example, the method 1200 can generate the one or more third scalar values each equal in magnitude to respective values of the one or more second scalar values.

[0713]

[0276] In an aspect, the method 1200 can generate the one or more third scalar values each equal in sign (e.g., direction) to the respective values of the one or more second scalar values, in response to a determination that the probability satisfies the threshold. For example, the method 1200 can generate the one or more third scalar values each equal in sign (e.g., direction) to the

[0714] 4869-4255-3171.8 83Atty. Dkt. 135589-0103

[0715] respective values of the one or more second scalar values, in response to a determination that the probability satisfies the threshold.

[0716]

[0277] In an aspect, the method 1200 can generate the one or more third scalar values each opposite in sign (e.g., direction) to the respective values of the one or more first scalar values, in response to a determination that the probability does not satisfy the threshold. For example, a coordinate assignment circuit (e.g., the coordinate assignment circuit 354 of FIG. 3) can generate the one or more third scalar values each opposite in sign (e.g., direction) to the respective values of the one or more first scalar values, in response to a determination that the probability does not satisfy the threshold.

[0717]

[0278] FIG. 13 depicts an example method of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform the method 1300. The method 1300 can be performed by other systems, components and / or combinations thereof. The method 1300 can provide output coordinates (e.g., scalar values) indicative of a second quantum state.

[0718]

[0279] At 1310, the method 1300 can provide a first output coordinate (e.g., a scalar value). The first output coordinate can be a member of a set of one or more output coordinates indicative of a second quantum state. In some embodiments, the second quantum state can be or be represented as a qubit vector. In some embodiments, the second quantum state can be represented in a data structure, such as a vector data structure. The first output coordinate can be a first coordinate of a qubit vector. At 1312, the method 1300 can provide the first output coordinate corresponding to or according to output of the quantum gate. At 1314, the method 1300 can provide the first output coordinate based on one or more first input coordinates of a first quantum state. At 1316, the method 1300 can provide the first output coordinate according to a first transform operation for a type of quantum gate.

[0719]

[0280] At 1320, the method 1300 can provide a second output coordinate (e.g., a scalar value). The second output coordinate can be a member of a set of one or more output coordinates indicative of the second quantum state. In some embodiments, the second output coordinate can be a second coordinate of a qubit vector. At 1322, the method 1300 can provide the second output coordinate based on one or more second input coordinates of the first quantum state. The second output coordinate can be a second coordinate of the qubit vector. At 1324, the method 1300 can provide the second output coordinate based on one or more second input coordinates of a first quantum state. At 1326, the method 1300 can provide the second output coordinate according to a second transform operation for the type of quantum gate.

[0720]

[0281] At 1330, the method 1300 can provide a third output coordinate (e.g., a scalar value). The third output coordinate can be a member of a set of one or more output coordinates indicative

[0721] 4869-4255-3171.8 84Atty. Dkt. 135589-0103

[0722] of the second quantum state. In some embodiments, the third output coordinate can be a third coordinate of a qubit vector. At 1332, the method 1300 can provide the second output coordinate based on one or more second input coordinates of the first quantum state. The second output coordinate can be a second coordinate of the qubit vector. At 1334, the method 1300 can provide the third output coordinate based on one or more third input coordinates of a first quantum state. At 1336, the method 1300 can provide the third output coordinate according to a third transform operation for the type of quantum gate.

[0723]

[0282] FIG. 14 depicts an example method of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1400. The method 1400 can be performed by other systems, components and / or combinations thereof. In an aspect, a multifunction quantum gate can perform the method 1400.

[0724]

[0283] At 1410, the method 1400 can determine a transform operation for a type of quantum gate. The method 1400 may determine the transform operation by receiving instructions indicating which quantum gate is to be implemented. The method 1400 may determine the transform operation by accessing a mapping of a quantum gate to corresponding transform instructions. In an aspect, the method 1400 may determine the transform operation by obtaining transform instructions (e.g. from a memory, storage) and loading them into memory of a multifunction quantum gate.

[0725]

[0284] At 1420, the method 1400 can generate one or more output coordinates indicative of a second quantum state. In an aspect, the one or more output coordinates may indicate a second qubit vector representative of the second quantum state. At 1422, the method 1400 can generate the output coordinates corresponding to output of the quantum gate. At 1424, the method 1400 can generate the output coordinates based on one or more input coordinates indicative of a first quantum state. In an aspect, the one or more input coordinates can indicate a first qubit vector representative of the first quantum state. At 1426, the method 1400 can generate the output coordinates according to the transform operation. For example, the method 1400 can generate output coordinates according to a transform operation of a quantum gate.

[0726]

[0285] In an aspect, the method 1400 can transform, according to a transform operation corresponding to the type of the quantum gate processor, at least one magnitude from at least one coordinate of one or more first scalar values of the input values, to a corresponding coordinate of one or more second scalar values of the output values. For example, the method 1400 can transform, according to a transform operation corresponding to the type of the quantum gate processor, at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values. In an aspect, the

[0727] 4869-4255-3171.8 85Atty. Dkt. 135589-0103

[0728] method 1400 can transform, according to the type of the quantum gate processor, at least one sign (e.g., direction) from at least one coordinate of the one or more first scalar values, to an opposite sign (e.g., direction) of a corresponding coordinate of the one or more second scalar values. For example, the method 1400 can transform, according to the type of the quantum gate processor, at least one sign (e.g., direction) from at least one coordinate of the one or more first scalar values, to an opposite sign (e.g., direction) of a corresponding coordinate of the one or more second scalar values. In an aspect, the type of the quantum gate corresponds to a quantum X gate, a quantum Z gate, a quantum H gate, a quantum R gate, or a quantum P gate. In an aspect, the type of quantum gate processor corresponds to any appropriate quantum gate.

[0729]

[0286] FIG. 15 depicts a flow diagram of an example method 1500 of executing instructions corresponding to a quantum computing system, according to one embodiment of the present disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform the method 1500. The method 1500 can be performed by other systems, components and / or combinations thereof. The method 1500 can determine a probability of two input qubits aligning. In an aspect, the method 1500 can generate a probabilistic representation (e.g., a density, a density matrix), of a likelihood of respective states in a coordinate space. In an aspect, the method 1500 can generate a probabilistic representation of relative directions between two input qubits (e.g., a density matrix). The method 1500 can provide a technical improvement to achieve behavior corresponding to quantum entanglement via a technical solution to generate one or more probabilities for each input qubit. In an aspect, the method 1500 can generate metrics indicative of relative probabilistic states. Thus, the method 1500 can achieve “mixing” between probabilistic states of two input probabilistic states to probabilistically model a quantum entanglement state.

