Systems and methods for controlled quantum information processing using transradical basis components

Abstract

Systems, methods, applications, and uses for trans-radix quantum information processing elements are disclosed. Specifically, embodiments as disclosed herein may provide radix-2 quantum operations utilizing a trans-radix quantum circuit comprising one or more radix-r circuit elements, where r>2. Embodiments of such quantum circuits may also include one or more radix-2 circuit elements, where input or output ports of the one or more radix-r quantum circuit elements are utilized to support the input and output of a radix-2 qubit transmission channel.

Description

[Technical Field]

[0001] (CROSS-REFERENCE TO RELATED APPLICATIONS) This application claims the benefit of priority under 35 U.S.C. § 119 of U.S. Provisional Patent Application No. 63 / 027,056, filed May 19, 2020, by Mitchell A. Thornton, entitled "SYSTEMS AND METHODS FOR CONTROLLED QUANTUM INFORMATION PROCESSING OPERATION WITH TRANS-RADIX BASIS COMPONENTS," which is hereby incorporated by reference in its entirety.

[0002] The present disclosure relates generally to quantum computing. Specifically, the present disclosure relates to embodiments of quantum information processing (QIP) operations suitable for use in dedicated quantum information science devices or general-purpose QIP devices, including, but not limited to, quantum computers (QCs). Even more specifically, the present disclosure relates to embodiments of systems and methods for implementing controlled-order operations, including devices capable of operations involving trans-radix quantum states. [Background technology]

[0003] Some computational problems, such as factoring large numbers, cannot be easily solved using conventional computers due to the time required to complete the computation. However, it has been shown that quantum computers can use non-classical algorithmic methods to provide efficient solutions to some of these types of computational problems.

[0004] The basic unit of quantum information in a quantum computer is called a quantum bit or qubit. Like traditional binary computers, quantum computers can use a binary representation of numbers. In addition, quantum systems can also utilize multi-valued logic and data, in which case atomic quantum data is referred to as a "qudit." Individual qubit or qudit data can be physically represented by the state of the quantum system. However, in quantum systems, data can be considered to be in more than one of several possible states at any single given time. Thus, in the case of a qubit, data can be in a state representing both a zero and a one at the same time. This state is called a superposition. Such quantum superposition is fundamentally different from classical data representations, even when classical probabilities are considered. Only when quantum data is observed does its value "collapse" into a single, well-defined state. This "collapse" can occur due to intentional observation or measurement, or it can occur due to environmental influences, called decoherence.

[0005] Thus, while a bit in the classical computing model always has a well-defined value (e.g., 0 or 1), a qubit in superposition has a joint probability of being in both of the two states representing 0 and 1. It is customary to denote the general state of a quantum system by |ψ>, with |0> and |1> representing the quantum states corresponding to the values ​​0 and 1, respectively. Quantum mechanics allows for the superposition of these two states, given by: |ψ>=α|0>+β|1> where α and β are complex numbers. In this case, the probability of observing the system in state |0> is α 2 and the probability of state |1> is β 2 is.

[0006] Quantum computers may utilize physical particles to represent or implement these qubits or qudits. One example is the rapid spinning of an electron, where an up or down spin can correspond to a superposition of states: 0, 1, or both up and down simultaneously. Performing a computation using an electron may essentially perform operations on both 0 and 1 simultaneously. Similarly, in a photonic approach to quantum computing, a "0" may be represented by the possibility of observing a single photon in a given path, while the potential of observing the same photon in a different path may represent a "1."

[0007] For example, consider a single photon passing through an interferometer with two paths, where phase shifts φ and φ are inserted into the two paths, respectively. A beam splitter gives a 50% probability that the photon will travel one path or the other. If a measurement is made to determine where the photon is located, it will be found in only one of the two paths. However, if no such measurement is made, a single photon can somehow experience both phase shifts φ and φ simultaneously. This suggests, in a sense, that a photon must be located in both paths simultaneously if no measurement is made to determine its location. This effect can be experimentally verified by observing the interference pattern resulting from the interaction of the two paths when only a single photon is allowed to pass through the device at a given time. Of course, if there is more than a single pair of possible photonic paths, the resulting system can be considered to represent a qudit.

[0008] It is therefore well known that control qudit operations are required to enable certain QIP processing tasks. As an example, many quantum communication protocols and processes require the generation of two entangled qudits. In this case, control qudit operations are commonly employed to serve as entanglement operators in QIP systems that include an entanglement generator. For at least this reason, there is motivation to devise efficient methods for realizing control qudit operations that are optimized (e.g., with respect to some desired physical property). Summary of the Invention [Means for solving the problem]

[0009] To address this need, attention is drawn to embodiments of systems and methods for implementing controlled-order operations, including, among other things, devices capable of operations involving trans-radix quantum states. Specifically, embodiments as disclosed herein may provide radix-2 quantum operations utilizing a trans-radix quantum circuit comprising one or more radix-r circuit elements, where r>2. Such quantum circuit embodiments may also include one or more radix-2 circuit elements, where input or output ports of the one or more radix-r quantum circuit elements are utilized to support the input and output of a radix-2 qubit transmission channel.

[0010] In particular, embodiments may utilize a radix-4 Chrestenson gate C4 in a transradix fashion. In particular, embodiments may utilize two qubits (e.g., photons) with the Chrestenson gate C4. Each qubit (e.g., photon) may be present at only two associated ports of the Chrestenson gate C4. In particular, a first qubit may be provided in the associated waveguide of the first or second port of the Chrestenson gate C4 such that the first qubit is in a ground state on the waveguide of the first port or on the waveguide of the second port, or such that the first qubit is superimposed on both the waveguide of the first port and the waveguide of the second port. Similarly, a second qubit may be provided at the associated third and fourth ports of Chrestenson gate C4 such that the second qubit is in a ground state on the third port waveguide or on the fourth port waveguide, or such that the second qubit is superimposed on both the first port waveguide and the second port waveguide. In many embodiments, it may be desirable for this presentation of the first and second qubits at their respective ports to be time-aligned.

[0011] Thus, based on the interaction of first and second qubits in the radix-4 Chrestenson gate C4, the first qubit may appear in a ground state on the waveguide of the first port or the waveguide of the second port, or may be superimposed on both waveguides from both the first and second ports of the Chrestenson gate C4, while the second qubit may appear in a ground state on the waveguide from the third port or the waveguide of the fourth port, or may be superimposed on both waveguides from both the third and fourth ports of the Chrestenson gate C4. Such trans-radix Chrestenson gate C4 embodiments may have up to a 100% success rate in achieving interaction (e.g., entanglement) of qubits utilized with such trans-radix Chrestenson gates.

[0012] As can be appreciated, such trans-radix Chrestenson gate C4 embodiments (e.g., embodiments utilizing QIP circuit element 550 in a trans-radix configuration) may have up to a 100% success rate in achieving interaction (e.g., entanglement) of two qubits utilized with such trans-radix Chrestenson gate. Thus, because such trans-radix Chrestenson gate C4 can be utilized as a type of controlled quantum circuit based on at least two qubits, such trans-radix Chrestenson gate C4 can be usefully employed in other (e.g., standard) controlled quantum circuit embodiments. For example, trans-radix Chrestenson gate C4 embodiments may be used to implement reliable controlled S-gate quantum circuit embodiments, reliable controlled Z-gate quantum circuit embodiments, reliable controlled X-gate quantum circuit embodiments, or more generally, reliable embodiments of quantum circuits in which entanglement or interaction of qubits is desired.

[0013] Also, as will be understood, the controlled S gate (e.g., or controlled X gate) may be a subset of all possible quantum operations in the binary domain. Thus, almost any desired quantum circuit may be constructed in a reliable manner using embodiments of the trans-radix Chrestenson gate C4, embodiments of the controlled S gate, embodiments of the controlled S gate, or embodiments of the controlled X gate as disclosed herein. Thus, embodiments herein may serve as a foundational change to the reliability of quantum circuits and quantum computing more generally.

[0014] Accordingly, embodiments of the disclosed systems and methods may also include one or more radix-4 QIP elements with characteristic transformation matrices in the form of Chrestenson transformation matrices, one or more radix-2 single-qubit QIP elements with characteristic transformation matrices in the form of unitary transformation matrices, and be adapted to operate as radix-2 control qubit circuit elements.

[0015] Certain embodiments of the present systems and methods may include one or more radix-4 QIP elements with characteristic transformation matrices in the form of Chrestenson transformation matrices, one or more radix-2 single-qubit QIP elements with characteristic transformation matrices in the form of Hadamard transformation matrices, and be adapted to operate as radix-2 controlled S QIP circuit elements.

[0016] Other embodiments of the present systems and methods may include one or more radix-4 QIP elements with characteristic transformation matrices in the form of Chrestenson transformation matrices, one or more radix-2 single-qubit QIP elements with characteristic transformation matrices in the form of Hadamard transform matrices, and be adapted to operate as radix-2 controlled-Z QIP circuit elements.

[0017] Some embodiments of the present systems and methods may include one or more radix-4 QIP elements with characteristic transformation matrices in the form of Chrestenson transformation matrices, one or more radix-2 single-qubit QIP elements with characteristic transformation matrices in the form of Hadamard transform matrices, and be adapted to operate as radix-2 controlled X QIP circuit elements.