[0730]

[0287] In an aspect, the method 1500 can measure a quantum state of one or more qubit cells to extract quantum information. The method 1500 can measure the quantum state of a first qubit and a target qubit, otherwise referred to as the “combined state.” For example, the method 1500 can measure the quantum state of a control qubit and a target qubit, which can be the “combined state.” The combined state can include either a mixed state or a pure state. In an aspect, the method 1500 can include taking a plurality of measurements of the control qubit and the target qubit for accumulating an ensemble. In an aspect, the method can correlate inputs and / or outputs by generating time tags to identify pairings (e.g., of outputs).

[0731]

[0288] At 1510, the method 1500 can identify one or more samples of one or more ensembles. In an aspect, at 1510, the method 1500 can identify the one or more samples of the one or more ensembles by measuring a state of one or more qubit cells. In an aspect, the state of the one or more qubit cells can be a quantum state, which can be a “combined state.” The combined state

[0732] 4869-4255-3171.8 86Atty. Dkt. 135589-0103

[0733] can include either a mixed state or a pure state. In an aspect, at 1510 of the method 1500, identifying one or more samples of one or more ensembles can include determining probability elements (e.g. samples) for each of an X-axis, a Y-axis, a Z-axis, and / or an arbitrary axis and accumulating a global ensemble representing a mapping of a probability space for a quantum state of the control qubit and the target qubit. The method 1500 can identify one or more samples of or for the global ensemble.

[0734]

[0289] At 1511, the method 1500 can identify one or more samples of one or more ensembles indicative of a first qubit and a second qubit. In an aspect, at 1511, the method 1500 can identify one or more samples of one or more ensembles indicative of a control qubit and a target qubit. At 1512, the method 1500 can identify one or more samples of one or more ensembles indicative of a first qubit. In an aspect, at 1512, the method 1500 can identify one or more samples of one or more ensembles indicative of a control qubit. At 1514, the method 1500 can identify one or more samples of one or more ensembles indicative of a second qubit. In an aspect, at 1514, the method 1500 can identify one or more samples of one or more ensembles indicative of a target qubit.

[0735]

[0290] In an aspect, the ensemble is indicative of a first qubit (e.g., a control qubit) and indicative of a second qubit (e.g., a target qubit) and all samples are of the same ensemble, or samples for creating, generating, or otherwise populating the same ensemble (e.g., a global ensemble). In an aspect, one or more samples of the ensemble indicative of the first qubit and one or more samples of the ensemble indicative of the second qubit are samples of or for creating, generating, or otherwise populating the same set of one or more ensembles (e.g., a global set of ensembles). Stated otherwise, the one or more ensembles may be a single set of one or more ensembles for which the samples are identified. In an aspect, identifying one or more samples includes accumulating an ensemble of the one or more samples, wherein the ensemble is indicative of the control qubit and the target qubit. In an aspect, identifying one or more samples includes accumulating an ensemble of the one or more samples, the ensemble indicative of a state (e.g., combined state, quantum state) of the control qubit and the target qubit. In an aspect, the ensemble includes or comprises a plurality of samples that have been accumulated and is indicative of a control qubit and a target qubit.

[0736]

[0291] In another embodiment, the method 1500 can include identifying samples of one or more ensembles indicative of a first qubit and identifying samples of one or more ensembles indicative of a second qubit, such that there are at least two sets of one or more ensembles. Stated otherwise, there may be a set of one or more ensembles corresponding to the first qubit and of or for which the sample(s) are identified. And there may be a set of one or more ensembles corresponding to the second qubit and of or for which the samples(s) are identified.

[0737] 4869-4255-3171.8 87Atty. Dkt. 135589-0103

[0738]

[0292] The method 1500 can identify one or more samples of one or more ensembles based on input scalar values. At 1516, the method 1500 can identify the sample(s) of or for the ensemble(s) based on a plurality of scalar values associated with a first qubit and in coordinate space. In an aspect, at 1516, the method 1500 can identify the sample(s) of or for the ensemble(s) based on a plurality of scalar values associated with a control qubit and in coordinate space. For example, a plurality of scalar values associated with the control qubit can be a first coordinate triplet. At 1518, the method 1500 can identify the sample(s) of or for the ensemble(s) based on a plurality of scalar values associated with a second qubit and in coordinate space. In an aspect, at 1518, the method 1500 can identify the sample(s) of or for the ensemble(s) based on a plurality of scalar values associated with a target qubit and in coordinate space. For example, a plurality of scalar values associated with the target qubit can be a second coordinate triplet. In an aspect, the plurality of scalar values can be associated with both the first qubit and the second qubit. In another aspect, a control plurality of scalar values can be associated with a control qubit in coordinate space and a target plurality of scalar values can be associated with a target qubit in coordinate space, such that the control plurality of scalar values and the target plurality of scalar values are separate or distinct.

[0739]

[0293] In an aspect, identifying one or more samples of or for one or more ensembles based on input scalar values includes accumulating the ensemble(s), wherein the ensemble(s) is / are indicative of the first qubit and the second qubit. In an aspect, identifying one or more samples of or for one or more ensembles based on input scalar values includes accumulating the ensemble(s), wherein the ensemble(s) is / are indicative of the control qubit and the target qubit. In an aspect, identifying the sample(s) based on input scalar values includes accumulating one or more ensembles, wherein the ensemble(s) is / are indicative of a state (e.g., combined state, quantum state) of the control qubit and the target qubit. In an aspect, the ensemble(s) include or comprise a plurality of samples that have been accumulated and are indicative of a control qubit and a target qubit.

[0740]

[0294] At 1520, the method 1500 can include determining one or more probabilities of alignment of the first qubit and the second qubit. In an aspect, at 1520, the method 1500 can include determining one or more probabilities of alignment of the control qubit and the target qubit. In an aspect, the sample(s) of the ensemble(s) can be processed to determine the one or more probabilities. In an aspect, at 1520, determining one or more probabilities of alignment can include determining one or more probability objects, according to probability elements. The probability objects can be values selected based on an ensemble (e.g., global ensemble). In an aspect, determining one or more probabilities of alignment can include determining, in

[0741] 4869-4255-3171.8 88Atty. Dkt. 135589-0103

[0742] accordance with the ensemble, one or more probabilities of the control qubit and the target qubit aligning.

[0743]

[0295] At 1522, the method 1500 can determine one or more probabilities of alignment in accordance with the samples of the ensemble(s). In an aspect, determining one or more probabilities of alignment in accordance with the sample(s) of the ensemble(s) can include each probability of alignment indicating likelihood of an alignment of the first qubit (e.g., control qubit) and the second qubit (e.g., target qubit). Alignment of the first qubit and the second qubit can include one or more axes of a control unit vector representing the first qubit (or quantum state thereof) aligning with the corresponding one or more axes of a target unit vector representing the second qubit (or quantum state thereof). In an aspect, at 1522, the method 1500 can generate or otherwise determine a plurality of probabilities each corresponding to an axis or direction. For example, at 1522 the method 1500 can generate a probability with respect to each axis or direction, and thus generate a plurality of probabilities, each indicative of a corresponding probability that a combined state of the first qubit and the second qubit is along a given axis or direction. In an aspect, the determining the one or more probabilities of alignment can include generating a probability for each direction (X-axis, Y-axis, Z-axis, arbitrary-axis) of a combined state of the control qubit input 710 and the target qubit input 712, with respect to the corresponding axis. In an aspect, the one or more probabilities can be provided as one or more probability object outputs, which correspond to probabilities of the control qubit and the target qubit aligning with the X-axis, Y-axis, and Z-axis.