[0018] In certain embodiments, the quantum Chrestenson gate may be implemented using photonics.

[0019] For example, some embodiments may comprise an optical four-port directional coupler configured to operate as a transformer radix-2 qubit QIP circuit element.

[0020] These and other aspects of the present disclosure will be better appreciated and understood when considered in conjunction with the following description and the accompanying drawings. It should be understood, however, that the following description, while indicating various embodiments of the present disclosure and numerous specific details thereof, is given by way of illustration and not by way of limitation. Many substitutions, modifications, additions, and / or rearrangements may be made within the scope of the present disclosure without departing from the spirit thereof, and the present disclosure includes all such substitutions, modifications, additions, and / or rearrangements. The present invention provides, for example, the following items. (Item 1) A transradix quantum circuit, comprising: Quantum circuit elements implementing the Chrestenson transformation matrix Equipped with the quantum circuit element is adapted to operate as a trans-radix-2 qubit quantum circuit element, and the two qubits include a first radical-2 qubit and a second radical-2 qubit; Transradical quantum circuits. (Item 2) Item 10. The transformer radix quantum circuit of item 1, further comprising one or more radix-2 quantum circuit elements coupled to the quantum circuit element, wherein the transformer radix quantum circuit is adapted to operate as a control quantum circuit. (Item 3) 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled S quantum circuit element. (Item 4) 4. The trans-radix quantum circuit of claim 3, wherein the radix-2 quantum element is applied to the first qubit before the first qubit is provided to the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element. (Item 5) 5. The trans-radix quantum circuit of claim 4, wherein the radix-2 quantum element is adapted to operate as the trans-radix-2 qubit quantum circuit element after the quantum circuit element is applied to the first qubit. (Item 6) 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled-Z quantum circuit element. (Item 7) 7. The trans-radix quantum circuit of claim 6, wherein the radix-2 quantum element is applied to the first qubit before the first qubit is provided to the quantum circuit element implementing the Chrestenson transformation matrix adapted to operate as the trans-radix-2 qubit quantum circuit element for the first time. (Item 8) 8. The trans-radix quantum circuit of claim 7, wherein the base-2 quantum element is applied to the first qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the first qubit for the first time and before the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the first qubit for a second time. (Item 9) Item 9. The trans-radix quantum circuit of item 8, wherein the radix-2 quantum element is applied to the second qubit before the second qubit is provided a second time to the quantum circuit element implementing the Chrestenson transformation matrix adapted to operate as the trans-radix-2 qubit quantum circuit. (Item 10) 10. The trans-radix quantum circuit of claim 9, wherein the radix-2 quantum element is adapted to operate as the trans-radix-2 qubit quantum circuit element after the quantum circuit element is applied to the second qubit a second time. (Item 11) 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled-X quantum circuit element. (Item 12) Item 12. The trans radix quantum circuit of item 11, wherein the base-2 quantum element is applied to the first qubit after the quantum circuit element adapted to operate as the trans radix-2 qubit quantum circuit element is applied to the first qubit for the first time and before the quantum circuit element adapted to operate as the trans radix-2 qubit quantum circuit element is applied to the first qubit for a second time; and the base-2 quantum element is applied to the second qubit after the quantum circuit element adapted to operate as the trans radix-2 qubit quantum circuit element is applied to the second qubit for the first time and before the quantum circuit element adapted to operate as the trans radix-2 qubit quantum circuit element is applied to the second qubit for the second time. (Item 13) Item 13. The trans radix-2 quantum circuit of item 12, wherein the base-2 quantum element is applied to the first qubit after the quantum circuit element adapted to operate as the trans radix-2 qubit quantum circuit element is applied to the first qubit a second time, and the base-2 quantum element is applied to the second qubit after the quantum circuit element adapted to operate as the trans radix-2 qubit quantum circuit element is applied to the second qubit a second time. (Item 14) Item 2. The trans-radix quantum circuit of item 1, wherein the quantum circuit element implementing the Chrestenson transformation matrix is ​​implemented in photonics. (Item 15) Item 15. The transformer-radix quantum circuit of item 14, wherein the quantum circuit element is an optical four-port directional coupler. (Item 16) Item 16. The trans-radix quantum circuit of item 15, wherein applying the quantum circuit element to the first radical-2 qubit and the second radical-2 qubit comprises providing the first radical-2 qubit to a first face or a second face of the optical four-port directional coupler and providing the second radical-2 qubit to a third face or a fourth face of the optical four-port directional coupler. (Item 17) 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled S quantum circuit element. (Item 18) 1. A method comprising: providing a first radix-2 qubit at a first port or a second port of an optical four-port directional coupler; providing a second radix-2 qubit at a third port or a fourth port of the optical four-port directional coupler; and operating the optical four-port directional coupler as a transformer radix-2 qubit QIP circuit element by A method comprising: (Item 19) Item 19. The method of item 18, wherein the first base-2 qubit and the second base-2 qubit are time-aligned at the first port, the second port, the third port, or the fourth port. (Item 20) A transradix quantum circuit, comprising: Quantum circuit elements that implement radix-2 quantum operations Equipped with A trans-radix quantum circuit, wherein the quantum circuit element comprises one or more radix-r circuit elements, where r>2. (Item 21) 21. The trans-radix quantum circuit of item 20, further comprising one or more radix-2 circuit elements, wherein an input or output port of the one or more radix-r quantum circuit elements supports the input or output of a radix-2 qubit transmission channel. [Brief explanation of the drawings]

[0021] The drawings accompanying and forming a part of this specification are included to depict certain aspects of the present disclosure. It should be noted that the features illustrated in the drawings are not necessarily drawn to scale. A more complete understanding of the present disclosure and its advantages may be obtained by reference to the following description considered in conjunction with the accompanying drawings, in which like reference numerals indicate like features.

[0022] [Figure 1] FIG. 1 illustrates a general diagram of a radix-r single-qubit operation with a generalized transformation matrix Ur in QIP circuit form.

[0023] [Figure 2] FIG. 2 illustrates a general diagram of a radix-r controlled qubit operation with a generalized target qubit transformation matrix Ur in QIP circuit form.

[0024] [Figure 3] 3 illustrates, in QIP circuit form, a schematic diagram of a radix-2 control qubit operation with a generalized target qubit transformation matrix U. The resulting overall transformation matrix for this control qubit operator is denoted as CU.

[0025] [Figure 4] 4 illustrates, in QIP circuit form, a schematic diagram of a radix-2 control qubit operation with a specific target qubit transformation matrix S. The resulting overall transformation matrix for this control qubit operator is denoted as CS.

[0026] [Figure 5a] FIG. 5a illustrates a schematic diagram of a radix-4 single-qudit operation, called a Chrestenson gate and denoted as C4, in QIP circuit form.

[0027] [Figure 5b] FIG. 5b illustrates, in schematic form, a quantum photonic implementation of a base-4 single-qudit operator, called a Chrestenson gate and denoted as C4, which utilizes the position observable of the photonic host particle as an information-carrying attribute when used within a QIP circuit.

[0028] [Figure 6] FIG. 6 illustrates one embodiment of a trans-radix operation in QIP circuit form, where two radix-2 qubits are applied to a single radix-4 Chrestenson gate C4.

[0029] [Figure 7]7a and 7b illustrate an embodiment of a transradix operator in QIP circuit form, where two radix-2 qubits are applied to a QIP system comprising radix-2 and radix-4 Chrestenson gates C4, resulting in an overall action equivalent to a two-qubit radix-2 control qubit operator.

[0030] [Figure 8a] FIG. 8a illustrates a prior art implementation of a radix-2 controlled Z circuit, consisting of two radix-2 controlled S circuits in series, in a QIP circuit configuration.

[0031] [Figure 8b] FIG. 8b illustrates a prior art representation of a Radix-2 Control Z circuit in QIP circuit form.

[0032] [Figure 9] 9a and 9b illustrate an embodiment of a transradix operator in QIP circuit form in which two radix-2 qubits are applied to a QIP system comprising four radix-2 and two radix-4 Chrestenson gates C4, resulting in an overall action equivalent to a two-qubit radix-2 controlled Z qubit operator with an equivalent specific target qubit transformation matrix Z resulting in an overall transformation matrix denoted as CZ.

[0033] [Figure 10a] FIG. 10a illustrates a prior art implementation of a radix-2 controlled X circuit, consisting of two radix-2 single-qubit Hadamard gates H and a controlled Z gate in QIP circuit form.

[0034] [Figure 10b] FIG. 10b illustrates a prior art representation of a radix-2 controlled X circuit in QIP circuit form.

[0035] [Figure 11a]FIG. 11a illustrates one embodiment of a trans-radix operator in QIP circuit form, where two radix-2 qubits are applied to a QIP system comprising four radix-2 Hadamard gates H and two radix-4 Chrestenson gates C4, resulting in an overall action equivalent to a two-qubit radix-2 controlled X qubit QIP circuit.

[0036] [Figure 11b] FIG. 11b illustrates one embodiment of a trans-radix operator in QIP circuit form, where two radix-2 qubits are applied to a QIP system comprising four radix-2 Hadamard gates H and two radix-4 Chrestenson gates C4, resulting in an overall action equivalent to a two-qubit radix-2 controlled X qubit QIP circuit. DETAILED DESCRIPTION OF THE INVENTION

[0037] Detailed Description The details of the present disclosure and its various features and advantages will be more fully explained by reference to non-limiting embodiments illustrated in the accompanying drawings and detailed in the following description. Descriptions of well-known starting materials, processing techniques, components, and equipment are omitted so as not to unnecessarily obscure the invention in detail. It should be understood, however, that the detailed description and specific examples, while indicating some embodiments of the present invention, are given by way of illustration only, not limitation. Various substitutions, modifications, additions, and / or rearrangements within the spirit and / or scope of the underlying inventive concept will become apparent to those skilled in the art from this disclosure.

[0038] Before discussing the embodiments in detail, it may be useful to provide a general overview of certain aspects related to the embodiments. As can be recalled from the above discussion, quantum information processing (QIP) comprises methods and systems that use the physical phenomena of quantum mechanics or quantum electrodynamics to represent and process data. Observable properties of microscopic, mesoscopic, or quasiparticles, hereafter referred to as "host particles," represent or encode data values ​​to be processed. The wave functions of the host particles evolve according to the axioms of quantum mechanics or electrodynamics. Within the QIP framework, a prescribed set of host particle wave function evolutions in time or space serves as a computation. In one implementation of a quantum computer (QC), a given sequence of distinct operations serves as a QC program, which is executed accordingly via the quantum mechanical evolution of the host particle wave functions. In this QC example, a user may specify a specific sequence of operations during the act of programming the QC. In one type of dedicated QIP device, a finite subset of sequences of distinct operations defines the dedicated functionality of the QIP device.

[0039] The quantum state of the host particle is [ka] It is understood that QIPs evolve according to well-known dynamical relations described by the wave equations, or the Liouville-von Neumann wave equation. The specific evolution is governed by one of many different possible solutions to these equations. A specific solution to the wave equation results in a complex exponential of the Hamiltonian operator, which is of the form, a unitary matrix. A specific Hamiltonian operator therefore defines a specific, distinct QIP operation. Mathematically, the unitary operator is applied to the initial conditions of the wave function, resulting in an evolved form due to the specific Hamiltonian on which the unitary operator depends.

[0040] Quasi-particles can be implemented in the form of integrated microwave electronic circuits. The host quasiparticles can evolve in time from an initial state to a different state via the application of microwave pulses of energy to the circuit. The circuit configuration, in terms of its frequency, phase, and amplitude, as well as the pulse configuration, define specific unitary matrices that serve as QIP operations. A set of different unitary matrices thereby represents a set of information processing operations that can be applied to the quasiparticles or the integrated microwave electronic circuits, accordingly.

[0041] Within the framework of a QIP, a collection of host particles abstractly represents a corresponding collection of quantum digits or "qudits." It is typical for a collection of qudits to be characterized by some pre-established order, thus allowing the collection to be considered a qudit register. The contents of the qudit register represent a complex value. The complex value is thus an abstracted view of the overall physical quantum state or wave function of the QIP system. The qudit register value can change as the QIP system quantum state evolves due to the application of operations. Therefore, a concrete QIP system computation is defined by a concrete unitary matrix that can transform the qudit register value.

[0042] QIP systems are typically implemented based on a pre-specified, fixed, non-zero integer value r, referred to as the system base or radix, which constrains the dimensionality of the quantum state vector or density matrix representing each host particle's wave function. The inherent discrete nature of quantum mechanical or quantum electrodynamic observables is controlled so that each host particle's wave function is not permitted to require a representation with a vector or matrix whose dimensionality exceeds r or r × r, respectively. One implementation of QIP utilizes host particle observables that can occupy only the two lowest quantum states, thus resulting in a binary system in which each host particle represents a quantum bit or "qubit."

[0043] For a QIP system implemented using n distinct host particles whose observables are limited to only r dimensions, the operations of the QIP system can be abstractly represented by concrete unitary matrices with dimensions rn×rn. Alternatively, the QIP system can be viewed as comprising a register of n individual distinct cardinality-r qudits. An operation performed on a single cardinality-r qudit, where the single qudit is abstractly represented as an r-dimensional vector or pure state, would then be represented by a corresponding r×r unitary matrix. Application of an operation evolves the single qudit into a new state that can be expressed as a Cartesian product of the r×r unitary matrix and the qudit's initial state, which is represented by the corresponding r-dimensional vector.

[0044] As can be appreciated by those skilled in the art, for any arbitrary cardinality r, a pure quantum state |Ψ r 〉 is the probability amplitude [ka] can be mathematically expressed as a linear combination of a set of calculated basis vectors scaled by [ka] represents a complex-valued field as follows: [ka] In the formula, |Ψ r > is a pure state. In addition, as is well known, the probability amplitude of a pure state must also satisfy the following: [ka]

[0045] Based on this representation of the pure states, the set of calculated basis vectors is a set of r orthonormal vectors denoted as |k〉 for k = 0 to k = r − 1, and the corresponding probability amplitude of each basis vector is a for i ≠ k. i =0 and a k = 1. Hence, the computational basis state is defined as the set {|0〉,|1〉,...,|r-2〉,|r-1〉}.

[0046] Similarly, the pure quantum state |Ψ r The unitary expansion matrix U applied to r can be viewed as comprising a set of column vectors representing all possible evolved states corresponding to an initial state equivalent to one of r distinct calculated basis states. r The leftmost column vector of corresponds to the evolved state of |0〉, and the rightmost column vector of corresponds to the evolved state of the basis vector |r-1〉.

[0047] From the perspective of a mathematical model of qudit evolution, operations expressed as unitary matrices can change one or more qudit values ​​in a register. One form of operation, known as a "control qudit operation," depends on the initial or current state of two qudits in a register. Note here that these operations can be extended from the binary (or radix-2) qubit case to the qudit case, and thus, more generally, these quantum operands can be referred to as either qubits (specifically, radix-2 encoded and used for this particular case) or qudits (which can be used to represent the more general case). Thus, in this particular case, operations on qubits are specifically mentioned, but all of this and subsequent operations illustrated herein can also be extended more generally to apply to qudits.

[0048] where n=2 and the corresponding unitary transformation matrix is ​​of dimension r 2 ×r 2The first of these two qubits is designated as the "control qubit," and the second of the two qubits is designated as the "target qubit." The control qubit operation conditionally evolves the target qubit depending on the value or current state of the control qubit. More specifically, control qubit operations refer to a set of two-qubit operations, where the first, or control qubit, exhibits specific, distinct forms of transformation applied to the second, target qubit, and the distinct forms of the target qubit transformation differ based on the state of the first, or control qubit. Thus, the two-qubit control qubit operation results in the conditional evolution of the target qubit. The complete family of control qubit operations and their use in synthesizing QIP systems is described in [TS+:08], referenced below.

[0049] More generally, a control qudit operation is defined by specifying a particular basis vector value, referred to herein as an "activation value," that, when satisfied by the control qudit, causes a single qudit transformation operator to be applied to the target qudit. The target qudit undergoes the specified single qudit transformation only if the control qudit satisfies the specified activation value criterion. If the control qudit does not satisfy the activation value criterion, the target qudit remains unchanged and its evolution is equivalent to that of an identity matrix operator. Since the control qudit can potentially be in a state of quantum superposition, the control qudit can similarly have a non-zero probability amplitude for the activation value component of its corresponding wave function. In this case, the control qudit operator can act as an entanglement gate under appropriate conditions, resulting in the evolution of the combined quantum state of the target and control qudits into a condition of quantum entanglement [ST:19].

[0050] An embodiment of a control qubit operation involves a first control qubit and a second, distinct target qubit, where the state of the control qubit governs which of two distinct single-qubit transformations is applied to the target qubit. In one particular embodiment of a control qubit operation, referred to as a "control S" operation, the control qubit either causes the target qubit to undergo a 90-degree relative phase shift between its two wavefunction components or to remain unchanged. In this embodiment, the control S unitary matrix is ​​a matrix of dimension 2 2 ×2 2 = 4 × 4. The target qubit either undergoes a single-qubit evolution represented by the 2 × 2 unitary matrix S, or alternatively, the target qubit remains unchanged, depending on the state of the control qubit and the prescribed activation basis value. It is typical for the control qubit operator to assume an activation value of |1〉 for the control qubit, where |1〉 is a member of the computation basis set. For QIP systems with r > 2, it is conventional for the control qubit to have prescribed activation basis values ​​from the set {|0〉,|1〉, ..., |r-2〉,|r-1〉} for a particular control qubit operation, where {|0〉,|1〉, ..., |r-2〉,|r-1〉} is a generalization of the computation basis in r-dimensional vector spaces.

[0051] Therefore, a control qudit operation is required to enable certain QIP processing tasks. As an example, many quantum communication protocols and processes require the generation of two entangled qudits. In this case, a control qudit operation is commonly employed to serve as an entanglement operator in a QIP system that comprises an entanglement generator. For this reason, there is an incentive to devise efficient methods for realizing a control qudit operation that is optimized with respect to certain desired physical properties, among others.