[0744]

[0296] At 1530, the method 1500 can determine one or more densities indicative of a likelihood of respective states. In an aspect, determining a density matrix (“density”) indicative of a likelihood of respective states can include determining density(ies) indicative of a likelihood of alignment of respective quantum states of the first qubit and the second qubit (or combined state thereof). In an aspect, determining a density matrix (“density”) indicative of a likelihood of respective states can include determining density(ies) indicative of a likelihood of alignment of respective quantum states of the control qubit and the target qubit (or combined state thereof). In an aspect, determining density(ies) indicative of a likelihood of respective states can include determining density(ies) indicative of a likelihood of alignment of respective quantum states of the control qubit and the target qubit (or combined state thereof). In an aspect, the determining of the density(ies) can include aggregating probabilities (of alignment of the control qubit and the target qubit). In an aspect, determining the one or more densities can include generating a linear system indicative of a likelihood of a combined state having one or more, or all potential states corresponding to a Bloch ball of Bloch sphere. Thus, determining the one or more densities can include generating a representation of a quantum state of the control qubit and the target qubit.

[0745] 4869-4255-3171.8 89Atty. Dkt. 135589-0103

[0746] Accordingly, density as discussed herein can correspond to a linear system (e.g., a matrix representation) that stores a likelihood that a combined state corresponds to any given probabilistic state.

[0747]

[0297] At 1532, the method 1500 can determine one or more densities (e.g., probability densities) according to the one or more probabilities (of alignment of the control qubit and the target qubit). In an aspect, at 1532, the method 1500 can determine a density according to a probability that the control qubit and the target qubit align. In an aspect, determining one or more densities according to the one or more probabilities can include determining a density based on a distribution of the one or more probabilities. In an aspect, determining the one or more densities according to the one or more probabilities can include generating a density for a combined state of the control qubit and the target qubit, based on probabilities of the combined state relative to each of the X-axis, Y-axis, Z-axis, and / or arbitrary-axis, and each given density can be indicative of a likelihood of states (e.g., probabilistic states). In an aspect, determining one or more densities according to the one or more probabilities can include generating a linear system of sixteen equations that correspond to a likelihood that the combined state corresponds to a probabilistic state aligned with the X-axis, Y-axis, Z-axis, and the arbitrary axis, and the linear system or density can be represented as a matrix, referred to as a density matrix.

[0748]

[0298] FIG. 16 depicts an example method of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1600. The method 1600 can be performed by other systems, components and / or combinations thereof. The method 1600 can transform a probabilistic representation of a likelihood of respective states in a coordinate space into one or more quantum states or representations thereof.

[0749]

[0299] At 1610, the method 1600 can transform one or more densities into one or more transformed densities. In an aspect, at 1610, the method can transform at least a portion of a received density and can generate a modified combined state (e.g., transformed density) that corresponds to a probability of a state of the control qubit and the target qubit in the coordinate space.

[0750]

[0300] At 1612, the method 1600 can transform one or more densities into one or more transformed densities corresponding to the probability(ies) of the state(s) of the control qubit and the target qubit. In an aspect, at 1612, the method 1600 can transform a density matrix into a transformed density matrix corresponding to a probability of a state of the control qubit and the target qubit. A modified combined state, represented as the transformed density, can correspond to a probability of a state of the control qubit and the target qubit. At 1614, the method 1600 can transform one or more densities into one or more transformed densities corresponding to the

[0751] 4869-4255-3171.8 90Atty. Dkt. 135589-0103

[0752] probability(ies) of the state(s) in a coordinate space. In an aspect, at 1614, the method 1600 can transform a density matrix into a transformed density matrix corresponding to a probability of a state in the coordinate space. In an aspect, at 1614, the method 1600 can transform a density matrix into a transformed density matrix corresponding to a probability of a state of the control qubit and the target qubit in the coordinate space.

[0753]

[0301] At 1620, the method 1600 can generate one or more coordinates in the coordinate space. In an aspect, at 1620, the method 1600 can generate one or more coordinates in the coordinate space by reducing a system of linear equations of the transformed density to a reduced system of linear equations. The linear system of equations (whether of the transformed density or the reduced linear system) can include multiple scalar values that, within the linear system of equations, indicate a probability of a state (e.g., probabilistic state) of the control qubit and the target qubit. Thus, the method 1600 can provide a technical improvement to identify output combined states that have probabilities corresponding to the quantum states of qubits according to the control qubit and the target qubit. In an aspect, at 1620, generating one or more coordinates in the coordinate space can include identifying an order for a plurality of output combined states that can be generated. In an aspect, the order can be according to a probability that the combined state corresponds to an output. In an aspect, at 1620, generating one or more coordinates in the coordinate space can include segmenting one or more probabilities of ranked direction probabilities outputs into a set that includes arrows whose likelihoods of occurrence correspond to the quantum probability. For example, this can include outputting unit vectors (e.g., arrows) representing samples of an ensemble corresponding to the state of the control qubit and the state of the target qubit based on the probabilities corresponding to the quantum states of qubits according to the first qubit (e.g., control qubit) and the second qubit (e.g., target qubit).

[0754]

[0302] At 1621, the method 1600 can include generating the coordinates indicative of a second quantum state (e.g. probabilistic state) of the first qubit (e.g., control qubit) and the second qubit (e.g., target qubit). At 1622, the method 1600 can include generating the coordinates indicative of a second quantum state (e.g. probabilistic state) of the first qubit. In an example, the first qubit can be a control qubit. At 1624, the method 1600 can include generating the coordinates indicative of a second quantum state (e.g. probabilistic state) of the second qubit. In an example, the second qubit can be a target qubit. At 1626, the method 1600 can generate the coordinates each indicative of corresponding coordinates relative to respective axes in the coordinate space. At 1628, the method 1600 can generate the coordinates based on each of the probabilities for each of the corresponding directions.

[0755]

[0303] FIG. 17 depicts a flow diagram of an example method 1700 of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the

[0756] 4869-4255-3171.8 91Atty. Dkt. 135589-0103

[0757] systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1700. The method 1700 can be performed by other systems, components and / or combinations thereof. At 1710, the method 1700 can obtain one or more first scalar values indicative of a first direction. At 1720, the method 1700 can obtain one or more second scalar values indicative of a second direction. At 1730, the method 1700 can generate a probability that the first direction corresponds to the second direction. At 1740, the method 1700 can provide an output determinative of a collapse of the quantum state.