[0052] To explain in more detail, in many cases, current QIP circuits (also referred to herein simply as quantum circuits) may have control operators (e.g., quantum circuits that are based on or can implement qudit entanglement). However, in many cases, these control operations may actually function to entangle such qudits (or otherwise allow such qudits to interact) only a small percentage of the time (e.g., 25% or the like). Such failures may result, for example, at least in part, from the information carriers used for such qudits. For example, when photons are utilized as the information carrier, it may be extremely difficult to cause photons to interact. Thus, in operations where such entanglement fails (e.g., 75% of the time), the entire quantum operation may fail. Such high failure rates (generally, both the entanglement operation and the resulting quantum circuit failure) act as a major obstacle to efficient QIP. Therefore, there is significant motivation to realize such control qudit operations with higher success rates, including control qudit operations that rely on qudit entanglement and such entanglement operations.

[0053] Some additional context may also prove useful. Turning now first to FIG. 1, a single qudit operation in the form of a QIP circuit 100 is depicted. The QIP circuit is depicted with a horizontal line 110, representing the qudit prior to its evolution due to QIP circuit gates 120. The progression of the horizontal line from left to right indicates the evolution of the qudit in time or space, with the qudit 110 remaining in a constant state along the progression of line 110. QIP gates 120 intersect the horizontal line representing the qudit and represent quantum state evolutions that are applied to the qudit, evolving it into a new state. Additionally, the symbol U annotates box 110. rrepresents a specific matrix operator that mathematically models the qudit evolution, thus providing a concrete form of the evolution as a transformation matrix. The subscript r represents the cardinality or dimension of the vectors and matrices that represent the wavefunctions and transformations. QIP circuit 100 is depicted with horizontal line 130, which represents the qudit after it has evolved due to QIP gate 120 and remained in a constant evolved state along the progression of line 130.

[0054] Mathematically, a cardinality-r qudit is typically represented either as a pure state wave function in the form of a column vector or ket of dimension r, or alternatively, the quantum state can be represented as an r × r density matrix. These two alternative forms of qudit representation are denoted by |Ψ, respectively. r 〉 and ρ r For pure states, the relationship between these two forms of quantum state representation is ρ r =|Ψ r 〉〈Ψ r More generally, a quantum state can be expressed as a k-member ensemble {p i ,|Ψ ri 〉}, and p i But, state |Ψ ri 〉 is the mixed state ρ r When expressing the objective probability of being in , this is expressed as a density matrix formed as follows: [ka]

[0055] The matrix operator U annotating the QIP gate 120 r Similarly, the pure state |Ψ r >, resulting in an evolved state. From a physical implementation perspective, the cardinality r typically represents the r lower discrete quantum levels to which the wave function is bounded for a given QIP system implementation.

[0056] Using these mathematical models, we can define an r × r unitary matrix U rThe time evolution of a single-cardinal r-qudit with respect to an operator described by is calculated as: [ka] Or, in terms of density matrix representation, calculated as: [ka]

[0057] Although pure state wave functions can be described using either column vectors or density matrices, the column vector formulation will be utilized herein without loss of generality.

[0058] Within these single qudit time evolutions, time t0 is r represents the state of the qudit before the operator represented by r For example, time t represents the state of the qudit on line 110, and time t represents the state of the qudit on line 130 after application of QIP gate 120. In general, the unitary transformation matrix U r =[u ij ] r×r is of dimension r×r and has components [ka] and [ka] represents a field of complex numbers.

[0059] The following pure state ket, [ka] and as a representative example for r=3 qudits represented as a general operator U3 defined as [ka] The resulting time evolution of |Ψ3〉 is calculated as follows: [ka]

[0060] FIG. 2 depicts a control qudit operation in the form of a QIP circuit 200. The QIP circuit 200 is depicted with horizontal lines 210 and 220, which represent separate and distinct qudits prior to their evolution due to any QIP gate. The progression of the horizontal lines 210 and 220 from left to right indicates the evolution of the qudits in time or space. The vertical symbol 230 indicates the presence of an operator. In FIG. 2, a single two-qudit operator is depicted by the vertical line and "U" r The presence of a circle 240 annotated with a value m that intersects the top qudit line 210 and a vertical line indicates that the top qudit 210 is the control qudit. The value m annotating the circle 240 denotes the activation basis value as |m〉. The presence of a circle 240 annotated with a value m that intersects the bottom qudit line 220 and a vertical line indicates that the top qudit 210 is the control qudit. The value m annotating the circle 240 denotes the activation basis value as |m〉. r " indicates that the lowest qudit 220 is the target qudit. Lines 260 and 270 represent the qudits after they have evolved due to QIP circuit gate 230. Line 260 represents the state of the control qudit, which remains unchanged from its form prior to the application of QIP circuit gate 230, and therefore has the same state as that of line 210. Line 270 represents the state of the target qudit (e.g., line 220) after it has evolved due to QIP circuit gate 230.

[0061] Symbol U annotating QIP gate 250 rrepresents the operator that describes the evolution of the target qubit 220 when the current state of the control qubit 210 satisfies the activation basis of |m〉 as specified by the annotation m in the circle 240. When the control qubit 210 does not satisfy the activation basis criterion as specified by the annotation m in the circle 240, the target qubit 220 remains unchanged and undergoes the activation of U r operator, and a single qudit operator is the r×r identity matrix I r It develops as if it were.

[0062] The qudits represented by lines 210, 220, 260, and 270 in Figure 2 may also be mathematically represented as evolving r-dimensional quantum state vectors or r x r density matrices. Lines 210 and 260 represent the same host particle that serves as a control qudit before and after the evolution due to QIP circuit gate 230. Similarly, lines 220 and 270 represent the same host particle that serves as a target qudit before and after the evolution due to QIP circuit gate 230. The progression of lines 210, 220, 260, and 270 from left to right indicates the evolution of the qudit in time or space, so that each point on lines 210, 220, 260, and 270 can be interpreted as representing a distinct instance of the time or space evolution of the corresponding quantum state vector or density matrix. Implemented by QIP gate 250, U r The single-qudit evolution operator, denoted as: is mathematically represented as an r×r unitary matrix that is conditionally applied to qudit 220 depending on the activation value of qudit 210, whose general form is: [ka]

[0063] A more general expression of the control qudit QIP operation, implemented by QIP circuit gate 230, is to define the overall two-qudit unitary matrix as C U r expressed as 2×r 2 The solution is to express it as a unitary matrix. C U The form of r is an r×r null matrix whose elements are all zero, and D rk can be expressed in the quadrant of submatrices, where k is the r × r single-qudit operator submatrix along the diagonal of the control qudit operator matrix. When k = m, the diagonal submatrix D rm is the single-qudit operator matrix U that evolves the target qudit quantum state into a new, distinct state. r Otherwise, when k ≠ m, each diagonal r × r submatrix is ​​an r × r density matrix I r In the quadrant form of the submatrix C U To express this, we use the following: [ka]

[0064] FIG. 3 depicts a general case of a control qubit operation in the form of a QIP circuit 300. The QIP circuit 300 is depicted with horizontal lines 310, 320, 360, and 370, representing separate and distinct qubits prior to and following the evolution due to the QIP circuit element 350. The progression of the horizontal lines 310 and 320 from left to right indicates the evolution of the qubits in time or space. The QIP circuit element 330 implements the QIP operation. In FIG. 3, a single two-qubit operator is depicted as a QIP gate 350, annotated with a vertical line and "U2." The presence of a black dot 340 intersecting the top qubit line 310 and the vertical line of the QIP circuit element 330 indicates that the top qubit 310 is the control qubit. As is conventional for QIP systems utilizing a binary basis, i.e., r=2, the activation basis value of the control qubit is assumed to be |1〉. The QIP gate 350 annotated with "U2," which intersects the least significant qubit shown as lines 320 and 370 and QIP gate 350 of QIP circuit element 330, indicates that the least significant qubit represented by lines 320 and 370 is the target qubit. Thus, when the target qubit 310 satisfies the activation value |1〉, the single-qubit operator U2 implemented by QIP gate 350 describes the evolution of the target qubit as line 370. When the control qubit 310 does not satisfy the activation basis value of |1〉, the target qubit (lines 320, 370) evolves with respect to the 2 × 2 identity operator I2, and its quantum state, represented by line 370, remains the same as that represented by line 320.

[0065] The operation of the control qubit QIP circuit 300 is the 4×4 operator C U can be mathematically described as an application of [ka]

[0066] FIG. 4 depicts a specific example of a control qubit operation in the form of a QIP circuit 400, known as a “controlled phase gate” or “controlled s gate.” QIP circuit 400 is depicted with horizontal lines 410, 420, 460, and 470, which represent two separate and distinct host particles that serve as qubits. Again, the progression of horizontal lines 410, 420, 460, and 470 from left to right indicates the evolution of the qubits in time or space. Lines 410 and 460 represent the same host particle that serves as the control qubit before and after the evolution due to QIP circuit element 430. Similarly, lines 420 and 470 represent the same host particle that serves as the target qubit before and after the evolution due to QIP circuit element 430. QIP circuit element 430 is referred to as a “phase gate.” In FIG. 4, a single two-qubit operator is depicted as a connected gate 450, annotated with a vertical line and an “S.” The presence of a black dot 440 intersecting the top-order qubit line 410 and the vertical line of the operator element 430 indicates that the top-order qubit 410 is the control qubit. As is conventional for QIP systems utilizing a binary basis, i.e., r=2, the activation basis value of the control qubit is assumed to be |1〉. A QIP gate 450 annotated with an "S" that intersects the bottom-order qubit, represented by lines 420 and 470, indicates that the bottom-order qubit is the target qubit.

[0067] Thus, when control qubit 410 satisfies activation value |1〉, the single-qubit operator implemented by gate 450 and annotated with S describes the evolution of target qubit 420 according to a phase gate transformation matrix S that results in a different state represented by line 470. When control qubit 410 does not satisfy the activation basis value of |1〉, the target qubit evolves with respect to the 2×2 identity operator I2, and its quantum state as represented by line 470 remains unchanged from its original state represented by line 420.

[0068] The operation of the control qubit QIP circuit 400 is C S The matrix can be mathematically described as the application of a 4×4 matrix operator: [ka]

[0069] Two-qubit transformation matrix C S The value i that appears in i 2 = -1. The action of the single-qubit phase operator S induces a relative phase shift of 90 degrees between the calculated basis components of the target qubit 420 when the control qubit 410 satisfies the activation basis criteria.

[0070] FIG. 5a depicts a specific case of a single radix-4 qudit operation in the form of a QIP circuit 500, known as a "radix-4 Chrestenson gate." QIP circuit 500 is depicted with horizontal lines 510 and 530 representing a single radix-4 qudit prior to and after the evolution due to the QIP circuit operator implemented by gate 520. The progression of horizontal lines 510 and 530 from left to right indicates the evolution of the radix-4 qubit in time or space. Gate 520 represents a single qudit operator that evolves the qudit in a specific manner according to a transformation matrix, represented by C4, as annotated inside gate 520. The radix-4 Chrestenson operator C4 is mathematically described as follows: [ka]

[0071] In general, the Chrestenson transformation matrix can be formulated for any base r where r > 2. For base r where r > 1, the Chrestenson transformation matrix C r The general form for is an r×r unitary matrix, with each matrix element expressed as: (ω p ) kis one of r distinct r-th roots of unity raised to integer powers in the range [0,r-1], denoted as [ka]

[0072] Radix r Chrestenson operator C r The explicit matrix form of is mathematically described as: [ka]

[0073] The C4 single-qudit operator implemented by gate 520 is a general radix-r Chrestenson gate C r where the parameters p and k in the formula for the r distinct roots of unity raised to an integer power are each restricted to have values ​​from the set {0, 1, 2, 3}. When the Chrestenson transform matrix is ​​formulated for the binary case, i.e., r=2, an operator well known to those skilled in the art, called the Hadamard transform matrix H, is obtained, given as follows: [ka]

[0074] The single-qudit Chrestenson operator is characterized by a transformation matrix that is derived as a discrete Fourier transform over abelian groups. The general theory of discrete Fourier transforms over abelian groups is explained with reference to [Ch:55][Vi:47]. Many useful applications of the Chrestenson transform in QIP systems are demonstrated in [ZK:02][ST:19].

[0075] Examples of physical implementations of the Radix-4 Chrestenson transformation are described in [SL+:18a][SL+:18b]. These physical implementations utilize photons as host particles, with their spatial location between designated groups of waveguides or in free space serving as information-carrying observables. Other implementations, such as superconducting qubits, are also possible and contemplated herein.

[0076] A C4 implementation in a quantum photonic integrated circuit (QPIC) utilizes four waveguides as the transmission medium for a single photon. Each of the four waveguides represents one of four computational basis vectors {|0〉, |1〉, |2〉, |3〉} and is referred to as a "basis waveguide." Relating this physical implementation to FIG. 5a would then equate horizontal lines 510 and 530 with the photon wave function at a time or space location between groups of four basis waveguides. Similarly, when a host particle, such as a photon, in such a system is in a state of equal superposition, one of its possible corresponding fully superposed wave functions is expressed as: [ka]

[0077] Projection measurements of the photon's location observables while in full superposition show that the probability of measuring the presence of a photon in any one of the four base waveguides, represented as lines 510 and 530 in QIP circuit 500, is equal to the square of the probability amplitude. Each probability amplitude is [ka] Therefore, the probability of measuring the presence of a photon in any one of the four base waveguides is 1 / 4.

[0078] A QIP circuit element characterized by a C4 transformation matrix can be employed to evolve a qudit initially in a basis state into a state of perfect superposition, where the squared magnitude of each component in the qudit state vector has the same value. As an example, consider qudit 510 is initialized in QIP circuit 500a as |Ψ4〉 = |0〉. When qudit 510 undergoes quantum state evolution due to C4 gate 520, it evolves into a state of perfect superposition, represented by line 520 as follows: [ka]

[0079] One example of a QIP circuit element with a characteristic transformation matrix C4 is described in [SL+:18a]. The device was originally conceived as a four-port coupler for application in classically operating optical circuits [MT+:04]. The original application for the device was to serve as a supporting element for optical photon filters in photonic integrated circuits (PICs), and it was later determined to function as a radix-4 Chrestenson gate when operating in the quantum photonic regime. Examples have been fabricated to operate across free-space channels as well as waveguide channels in integrated circuits.

[0080] The geometry of one example of such a QIP circuit element is depicted in FIG. 5b as element 550 depicting bidirectional ports 560, 570, 580, and 590 on each side or face of element 550, annotated as "North (N)" 560, "East (E)" 570, "South (S)" 580, and "West (W)" 590, respectively. When this exemplary element 550 is utilized as element 520 in a QPIC, waveguides are coupled to bidirectional ports 560, 570, 580, and 590. For example, this group of four waveguides may serve as lines 510 and 530 as depicted in FIG. 5a, where photons incident on 520, whose state is represented by line 510, correspond to waveguides to bidirectional ports 560, 570, 580, and 590. Similarly, a photon emerging from element 550 after undergoing a C4 operation propagates away from element 550 along the same group of four waveguides 560, 570, 580, and 590 to bidirectional ports 560, 570, 580, and 590 (e.g., as represented in QIP circuit 500 using line 530). In other words, a single photon may be on a single waveguide to a single port 560, 570, 580, and 590, or may be superimposed on all waveguides to all ports 560, 570, 580, and 590. Reflections of the photon can then be emitted from element 550 on those waveguides from ports 560, 570, 580, and 590. Thus, a radix-4 Chrestenson gate C4 (which may be implemented using QIP circuit element 550, for example) can be utilized to implement a single radix-4 qudit operation.

[0081] As discussed above, in many cases, QIPs may require the use of control operations, which in turn may rely on entanglement or interactions of qudits. However, in many cases, these control operations may actually function to entangle such qudits (or otherwise rely on the interaction of such qudits) only a small percentage of the time (e.g., 25% or equivalent). Such failures may result, for example, at least in part, from the information carriers used for such qudits. For example, when photons are utilized as information carriers, it may be extremely difficult to get photons to interact. Thus, in operations where such entanglement fails (e.g., 75% of the time), the entire quantum operation may fail. QIP circuits that exhibit such high failure rates are described in Knill et al., "A Scheme for Efficient Quantum Computation with Linear Optics," Nature Vol. 409, January 4, 2001, and Okamoto et al., "Realization of a Knill-Laflamme-Milburn C-NOT gate - a photonic quantum circuit combining effective optical nonlinearities," [arXiv:1006.4743v1], June 24, 2010, both of which are incorporated herein by reference in their entireties. Such high failure rates (both of entanglement operations and the resulting quantum circuit failures in general) act as a major obstacle to efficient QIP. Therefore, there is significant motivation to realize such control qudit operations with higher success rates, including entanglement of qudits and control qudit operations that rely on such entanglement operations.

[0082] To address these desires, among other objectives, embodiments as disclosed herein may provide radix-2 quantum operations utilizing a trans-radix quantum circuit comprising one or more radix-r circuit elements, where r > 2. Embodiments of such quantum circuits may also include one or more radix-2 circuit elements, where input or output ports of the one or more radix-r quantum circuit elements are utilized to support the input and output of a radix-2 qubit transmission channel.

[0083] Specifically, embodiments may utilize a radix-4 Chrestenson gate C4 in a transradix manner. For example, embodiments may utilize two qudits (e.g., photons) with the Chrestenson gate C4. Each qudit (e.g., photon) may be present at only two associated ports of the Chrestenson gate C4. In particular, a first qudit may be provided in the associated waveguide of the first or second port of the Chrestenson gate C4 such that the first qudit is in a ground state on the waveguide of the first port or on the waveguide of the second port, or such that the first qudit is superimposed on both the waveguide of the first port and the waveguide of the second port. Similarly, a second qudit may be provided at the associated third and fourth ports of Chrestenson gate C4 such that the second qudit is in its ground state on the waveguide of the third port or on the waveguide of the fourth port, or such that the second qudit is superimposed on both the waveguide of the first port and the waveguide of the second port. In many embodiments, it may be desirable for this presentation of the first and second qudits at their respective ports to be time-aligned.