[0758]

[0304] FIG. 18 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1800. The method 1800 can be performed by other systems, components and / or combinations thereof. At 1810, the method 1800 can provide a first output coordinate indicative of a second quantum state. At 1820, the method 1800 can provide a second output coordinate indicative of the second quantum state.

[0759]

[0305] FIG. 19 depicts a flow diagram of an example method of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform method 1900. The method 1900 can be performed by other systems, components and / or combinations thereof. At 1910, the method 1900 can determine a transform operation for a type of quantum gate. At 1920, the method 1900 can generate one or more output coordinates indicative of a second quantum state.

[0760]

[0306] FIG. 20 depicts an example method 2000 of executing instructions corresponding to a quantum computing system, according to this disclosure. At least one of the systems of FIGs. 1-10, or any component thereof, or any combination thereof, can perform the method 2000. The method 2000 can be performed by other systems, components and / or combinations thereof. At 2010, the method 2000 can include identifying sample(s) of ensemble(s). At 2020, the method 2000 can include determining probability(ies) of alignment of a first qubit and a second qubit. At 2030, the method 2000 can include determining density (ies) indicative of a likelihood of states (combined states of the first qubit and the second qubit). At 2040, the method 2000 can include transforming density(ies) into transformed density(ies). At 2050, the method 2000 can include generating one or more coordinates in the coordinate space.

[0761]

[0307] Example Embodiments

[0762] 4869-4255-3171.8 92Atty. Dkt. 135589-0103

[0763]

[0308] The following are some example embodiments within the scope of the disclosure. In order to avoid complexity in providing the disclosure, not all of the examples listed below are separately and explicitly disclosed as having been contemplated herein as combinable with all of the others of the examples listed below and other embodiments disclosed hereinabove. Unless one of ordinary skill in the art would understand that these examples listed below (and the above disclosed embodiments) are not combinable, it is contemplated within the scope of the disclosure that such examples and embodiments are combinable.

[0764]

[0309] Example 1. In an aspect, a system according to the present disclosure can comprise one or more processors to: obtain one or more first values (e.g., first scalar values) indicative of a first direction corresponding to a quantum state (e.g., pure quantum state, mixed quantum state); obtain one or more second values (e.g., second scalar values) indicative of a second direction corresponding to a measurement direction for the quantum state; generate, based on the one or more first values and the one or more second values, a probability that the first direction corresponds to the second direction, the probability including a scalar value; and provide an output determinative of a collapse of the quantum state, the output indicative of the scalar value of the probability.

[0765]

[0310] Example 2. The system of Example 1, wherein the one or more processors are further to aggregate, into one or more aggregated coordinates, first coordinates of the one or more first values with second coordinates of the one or more second values.

[0766]

[0311] Example 3. The system of Example 1, wherein the one or more processors are further to combine the one or more aggregated coordinates into the probability.

[0767]

[0312] Example 4. The system of Example 1, wherein the one or more processors are further to generate, according to the probability satisfying a threshold, one or more third values (e.g., third scalar values) based at least partially on the measurement direction for the quantum state.

[0768]

[0313] Example 5. The system of Example 4, wherein the one or more third values are indicative of a third direction, and the third direction corresponds to the collapse of the quantum state.

[0769]

[0314] Example 6. The system of Example 4, wherein the threshold is based on at least one of a random value, a pseudorandom value, or a quasi-random value.

[0770]

[0315] Example 7. The system of Example 4, wherein the one or more processors are further to generate the one or more third values each equal in magnitude to respective values of the one or more second values.

[0771]

[0316] Example 8. The system of Example 7, wherein the one or more processors are further to generate the one or more third values each equal in sign (e.g., direction) to the respective

[0772] 4869-4255-3171.8 93Atty. Dkt. 135589-0103

[0773] values of the one or more second values, in response to a determination that the probability satisfies the threshold.

[0774]

[0317] Example 9. The system of Example 7, wherein the one or more processors are further to generate the one or more third values each opposite in sign (e.g., opposite in direction) to the respective values of the one or more second values, in response to a determination that the probability does not satisfy the threshold.

[0775]

[0318] Example 10. The system of Example 9, the one or more first values corresponding to one or more first coordinates defining a first vector in a three-dimensional space corresponding to a Hilbert space, the first vector having the first direction.

[0776]

[0319] Example 11. The system of Example 10, the one or more second values corresponding to one or more second coordinates defining a second vector in the three-dimensional space corresponding to the Hilbert space, the second vector having the second direction.

[0777]

[0320] Example 12. The system of Example 11, the one or more third values corresponding to one or more third coordinates defining a third vector in the three-dimensional space corresponding to the Hilbert space, the third vector having a third direction.

[0778]

[0321] Example 13. The system of Example 1, the one or more first values corresponding to one or more first coordinates defining a first vector in an N-dimensional space, the first vector having the first direction.

[0779]

[0322] Example 14. The system of Example 13, the one or more second values corresponding to one or more second coordinates defining a second vector in the N-dimensional space corresponding to a Hilbert space, the second vector having the second direction.

[0780]

[0323] Example 15. The system of Example 13, the one or more third values corresponding to one or more third coordinates defining a third vector in the N-dimensional space corresponding to a Hilbert space, the third vector having a third direction.

[0781]

[0324] Example 16. The system of Example 1, wherein the one or more processors are further to obtain the one or more first values as output of a quantum gate processor configured to transform, according to a type of the quantum gate processor, one or more input values into the one or more first values.

[0782]

[0325] Example 17. The system of Example 1, wherein the one or more processors are further to obtain the one or more first values from a source for qubits.

[0783]

[0326] Example 18. The system of Example 16, wherein the one or more first values correspond to a first triplet of coordinates, and the one or more input values correspond to a second triplet of coordinates.

[0784] 4869-4255-3171.8 94Atty. Dkt. 135589-0103

[0785]

[0327] Example 19. The system of Example 16, wherein the quantum gate processor is configured to: copy at least one magnitude from at least one coordinate of the one or more first values, to a corresponding coordinate of the one or more second values.

[0786]

[0328] Example 20. The system of Example 16, wherein the quantum gate processor is configured to transform, according to a transform operation corresponding to the type of the quantum gate processor, at least one magnitude from at least one coordinate of the one or more first values, to a corresponding coordinate of the one or more second values.

[0787]

[0329] Example 21. The system of Example 16, wherein the quantum gate processor is configured to: copy at least one direction from at least one coordinate of the one or more first values, to a corresponding coordinate of the one or more second values.

[0788]

[0330] Example 22. The system of Example 16, wherein the quantum gate processor is configured to transform, according to the type of the quantum gate processor, at least one sign (e.g. direction) from at least one coordinate of the one or more first values, to an opposite sign (e.g. opposite direction) of a corresponding coordinate of the one or more second values.

[0789]

[0331] Example 23. The system of Example 16, wherein the type of the quantum gate processor corresponds to a quantum X gate, a quantum Z gate, a quantum H gate, a quantum R gate, or a quantum P gate.