[0084] Thus, based on the interaction of the first and second qudits in the radix-4 Chrestenson gate C4, the first qudit may appear in the ground state on the waveguide of the first port or the waveguide of the second port, or may be superimposed on both waveguides from both the first and second ports of the Chrestenson gate C4, while the second qudit may appear in the ground state on the waveguide from the third port or the waveguide of the fourth port, or may be superimposed on both waveguides from both the third and fourth ports of the Chrestenson gate C4. Such trans-radix Chrestenson gate C4 embodiments may have up to a 100% success rate in achieving interaction (e.g., entanglement) of qudits utilized with such trans-radix Chrestenson gates.

[0085] Referring now to FIG. 6, one embodiment of a radix-4 Chrestenson gate C4 adapted to operate in a transradix manner is depicted in QIP circuit 600. Horizontal lines 610, 620, 640, and 650 represent radix-2 qubits, while C4 gate 630 is a radix-4 single-qudit operator (e.g., annotated accordingly as “C4”). Certain physical implementations of the C4 operator in a QIP system allow the input to the C4 operator to be physically separated into its computational basis components, such as the quantum photonic implementation described with respect to FIG. 5b, where the position observable serves as the information-carrying attribute. In a similar manner, a photonic qubit employed with a position observable would utilize two waveguides representing the basis states {|0〉,|1〉} in an implementation known as a “dual-rail” configuration within a QIP system. A transradix configuration can be achieved when four waveguides supporting a radix-4 qudit and representing the basis states {|0〉,|1〉,|2〉,|3〉} are partitioned into two sets of two waveguides, with the first two waveguides serving as the medium for the first qubit and the second set of waveguides serving as the medium for the second qubit.

[0086] Although many physical implementations may be utilized for the Chrestenson gate C4630 in the trans-radix operation of such a gate, in one embodiment, a C4 circuit operating in trans-radix mode may utilize a physical implementation of element 550 as depicted in FIG. 5b as a Chrestenson gate C4630. In such an embodiment, waveguides coupled to ports annotated as "North" and "East" illustrated in FIG. 5b as 560 and 570 serve as two basis waveguides for the first qubit, represented as most significant qubits 610 and 640 on the Chrestenson gate C4630 (jointly referred to as "NE"). Similarly, the least significant qubits 620 and 650 are transmitted in two waveguides coupled to element 550 through waveguides 580 and 590 at locations "South" and "West" on the Chrestenson gate C4630 (jointly referred to as "SW").

[0087] In other words, according to one embodiment, one qubit 610 may be provided in the associated waveguide of the first port (e.g., 560, North) or the second port (e.g., 570, East) of Chrestenson gate C4 (e.g., 550) such that the first qudit is in the ground state on the waveguide of the first port (e.g., 560, North) or on the waveguide of the second port (e.g., 570, East), or such that the first qudit is superimposed on both the waveguide of the first port and the waveguide of the second port (e.g., both 560, North and 570, East). Similarly, a second qudit may be provided at the associated third port (e.g., 580, south) and fourth port (e.g., 590, west) of Chrestenson gate C4 (e.g., 550) such that the second qudit is in the ground state on the waveguide of the third port (e.g., 580, south) or on the waveguide of the fourth port (e.g., 590, west), or such that the second qudit is superimposed on both the waveguide of the first port and the waveguide of the second port (e.g., both 580, south and 590, west). Again, in many embodiments, it may be desirable for this presentation of the first and second qudits at their respective ports to be time-aligned.

[0088] When the C4 gate 630 is used as a trans-radix operator with two qubits (e.g., instead of a single qudit), it functions as a two-qubit operator. Each pair of waveguides (e.g., the first pair is north and east (NE) and the second pair is south and west (SW)) that serves as a medium for the two qubits represents a binary number {|0〉,|1〉}. The overall quantum state of the trans-radix QIP circuit 600 before the C4 operator 630 is applied to the pair of qubits 610 and 620 is calculated as the tensor product of the qubits. As an example, consider the following: a top-order qubit 610 with an initial state denoted as |Ψ〉 and a bottom-order qubit 620 denoted as |Φ〉: [ka]

[0089] The overall state is then expressed as the tensor product [ka] which is calculated as: [ka]

[0090] Note that the basis vector values ​​utilize the subscript "2" to indicate that they are base 2 or binary values, and not decimal values.

[0091] Once the composite state of qubits 610 and 620 is described in this general form, a C4 operator 630 can be applied to yield the evolved state of the pair of qubits 640 and 650 as: [ka]

[0092] When two qubits 610 and 620 are initialized in all possible combinations of calculated basis states |Ψ2Φ2〉={|00〉,|01〉,|10〉,|11〉}, four different evolutions occur in qubits 640 and 650 when C4 operator 630 is applied as follows: [ka]

[0093] These calculations show that when two base-2 qubits 610 and 620 are initialized to one of their respective composite basis states, the C4 operator 630 operates to generate a perfect base-4 superposition of composite states formed with the pair of base-2 qubits |ΨΦ〉 640 and 650. In general, trans-radix operations in higher-radix QIP circuits enable the use of lower-radix information-carrying qudits whose composite states represent the higher-radix overall quantum state.

[0094] As can be appreciated, such trans-radix Chrestenson gate C4 embodiments (e.g., embodiments utilizing QIP circuit element 550 in a trans-radix configuration) may have up to a 100% success rate in achieving interaction (e.g., entanglement) of two qudits utilized with such trans-radix Chrestenson gate. Thus, because such trans-radix Chrestenson gate C4 may be utilized as a type of controlled quantum circuit based on at least two qudits, such trans-radix Chrestenson gate C4 may be usefully employed in other (e.g., standard) controlled quantum circuit embodiments. For example, trans-radix Chrestenson gate C4 embodiments may be used to implement reliable controlled-S gate quantum circuit embodiments, reliable controlled-Z quantum circuit embodiments, reliable controlled-X quantum circuit embodiments, or more generally, reliable embodiments of quantum circuits in which entanglement or interaction of qubits is desired.

[0095] Also, as will be understood, the controlled S gate (e.g., or controlled X gate) may be a subset of all possible quantum operations in the binary domain. Thus, almost any desired quantum circuit may be constructed in a reliable manner using embodiments of the trans-radix Chrestenson gate C4, embodiments of the controlled S gate, embodiments of the controlled S gate, or embodiments of the controlled X gate as disclosed herein. Thus, embodiments herein may serve as a foundational change to the reliability of quantum circuits and quantum computing more generally.

[0096]

[0047] Discussing a first embodiment of a controlled S gate utilizing a trans-radix Chrestenson gate C4, it will be recalled from the above discussion of Figure 4 that a controlled S gate (e.g., a single-qubit phase operator S) induces a 90 degree relative phase shift between the calculated basis components of a target qubit when the control qubit satisfies the activation basis criteria. Such a controlled S gate may be reliably implemented using a trans-radix Chrestenson gate C4.

[0097] Turning now to FIG. 7a, one embodiment of a trans-radix QIP circuit 700 for implementing a controlled S-gate is depicted. Quantum circuit 700 operates on two radix-2 qubits at the input, represented as lines 710a and 720a, and |Ψ〉 and |Φ〉, respectively. In addition, QIP circuit 700 comprises an exchange or swap operation (e.g., element) 730, depicted as the exchange of lines 710a and 720a. Swap operation 730 may not be required in a physical implementation because it symbolically represents a renaming operation; however, it is included in QIP circuit diagram 700 to preserve the mathematical modeling convention of utilizing the most significant qubit as the leftmost operand when a non-commutative tensor product is computed during mathematical analysis of the QIP circuit.

[0098] The QIP circuit 700 also includes a radix-2 Hadamard gate 740a annotated as H, a transformer radix Chrestenson gate C4 750a annotated as C4 (e.g., a Chrestenson gate C4 operating as a transformer radix Chrestenson gate as discussed), and another radix-2 Hadamard gate 760a annotated as H.

[0099] Here, T SWAP After passing through 730, the radix-2 qubit |Ψ〉 on line 710a may be provided to two ports of a trans-radix Chrestenson gate C4750a (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C4750). The radix-2 qubit |Φ〉 on line 720a is provided to T SWAP After passing through 730, it is provided to Hadamard gate 740a, and from Hadamard gate 740a to two other ports of transformer radix Chrestenson gate C4750a (e.g., the north (N) port and the east (E) port of transformer radix Chrestenson gate C4750a). The waveguides from these ports (e.g., the north (N) port and the east (E) port of transformer radix Chrestenson gate C4750a) are provided to Hadamard gate 760a.

[0100] The overall transfer characteristic of the QIP circuit 700 is T CS and can be computed as a Cartesian product of operators in circuit 700 as follows: [ka]

[0101] In the formula, each term is {H A ,H B ,C4} gate, and T SWAPdenotes a swap operation 730, which may actually be physically implemented simply as a qubit renaming operation. As mentioned above, the Hadamard operator is a binary basis case of the radix-r Chrestenson operation, with r=2. T SWAP The transfer matrix is: [ka]

[0102] Therefore, the overall transformation matrix T of the QIP circuit 700 CS can be calculated explicitly as: [ka]

[0103] Note that the overall transfer matrix of the transformer radix QIP circuit 700 is the same as the overall transfer matrix describing the control S QIP circuit 400. CS =C S , it is clear that the transformer radix QIP circuit 700 is equivalent to the QIP circuit 400 and realizes a two-qubit radix-2 controlled S-operator.

[0104] 7b depicts another embodiment of a trans-radix QIP circuit 702 for implementing a controlled S gate. The quantum circuit 702 operates on two radix-2 qubits at the input, represented as lines 710b and 720b, and represented as |Ψ〉 and |Φ〉, respectively. The QIP circuit 702 also comprises a radix-2 Hadamard gate 740b annotated as H, a trans-radix Chrestenson gate C4 750b annotated as C4 (e.g., a Chrestenson gate C4 operating as a trans-radix Chrestenson gate as discussed), and another radix-2 Hadamard gate 760b annotated as H.

[0105] Here, the radix-2 qubit |Ψ〉 on line 710b may be provided to two ports of the trans-radix Chrestenson gate C4750b (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C4750b). The radix-2 qubit |Φ〉 on line 720b is provided to Hadamard gate 740b and from there to two other ports of the trans-radix Chrestenson gate C4750b (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C4750b). Waveguides from these ports (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C4750b) are provided to Hadamard gate 760b.

[0106] Therefore, the overall transformation matrix T of the QIP circuit 702 CS can be calculated explicitly as: [ka]

[0107] Again, note that the overall transfer matrix of the transformer radix QIP circuit 702 is the same as the overall transfer matrix describing the control S QIP circuit 400. CS =C S , it is clear that the transformer radix QIP circuit 702 is equivalent to the QIP circuit 400 and realizes a two-qubit radix-2 controlled S-operator.

[0108] As is known in the art, several other controlled qubit operators may be constructed utilizing the controlled S operator. For example, Figure 8a depicts a high-level diagram of a prior art QIP circuit 800a consisting of two base-2 qubits at the input, represented as lines 810a and 820a, and denoted as |Ψ〉 and |Φ〉, respectively. In addition, QIP circuit 800a consists of two controlled S gates, labeled 830a and 840a. The overall transfer matrix for QIP circuit 800a is the characteristic transformation matrix C ZThe transformation matrix C characterizing the Radix-2 QIP Controlled-Z circuit 800b is equivalent to that of the single Radix-2 Controlled-Z QIP circuit depicted as circuit 800b in FIG. Z is as follows: [ka]

[0109] The overall transformation matrix for QIP circuit 800a is shown to be equivalent to the following radix-2 control Z-transform matrix: [ka]

[0110] The overall transformation matrix for QIP circuit 800a is C Z Since it is equivalent to QIP circuit 800b, it is clear that radix-2 QIP circuit 800a is equivalent to QIP circuit 800b and realizes a two-qubit radix-2 controlled Z operator.

[0111] In other words, a controlled-Z operator may be obtained by chaining two controlled-S operators. Thus, embodiments herein may achieve a reliable controlled-Z quantum circuit using an embodiment of a reliable controlled-S circuit. Accordingly, reference is made to FIG. 9a, which depicts an embodiment of a trans-radix quantum circuit 900 for implementing a controlled-Z operation. The trans-radix QIP circuit 900 is adapted to operate on two radix-2 qubits at the input, represented as lines 910a and 920a, and represented as |Ψ〉 and |Φ〉, respectively. Additionally, the QIP circuit 900 consists of four single radix-2 Hadamard gates H, as shown by boxes 930a, 950a, 960a, and 980a. The QIP circuit 900 also includes two radix-4 Chrestenson gates C4 operating in trans-radix mode, as shown by boxes 940a and 970a.

[0112] Here, T SWAPAfter passing through 912, the radix-2 qubit |Ψ〉 on line 910a may be provided to two ports of a trans-radix Chrestenson gate C4940a (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C4940a). The radix-2 qubit |Φ〉 on line 920a is provided to T SWAP After passing through 912, it is provided to Hadamard gate 930a, and from Hadamard gate 930a to two other ports of transformer radix Chrestenson gate C4940a (e.g., the north (N) port and the east (E) port of transformer radix Chrestenson gate C4940a). The waveguides from these ports (e.g., the north (N) port and the east (E) port of transformer radix Chrestenson gate C4940a) are provided to Hadamard gate 950a.

[0113] T SWAP After passing through 914, the radix-2 qubit |Ψ〉 on line 910a may be provided to Hadamard gate 960a and from Hadamard gate 960a to two ports (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C4970a) of a trans-radix Chrestenson gate C4970a. Waveguides from these ports (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C4970a) are provided to Hadamard gate 980a. Similarly, the radix-2 qubit |Φ〉 on line 920a may be provided to Hadamard gate 980a via T SWAP After passing through 914, it may be provided to two other ports of the transformer radix Chrestenson gate C4970a (eg, the south (S) port and the west (W) port of the transformer radix Chrestenson gate C4970a).

[0114] The overall evolution of the qubits on lines 910a and 920a due to their propagation through trans-radix QIP circuit 900 is given by the overall transformation matrix T CZ The characteristic matrix C, as verified by calculating ZThe embodiment of the QIP circuit 900 comprises two instances of the embodiment of the QIP circuit 700 in cascade, and thus T CZ =T CS T CS Note that, and is calculated as: [ka]

[0115] T CZ =C Z , it should be apparent that the embodiment of transformer radix QIP circuit 900 is equivalent to QIP circuit 800b and implements a radix-2 controlled-Z operator.

[0116] Figure 9b depicts another embodiment of a transformer radix QIP circuit 902 for implementing a controlled z-gate. The transformer radix QIP circuit 902 is adapted to operate on two radix-2 qubits at the input, represented as lines 910b and 920b, and represented as |Ψ〉 and |Φ〉, respectively. In addition, the QIP circuit 902 consists of four single radix-2 Hadamard gates H, as shown by boxes 930b, 950b, 960b, and 980b. The QIP circuit 902 also includes two radix-4 Chrestenson gates C4 operating in transformer radix mode, as shown by boxes 940b and 970b.

[0117] Here, the radix-2 qubit |Ψ〉 on line 910b may be provided to two ports of a trans-radix Chrestenson gate C4940b (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C4940b). Waveguides from these ports (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C4940b) are provided to a Hadamard gate 960b. The radix-2 qubit |Ψ〉 on line 910b proceeds from the Hadamard gate 960b to two ports of a trans-radix Chrestenson gate C4970b (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C4970b). Waveguides from these ports (eg, the north (N) and east (E) ports of transformer-radix Chrestenson gate C4970b) are provided to Hadamard gate 980b.

[0118] The radix-2 qubit |Φ〉 on line 920b is provided to Hadamard gate 930b, and from Hadamard gate 930b to two other ports of trans-radix Chrestenson gate C4940b (e.g., the north (N) port and east (E) port of trans-radix Chrestenson gate C4940a). Waveguides from these ports (e.g., the north (N) port and east (E) port of trans-radix Chrestenson gate C4940a) are provided to Hadamard gate 950b. The radix-2 qubit |Φ〉 on line 920b proceeds from Hadamard gate 950b to two ports of trans-radix Chrestenson gate C4970b (e.g., the south (S) port and west (W) port of trans-radix Chrestenson gate C4970b).

[0119] The overall evolution of the qubits on lines 910b and 920b due to their propagation through transradix QIP circuit 902 is given by the overall transformation matrix T CZ The characteristic matrix C, as verified by calculating ZThe embodiment of the QIP circuit 902 comprises two instances of the embodiment of the QIP circuit 702 in cascade, and thus T CZ =T CS T CS Note that, and is calculated as: [ka]

[0120] T CZ =C Z , it should be apparent that the embodiment of transformer radix QIP circuit 902 is equivalent to QIP circuit 800b and implements a radix-2 controlled-Z operator.

[0121] Again, other controlled qubit operators may be constructed using controlled S or controlled Z operators, as is known in the art. For example, Figure 10a depicts a prior art QIP circuit 1000a consisting of two radix-2 qubits at the input, represented as lines 1010a and 1020a, and |Ψ〉 and |Φ〉, respectively. In addition, QIP circuit 1000a consists of a single controlled Z gate, labeled 1030a, and two single-qubit radix-2 Hadamard gates 1040a and 1050a. The overall transfer matrix for QIP circuit 1000a is the characteristic transformation matrix C X The transformation matrix C characterizing the radix-2 QIP controlled X circuit 1000b is equivalent to that of a single radix-2 controlled X QIP circuit depicted as circuit 1000b with X is as follows: [ka]

[0122] The overall transformation matrix for QIP circuit 1000a is shown to be equivalent to the following radix-2 control X transformation matrix: [ka]

[0123] T CX =C X Therefore, it is clear that radix-2 QIP circuit 1000a is equivalent to QIP circuit 1000b and realizes a two-qubit radix-2 controlled X operator.

[0124] In other words, a Controlled-X operator may be obtained utilizing a Controlled-Z operator. Additionally, as discussed above, a Controlled-X operator may be obtained using chained Controlled-S operators. Thus, embodiments herein may achieve a reliable Controlled-X quantum circuit using embodiments of a reliable Controlled-S or a reliable Controlled-Z circuit as discussed.