[0790]

[0332] Example 24. The system of Example 1, the one or more first values being input scalar values corresponding to one or more input coordinates defining an input vector in a multidimensional space corresponding to a Hilbert space, the input vector having the first direction.

[0791]

[0333] Example 25. The system of Example 24, the one or more second values being measurement scalar values corresponding to one or more measurement coordinates defining a measurement vector in the multidimensional space corresponding to the Hilbert space, the measurement vector having the second direction.

[0792]

[0334] Example 26. The system of Example 24, the one or more processors to: generate, according to the probability satisfying a threshold, one or more third values based at least partially on the measurement direction for the quantum state, wherein the one or more third values are output scalar values corresponding to one or more output coordinates defining an output vector in the multidimensional space corresponding to the Hilbert space, the output vector having a third direction.

[0793]

[0335] Example 27. In an aspect, a method of measuring according to the present disclosure can comprise: obtaining one or more first values (e.g., first scalar values) indicative of a first direction corresponding to a quantum state; obtaining one or more second values (e.g., second scalar values) indicative of a second direction corresponding to a measurement direction for the

[0794] 4869-4255-3171.8 95Atty. Dkt. 135589-0103

[0795] quantum state; generating, based on the one or more first values and the one or more second values, a probability that the first direction corresponds to the second direction, the probability including a scalar value; and provide an output determinative of a collapse of the quantum state, the output indicative of the scalar value of the probability.

[0796]

[0336] Example 28. The method of Example 27, further comprising: aggregating, into one or more aggregated coordinates, first coordinates of the one or more first scalar values with second coordinates of the one or more second scalar values.

[0797]

[0337] Example 29. The method of Example 28, further comprising: combining the one or more aggregated coordinates into the probability.

[0798]

[0338] Example 30. The method of Example 27, further comprising: generating, according to the probability satisfying a threshold, one or more third values (e.g., third scalar values) based at least partially on the measurement direction for the quantum state.

[0799]

[0339] Example 31. The method of Example 30, wherein generating the one or more third scalar values includes generating the one or more third scalar values to be indicative of a third direction, and the third direction corresponds to the collapse of the quantum state.

[0800]

[0340] Example 32. The method of Example 30, wherein the threshold is based on at least one of a random value, a pseudorandom value, or a quasi-random value.

[0801]

[0341] Example 33. The method of Example 30, wherein generating the one or more third scalar values includes generating the one or more third scalar values to each be equal in magnitude to respective values of the one or more second scalar values.

[0802]

[0342] Example 34. The method of Example 33, wherein generating the one or more third scalar values includes generating the one or more third values to each have a sign (e.g., a direction) equal to the respective values of the one or more second scalar values, in response to a determination that the probability satisfies the threshold.

[0803]

[0343] Example 35. The method of claim 34, wherein generating the one or more third scalar values includes generating the one or more third values to each be opposite in sign (e.g., opposite in direction) to the respective values of the one or more second scalar values, in response to a determination that the probability does not satisfy the threshold.

[0804]

[0344] Example 36. The method of Example 27, the one or more first values corresponding to one or more first coordinates defining a first vector in a three-dimensional space corresponding to a Hilbert space, the first vector having the first direction.

[0805]

[0345] Example 37. The method of Example 36, the one or more second values corresponding to one or more second coordinates defining a second vector in the three-dimensional space corresponding to the Hilbert space, the second vector having the second direction.

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[0807]

[0346] Example 38. The method of Example 37, further comprising: generating, according to the probability satisfying a threshold, one or more third values (e.g., third scalar values) based at least partially on the measurement direction for the quantum state, wherein the one or more third scalar values correspond to one or more third coordinates defining a third vector in the three-dimensional space corresponding to the Hilbert space, the third vector having a third direction.

[0808]

[0347] Example 39. The method of Example 27, the one or more first values corresponding to one or more first coordinates defining a first vector in an N-dimensional space, the first vector having the first direction.

[0809]

[0348] Example 40. The method of Example 39, the one or more second values corresponding to one or more second coordinates defining a second vector in the N-dimensional space corresponding to a Hilbert space, the second vector having the second direction.

[0810]

[0349] Example 41. The method of Example 27, further comprising: obtaining the one or more first values as output of a quantum gate configured to transform, according to a type of the quantum gate, one or more input scalar values into the one or more first values.

[0811]

[0350] Example 42. The method of Example 27, further comprising obtain the one or more first values from a source for qubits.

[0812]

[0351] Example 43. The method of Example 41, wherein the one or more first values correspond to a first triplet of coordinates, and the one or more input scalar values correspond to a second triplet of coordinates.

[0813]

[0352] Example 44. The method of Example 41, wherein the quantum gate is configured to: copy at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second values.

[0814]

[0353] Example 45. The method of Example 41, wherein the quantum gate is configured to: transform, according to a transform operation corresponding to the type of the quantum gate, at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second values.

[0815]

[0354] Example 46. The method of Example 41, wherein the quantum gate is configured to: copy at least one sign (e.g., direction) from at least one coordinate of the one or more first values, to a corresponding coordinate of the one or more second values.

[0816]

[0355] Example 47. The method of Example 41, wherein the quantum gate is configured to: transform, according to the type of the quantum gate, at least one sign (e.g., direction) from at least one coordinate of the one or more first values, to an opposite sign (e.g., direction) of a corresponding coordinate of the one or more second values.

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[0818]

[0356] Example 48. The method of Example 41, wherein the type of the quantum gate corresponds to a quantum X gate, a quantum Z gate, a quantum H gate, a quantum R gate, or a quantum P gate.

[0819]

[0357] Example 49. The method of Example 27, the one or more first values being input scalar values corresponding to one or more input coordinates defining an input vector in a multidimensional space corresponding to a Hilbert space, the input vector having the first direction.

[0820]

[0358] Example 50. The method of Example 49, the one or more second values being measurement scalar values corresponding to one or more measurement coordinates defining a measurement vector in the multidimensional space corresponding to the Hilbert space, the measurement vector having the second direction.

[0821]

[0359] Example 51. The method of Example 50, further comprising: generating, according to the probability satisfying a threshold, one or more third values (e.g., third scalar values) based at least partially on the measurement direction for the quantum state, the one or more third values being output scalar values corresponding to one or more output coordinates defining an output vector in the multidimensional space corresponding to the Hilbert space, the output vector having a third direction.

[0822]

[0360] Example 52. A device corresponding to a quantum gate, the device comprising: a circuit to output, based on one or more input coordinates indicative of a quantum state (e.g., a first quantum state) and according to a transform operation associated with a type of quantum gate, one or more output coordinates indicative of a new quantum state (e.g., a second quantum state) corresponding to output of the quantum gate.

[0823]

[0361] Example 53. The device of Example 52, further comprising: a memory coupled with the circuit and storing one or more transform instructions corresponding to the transform operation.

[0824]

[0362] Example 54. The device of Example 53, wherein the circuit is to obtain the one or more transform instructions from the memory.

[0825]

[0363] Example 55. The device of Example 52, further comprising: an input circuit to receive the one or more input coordinates.

[0826]

[0364] Example 56. The device of Example 52, wherein the one or more input coordinates include a first coordinate of the quantum state, a second coordinate of the quantum state, and a third coordinate of the quantum state.

[0827]

[0365] Example 57. The device of Example 56, wherein the type of quantum gate is a X gate, and wherein the transform operation corresponds to: generating a first output coordinate (of the one or more output coordinates) having a magnitude equal to a magnitude of the first coordinate

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[0829] and having a direction equal to a direction of the first coordinate; generating a second output coordinate (of the one or more output coordinates) having a magnitude equal to a magnitude of the second coordinate and having a sign (e.g., direction) opposite to a sign of the second coordinate; and generating a third output coordinate (of the one or more output coordinates) having a magnitude equal to a magnitude of the third coordinate and having a sign opposite to a sign of the third coordinate.

[0830]

[0366] Example 58. The device of Example 56, wherein the type of quantum gate is a Z gate, and wherein the transform operation corresponds to: generating a first output coordinate having a magnitude equal to a magnitude of the first coordinate and having a sign opposite to a sign of the first coordinate; generating a second output coordinate having a magnitude equal to a magnitude of the second coordinate and having a sign opposite to a sign of the second coordinate; and generating a third output coordinate having a magnitude equal to a magnitude of the third coordinate and having a direction equal to a direction of the third coordinate.

[0831]

[0367] Example 59 The device of Example 56, wherein the type of quantum gate is a T gate, and wherein the transform operation corresponds to: to generating a first output coordinate as an arithmetic product of a constant with a difference between the first coordinate and the second coordinate; generating the second output coordinate as an arithmetic product of a constant with a sum between the first coordinate and the second coordinate; and generating the third output coordinate having a magnitude equal to a magnitude of the third coordinate, and having a direction equal to a direction of the third coordinate.

[0832]

[0368] Example 60. The device of Example 56, wherein the type of quantum gate is an H gate, and wherein the transform operation corresponds to: generating the first output coordinate having a magnitude equal to a magnitude of the third coordinate, and having a direction equal to a direction of the third coordinate; generating the second output coordinate having a magnitude equal to a magnitude of the second coordinate, and having a sign opposite to a sign of the second coordinate; generating the third output coordinate having a magnitude equal to a magnitude of the first coordinate, and having a direction equal to a direction of the first coordinate.

[0833]

[0369] Example 61. The device of Example 56, wherein the type of quantum gate is a P gate or an R gate, and wherein the transform operation corresponds to: generating the first output as a difference between a first trigonometric operation and a second trigonometric operation, the first trigonometric operation based on the first coordinate and the second trigonometric operation based on the second coordinate.

[0834]

[0370] Example 62. The device of Example 61, wherein the type of quantum gate is the R gate, the first trigonometric operation is based on a constant, the second trigonometric operation is based on the constant, and the constant is according to the R gate.

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[0836]

[0371] Example 63. The device of Example 61, wherein the transform operation further corresponds to: generating the second output coordinate as a sum of a first trigonometric operation and a second trigonometric operation, the first trigonometric operation based on the first coordinate and the second trigonometric operation based on the second coordinate.

[0837]

[0372] Example 64. The device of Example 63, wherein the type of quantum gate is the R gate, the first trigonometric operation is based on a constant, the second trigonometric operation is based on the constant, and the constant is according to the R gate.

[0838]

[0373] Example 65. The device of Example 61, wherein the transform operation further corresponds to: generating the third output coordinate having a magnitude equal to a magnitude of the third coordinate, and having a direction equal to a direction of the third coordinate.

[0839]

[0374] Example 66. A device corresponding to a quantum gate, the device comprising: a first circuit to output, based on one or more input coordinates indicative of a first quantum state (e.g. a vector) and according to a first transform operation associated with a type of quantum gate, a first output coordinate indicative of a second quantum state (e.g., a new quantum state) (e.g. a vector) corresponding to output of the quantum gate; and a second circuit to output, based on one or more second input coordinates of the first quantum state and according to a second transform operation associated with the type of quantum gate, a second output coordinate indicative of the second quantum state.

[0840]

[0375] Example 67. The device of Example 66, further comprising: a first memory coupled with the first circuit and storing one or more first transform instructions corresponding to the first transform operation.

[0841]

[0376] Example 68. The device of Example 67, wherein the first circuit is to obtain the one or more first transform instructions from the first memory.

[0842]

[0377] Example 69. The device of Example 66, further comprising: a second memory coupled with the second circuit and storing one or more second transform instructions corresponding to the second transform operation.

[0843]

[0378] Example 70. The device of Example 69, wherein the second circuit is to obtain the one or more second transform instructions from the second memory.

[0844]

[0379] Example 71. The device of Example 66, further comprising a third circuit to output, based on one or more third input coordinates of the first quantum state and according to a third transform operation associated with the type of quantum gate, a third output coordinate indicative of the second quantum state.

[0845]

[0380] Example 72. The device of Example 71, further comprising a third memory coupled with the third circuit and storing one or more third transform instructions corresponding to the third transform operation.

[0846] 4869-4255-3171.8 100Atty. Dkt. 135589-0103

[0847]

[0381] Example 73. The device of Example 72, wherein the third circuit is configured to obtain the one or more third transform instructions from the third memory.

[0848]

[0382] Example 74. The device of Example 71, wherein the type of quantum gate is a Z gate, the one or more first input coordinates include a first coordinate of the first quantum state, the one or more second input coordinates include a second coordinate of the first quantum state, and the one or more third input coordinates include a third coordinate of the first quantum state.

[0849]

[0383] Example 75. The device of Example 74, wherein the first transform operation corresponds to generating the first output having a magnitude equal to a magnitude of the first coordinate, and having a sign opposite to a direction of the first coordinate.

[0850]

[0384] Example 76. The device of Example 74, wherein the second transform operation corresponds to generating the second output having a magnitude equal to a magnitude of the second coordinate, and having a sign opposite to a sign of the second coordinate.

[0851]

[0385] Example 77. The device of Example 74, wherein the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the third coordinate, and having a direction equal to a direction of the third coordinate.

[0852]

[0386] Example 78. The device of Example 71, wherein the type of quantum gate is a T gate, the one or more first input coordinates include a first coordinate of the first quantum state and a second coordinate of the first quantum state, the one or more second input coordinates include the first coordinate of the quantum state and the second coordinate of the first quantum state, and the one or more third input coordinates include a third coordinate of the first quantum state.

[0853]

[0387] Example 79. The device of claim Example 78, wherein the first transform operation corresponds to generating the first output as an arithmetic product of a constant with a difference between the first coordinate and the second coordinate.

[0854]

[0388] Example 80. The device of Example 78, wherein the second transform operation corresponds to generating the second output as an arithmetic product of a constant with a sum between the first coordinate and the second coordinate.

[0855]

[0389] Example 81. The device of Example 78, wherein the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the second coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate.

[0856]

[0390] Example 82. The device of Example 71, wherein the type of quantum gate is an H gate, the one or more first input coordinates include a first coordinate of the first quantum state, the one or more second input coordinates include a second coordinate of the first quantum state, and the one or more third input coordinates include a third coordinate of the first quantum state.

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[0858]

[0391] Example 83. The device of Example 82, wherein the first transform operation corresponds to generating the first output having a magnitude equal to a magnitude of the third coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate.

[0859]

[0392] Example 84. The device of Example 82, wherein the second transform operation corresponds to generating the second output having a magnitude equal to a magnitude of the second coordinate, and having a sign (e.g., direction) opposite to a sign (e.g., direction) of the second coordinate.

[0860]

[0393] Example 85. The device of Example 82, wherein the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the first coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the first coordinate.

[0861]

[0394] Example 86. The device of Example 71, wherein the type of quantum gate is a P gate or an R gate, the one or more first input coordinates include a first coordinate of the first quantum state and a second coordinate of the first quantum state, the one or more second input coordinates include the first coordinate of the first quantum state and the second coordinate of the first quantum state, and the one or more third input coordinates include a third coordinate of the first quantum state.

[0862]

[0395] Example 87. The device of Example 86, wherein the first transform operation corresponds to generating the first output as a difference between a first trigonometric operation and a second trigonometric operation, the first trigonometric operation based on the first coordinate and the second trigonometric operation based on the second coordinate.

[0863]

[0396] Example 88. The device of Example 87, wherein the type of quantum gate is the R gate, the first trigonometric operation is based on a constant, the second trigonometric operation is based on the constant, and the constant is according to the R gate.

[0864]

[0397] Example 89. The device of Example 86, wherein the second transform operation corresponds to generating the second output as a sum of a first trigonometric operation and a second trigonometric operation, the first trigonometric operation based on the first coordinate and the second trigonometric operation based on the second coordinate.

[0865]

[0398] Example 90. The device of Example 89, wherein the type of quantum gate is the R gate, the first trigonometric operation is based on a constant, the second trigonometric operation is based on the constant, and the constant is according to the R gate.

[0866]

[0399] Example 91. The device of Example 86, wherein the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the third

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[0868] coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the third coordinate.

[0869]

[0400] Example 92. The device of Example 71, wherein the type of quantum gate is a X gate, the one or more first input coordinates include a first coordinate of the first quantum state, the one or more second input coordinates include a second coordinate of the first quantum state, and the one or more third input coordinates include a third coordinate of the first quantum state.

[0870]

[0401] Example 93. The device of Example 92, wherein the first transform operation corresponds to generating the first output having a magnitude equal to a magnitude of the first coordinate, and having a sign (e.g., direction) equal to a sign (e.g., direction) of the first coordinate.

[0871]

[0402] Example 94. The device of Example 92, wherein the second transform operation corresponds to generating the second output having a magnitude equal to a magnitude of the second coordinate, and having a sign (e.g., direction) opposite to a sign (e.g., direction) of the second coordinate.

[0872]

[0403] Example 95. The device of Example 94, wherein the third transform operation corresponds to generating the third output having a magnitude equal to a magnitude of the third coordinate, and having a sign (e.g., direction) opposite to a sign (e.g., direction) of the third coordinate.

[0873]

[0404] Example 96. A device corresponding to a quantum gate, the device comprising: a memory and one or more processors to: determine a transform operation associated with a type of quantum gate; and generate, based on one or more input coordinates indicative of a first quantum state (e.g. a vector) and according to the transform operation, one or more output coordinates indicative of a second quantum state (e.g., a vector) corresponding to output of the quantum gate.

[0874]

[0405] Example 97. The device of Example 96, the one or more processors further to: select the transform operation that corresponds to the type of quantum gate; and obtain the transform operation from a transform storage of the memory.

[0875]

[0406] Example 98. The device of Example 96, wherein the transform storage stores a plurality of transform operations including the transform operation.

[0876]

[0407] Example 99. The device of Example 98, wherein the plurality of transform operations corresponds to a plurality of types of quantum gates including the type of quantum gate.

[0877]

[0408] Example 100. The device of Example 96, wherein the one or more input coordinates correspond to a first triplet of coordinates in a three-dimensional space corresponding to a Hilbert space, and the one or more output coordinates correspond to a second triplet of coordinates in the three-dimensional space corresponding to the Hilbert space.

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[0879]

[0409] Example 101. In an aspect, a non-transitory computer readable medium according to the present disclosure can include one or more instructions stored thereon and executable by a processor to: determine, by the processor, a transform operation associated with a type of quantum gate; and generate, by the processor and based on one or more input coordinates indicative of a first quantum state (e.g., vector) and according to the transform operation, one or more output coordinates indicative of a second quantum state (e.g., vector) corresponding to output of ...

Claims

Atty. Dkt. 135589-0103WHAT IS CLAIMED IS:

1. A system, comprising:one or more processors to:obtain one or more first scalar values indicative of a first direction corresponding to a quantum state;obtain one or more second scalar values indicative of a second direction corresponding to a measurement direction for the quantum state;generate, based on the one or more first scalar values and the one or more second scalar values, a probability that the first direction corresponds to the second direction, the probability including a scalar value; andprovide an output determinative of a collapse of the quantum state, the output indicative of the scalar value of the probability.

2. The system of claim 1, the one or more processors to:aggregate, into one or more aggregated coordinates, first coordinates of the one or more first scalar values with second coordinates of the one or more second scalar values.

3. The system of claim 2, the one or more processors to:combine the one or more aggregated coordinates into the probability.

4. The system of claim 1, the one or more processors to:generate, according to the probability satisfying a threshold, one or more third scalar values based at least partially on the measurement direction for the quantum state.

5. The system of claim 4, wherein the one or more third scalar values are indicative of a third direction, and the third direction corresponds to the collapse of the quantum state.

6. The system of claim 4, wherein the threshold is based on at least one of a random value, a pseudorandom value, or a quasi-random value.

7. The system of claim 4, the one or more processors to:generate the one or more third scalar values each equal in magnitude to respective values of the one or more second scalar values.

8. The system of claim 7, the one or more processors to:4869-4255-3171.8 114Atty. Dkt. 135589-0103generate the one or more third scalar values each having a sign equal to the respective values of the one or more second scalar values, in response to a determination that the probability satisfies the threshold.

9. The system of claim 7, the one or more processors to:generate the one or more third scalar values each having a sign opposite to the respective values of the one or more second scalar values, in response to a determination that the probability does not satisfy the threshold.

10. The system of claim 9, the one or more first scalar values corresponding to one or more first coordinates defining a first vector in a three-dimensional space corresponding to a Hilbert space, the first vector having the first direction.

11. The system of claim 10, the one or more second scalar values corresponding to one or more second coordinates defining a second vector in the three-dimensional space corresponding to the Hilbert space, the second vector having the second direction.

12. The system of claim 11, the one or more third scalar values corresponding to one or more third coordinates defining a third vector in the three-dimensional space corresponding to the Hilbert space, the third vector having a third direction.

13. The system of claim 7, the one or more first scalar values corresponding to one or more first coordinates defining a first vector in an N-dimensional space, the first vector having the first direction.

14. The system of claim 13, the one or more second scalar values corresponding to one or more second coordinates defining a second vector in the N-dimensional space corresponding to a Hilbert space, the second vector having the second direction.

15. The system of claim 14, the one or more third scalar values corresponding to one or more third coordinates defining a third vector in the N-dimensional space corresponding to a Hilbert space, the third vector having a third direction.

16. The system of claim 1, the one or more processors to:4869-4255-3171.8 115Atty. Dkt. 135589-0103obtain the one or more first scalar values as output of a quantum gate processor configured to transform, according to a type of the quantum gate processor, one or more input scalar values into the one or more first scalar values.

17. The system of claim 16, wherein the one or more first scalar values correspond to a first triplet of coordinates, and the one or more input scalar values correspond to a second triplet of coordinates.

18. The system of claim 16, wherein the quantum gate processor is configured to:copy at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values.

19. The system of claim 16, wherein the quantum gate processor is configured to:transform, according to a transform operation corresponding to the type of the quantum gate processor, at least one magnitude from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values.

20. The system of claim 16, wherein the quantum gate processor is configured to:copy at least one direction from at least one coordinate of the one or more first scalar values, to a corresponding coordinate of the one or more second scalar values.

21. The system of claim 16, wherein the quantum gate processor is configured to:transform, according to the type of the quantum gate processor, at least one sign from at least one coordinate of the one or more first scalar values, to an opposite sign of a corresponding coordinate of the one or more second scalar values.

22. The system of claim 16, wherein the type of the quantum gate processor corresponds to a quantum X gate, a quantum Z gate, a quantum H gate, a quantum R gate, or a quantum P gate.

23. The system of claim 1, the one or more first scalar values being input scalar values corresponding to one or more input coordinates defining an input vector in a multidimensional space corresponding to a Hilbert space, the input vector having the first direction.

24. The system of claim 23, the one or more second scalar values being measurement scalar values corresponding to one or more measurement coordinates defining a measurement vector in4869-4255-3171.8 116Atty. Dkt. 135589-0103the multidimensional space corresponding to the Hilbert space, the measurement vector having the second direction.

25. The system of claim 23, the one or more processors to:generate, according to the probability satisfying a threshold, one or more third scalar values based at least partially on the measurement direction for the quantum state,wherein the one or more third scalar values are output scalar values corresponding to one or more output coordinates defining an output vector in the multidimensional space corresponding to the Hilbert space, the output vector having a third direction.

26. A method of measuring, comprising:obtaining one or more first values indicative of a first direction corresponding to a quantum state;obtaining one or more second values indicative of a second direction corresponding to a measurement direction for the quantum state;generating, based on the one or more first values and the one or more second values, a probability that the first direction corresponds to the second direction, the probability including a scalar value; andprovide an output determinative of a collapse of the quantum state, the output indicative of the scalar value of the probability.

27. A system, comprising:one or more processors, coupled with memory, to:identify, based on a first plurality of scalar values associated with a control qubit in a coordinate space and a target qubit in the coordinate space, a sample of an ensemble indicative of the control qubit and the target qubit;determine, in accordance with the sample of the ensemble, one or more probabilities of the control qubit and the target qubit aligning;determine, for each of the one or more probabilities, a density indicative of a likelihood of respective states in the coordinate space;transform the density into a transformed density corresponding to a probability of a state of the control qubit and the target qubit in the coordinate space; andgenerate, based on the transformed density, a second plurality of scalar values in the coordinate space indicative of a State of the control qubit and the target qubit.4869-4255-3171.8 117Atty. Dkt. 135589-010328. The system of claim 27, the one or more processors to:receive the first plurality of scalar values, including first axial scalar values indicative of a first axial direction in the coordinate space, second axial scalar values indicative of a second axial direction in the coordinate space, and third axial scalar values indicative of a third axial direction in the coordinate space.

29. The system of claim 28, the one or more processors to:determine, based on a first orientation corresponding to the first axial direction, a first probability of the control qubit and the target qubit aligning.

30. The system of claim 29, the one or more processors to:determine, based on a second orientation corresponding to the second axial direction, a second probability of the control qubit and the target qubit aligning.

31. The system of claim 30, the one or more processors to:determine, based on a third orientation corresponding to the third axial direction, a third probability of the control qubit and the target qubit aligning.

32. The system of claim 28, wherein the coordinate space is defined based on the first axial direction, the second axial direction, and the third axial direction.

33. The system of claim 28, wherein the coordinate space is a three-dimensional coordinate space.

34. The system of claim 28, the one or more processors to:receive the first scalar values including independent scalar values indicative of a fourth axial direction in the coordinate space, the fourth axial direction distinct from one or more axes of the coordinate space.

35. The system of claim 34, the one or more processors to:determine, based on a fourth orientation of the orientations corresponding to the fourth axial direction, a fourth probability with respect to the fourth axial direction.

36. The system of claim 34, wherein the fourth axial direction is an arbitrary direction in the coordinate space.4869-4255-3171.8 118Atty. Dkt. 135589-010337. The system of claim 34, the one or more processors to:generate, based on input including the transformed density, a probability of states relative to the coordinate space, the transformed density having a first number of parameters, and the probability of states having a second number of parameters.

38. The system of claim 37, wherein the second number is less than the first number.

39. The system of claim 34, the one or more processors to:order the corresponding probabilities of states that are relative to the coordinate space.

40. The system of claim 34, the one or more processors to:transform each of the probabilities for each of the corresponding directions into the corresponding probabilities of states in the coordinate space, based on at least one of a respective random value, a respective pseudorandom value, or a respective quasi-random value.

41. A method, comprising:identifying, based on a first plurality of scalar values associated with a control qubit in a coordinate space and a target qubit in the coordinate space, a sample of an ensemble indicative of the control qubit and the target qubit;determining, in accordance with the sample of the ensemble, one or more probabilities of alignment of the control qubit and the target qubit;determining, for each of the one or more probabilities, a first density indicative of a likelihood of respective states in the coordinate space;transforming the first density into a transformed density corresponding to a probability of a state of the control qubit and the target qubit in the coordinate space; andgenerating, based on the transformed density, a second plurality of scalar values in the coordinate space indicative of a second state of the control qubit and a second state of the target qubit.4869-4255-3171.8 119

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