[0125] Turning now to Figure 11a, one embodiment of trans-radix QIP circuit 1100a consists of two radix-2 qubits at the input, represented as lines 1110a and 1120a, and represented as |Ψ〉 and |Φ〉, respectively. In addition, QIP circuit 1100a consists of four single radix-2 Hadamard gates H, as shown by boxes 1150a, 1170a, 1192a, and 1194a. QIP circuit 1100a also includes two radix-4 Chrestenson gates C4 operating in trans-radix mode, as shown by boxes 1140a and 1180a. QIP circuit 1100a also includes two swap operations T SWAP 1140a and 1160a. The swap operations 1140a and 1160a simply apply the overall QIP circuit transformation matrix T CX It exists for mathematical convenience when analyzing and does not represent an actual physical implementation.

[0126] Here, T SWAPAfter passing through 1130, the radix-2 qubit |Ψ〉 on line 1110a may be provided to two ports of a trans-radix Chrestenson gate C41140a (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C41140a). The waveguides from these ports (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C41140a) are connected to the T SWAP 1160 to Hadamard gate 1170a, and from Hadamard gate 1170a to two ports (e.g., the north (N) port and the east (E) port of trans-radix Chrestenson gate C41180a) of trans-radix Chrestenson gate C41180a. From these two ports (e.g., the north (N) port and the east (E) port of trans-radix Chrestenson gate C41180a), the radix-2 qubit |Ψ〉 on line 1110a may be provided to Hadamard gate 1192a.

[0127] The base-2 qubit |Φ2〉 on line 1120a is T SWAP After passing through 1130, the waveguides are provided to two other ports of the transformer radix Chrestenson gate C4 1140a (e.g., the north (N) port and the east (E) port of the transformer radix Chrestenson gate C4 1140a). The waveguides from these ports (e.g., the north (N) port and the east (E) port of the transformer radix Chrestenson gate C4 1140a) are provided to a Hadamard gate 1150a, from which the T SWAP 1160 to two ports (e.g., the south (S) port and the west (W) port of the transformer radix Chrestenson gate C41180a) of the transformer radix Chrestenson gate C41180a. From these two ports (e.g., the south (S) port and the west (W) port of the transformer radix Chrestenson gate C41180a), the radix-2 qubit |Φ〉 on line 1120a may be provided to Hadamard gate 1194a.

[0128] The overall evolution of qubits 1110a and 1120a due to their propagation through trans-radix QIP circuit 1100a is given by the overall transformation matrix T CX The characteristic matrix C, as verified by calculating X The overall transformation matrix for the transformer radix QIP circuit 1100a is: [ka]

[0129] T CX =C X Therefore, it is clear that the transformer radix QIP circuit 1100a is equivalent to the QIP circuit 1000b and implements the radix-2 controlled X operator.

[0130] Figure 11b depicts an embodiment of a trans-radix QIP circuit 1102 consisting of two radix-2 qubits at the input, represented as lines 1110b and 1120b, and |Ψ〉 and |Φ〉, respectively. In addition, QIP circuit 1100b consists of four single radix-2 Hadamard gates H, as shown by boxes 1150b, 1170b, 1192b, and 1194b. QIP circuit 1100b also includes two radix-4 Chrestenson gates C4 operating in trans-radix mode, as shown by boxes 1140b and 1180b. The transformer radix Chrestenson gate 1140b is shown with ports designated as "SW" at the top portion of the device (e.g., the south (S) and west (W) ports of the transformer radix Chrestenson gate C41140b) and "NE" at the bottom (e.g., the north (N) and east (E) ports of the transformer radix Chrestenson gate C41140b).

[0131] In this embodiment, the radix-2 qubit |Ψ〉 on line 1110b may be provided to two ports of a trans-radix Chrestenson gate C41140b (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C41140b). Waveguides from these ports (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C41140b) are provided to a Hadamard gate 1170b, and from the Hadamard gate 1170b, they are provided to two ports of a trans-radix Chrestenson gate C41180b (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C41180b). From these two ports (eg, the north (N) and east (E) ports of the trans-radix Chrestenson gate C41180b), the radix-2 qubit |Ψ2〉 on line 1110b may be provided to Hadamard gate 1192b.

[0132] The radix-2 qubit |Φ〉 on line 1120b is provided to two other ports of the trans-radix Chrestenson gate C41140b (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C41140b). Waveguides from these ports (e.g., the north (N) port and the east (E) port of the trans-radix Chrestenson gate C41140b) are provided to the Hadamard gate 1150b, and from the Hadamard gate 1150b, to two ports of the trans-radix Chrestenson gate C41180b (e.g., the south (S) port and the west (W) port of the trans-radix Chrestenson gate C41180b). From these two ports (eg, the south (S) and west (W) ports of the trans-radix Chrestenson gate C41180b), the radix-2 qubit |Φ2〉 on line 1120b may be provided to Hadamard gate 1194b.

[0133] The arrangement of circuit 1102 allows the swap operation of QIP circuit 1100 to be eliminated, thus making QIP circuit 1102 structurally similar to QIP circuit 1100. The swap shown in QIP circuit 1100 does not affect the physical realization and is included only to remain consistent with the convention of mathematical analysis that dictates that the most significant qubit in a QIP circuit diagram is the leftmost operator when computing a tensor product. However, this mathematical convention does not affect the physical implementation, and thus trans-radix QIP circuit 1102 also implements a radix-2 controlled-X operation, also known as an uncontrolled gate.

[0134] The embodiments as described herein can be understood by reference to the following descriptions, with the understanding that these descriptions describe particular embodiments, and that any language used therein (e.g., "must," "should," "requires," etc.) is utilized with respect to those embodiments only and should not be taken in any way as imposing any restrictions, requirements, or limitations on other embodiments.

[0135] The invention has now generally been described with reference to specific embodiments. However, those skilled in the art will recognize that various modifications and changes can be made without departing from the scope of the invention. Accordingly, the description herein is to be regarded in an illustrative rather than a restrictive sense, and all such modifications are intended to be included within the scope of the invention.

[0136] Although the present invention has been described with reference to specific embodiments thereof, these embodiments are merely illustrative and not limiting of the present invention. The description herein of illustrative embodiments of the present invention is not intended to be exhaustive or to limit the present invention to the precise form disclosed herein (in particular, the inclusion of any particular embodiment, feature, or function is not intended to limit the scope of the present invention to such embodiment, feature, or function). Rather, the description is intended to describe illustrative embodiments, features, and functions to provide those skilled in the art with a context for understanding the present invention, without limiting the present invention to any particularly described embodiment, feature, or function. Specific embodiments of the present invention and examples thereof are described herein for illustrative purposes only; however, as those skilled in the art will recognize and appreciate, various equivalent modifications are possible within the spirit and scope of the present invention. As indicated, these modifications can be made herein in light of the foregoing description of illustrative embodiments of the present invention and are intended to be included within the spirit and scope of the present invention.

[0137] References throughout this specification to “one embodiment,” “an embodiment,” or “a specific embodiment,” “a specific implementation,” or similar terminology mean that a particular feature, structure, or characteristic described in connection with an embodiment is included in at least one embodiment and may not necessarily be present in all embodiments. Thus, individual appearances of the phrases “in one embodiment,” “in an embodiment,” or “in a specific embodiment,” or similar terminology in various places throughout this specification do not necessarily refer to the same embodiment. Furthermore, the particular features, structures, or characteristics of any particular embodiment may be combined in any suitable manner with one or more other embodiments. It should be understood that other variations and modifications of the embodiments described and illustrated herein are possible in light of the teachings herein and are considered to be part of the spirit and scope of the invention.

[0138] In the description, numerous specific details, such as examples of components or methods, are provided to provide a thorough understanding of embodiments of the present invention. However, those skilled in the art will recognize that an embodiment may be practiced without one or more of the specific details, or in conjunction with other devices, systems, assemblies, methods, components, materials, parts, or equivalents. In other instances, well-known structures, components, systems, materials, or operations are not shown or described in specific detail to avoid obscuring aspects of embodiments of the present invention. While the present invention may be illustrated using specific embodiments, this is not, and does not, limit the present invention to any particular embodiment, and those skilled in the art will recognize that additional embodiments are readily apparent and are part of the present invention.

[0139] Additionally, as used herein, the term "or" is generally intended to mean "and / or" unless otherwise indicated. As used herein, terms preceded by "a" or "an" (and "the" when the antecedent is "a" or "an") include both the singular and plural of such term (i.e., the reference to "a" or "an" clearly indicates only the singular or only the plural). Also, as used in the description herein, the meaning of "in" includes "in" and "on," unless the context clearly dictates otherwise. References

[0140] The following references may be useful towards an understanding of the embodiments depicted herein, and are fully incorporated herein by reference in their entirety for all purposes. [Table 1]

Claims

1. A transradix quantum circuit, comprising: Quantum circuit element with four waveguides and implementing the Chrestenson transformation matrix Equipped with the quantum circuit element is adapted to operate as a transformer radical-2 qubit quantum circuit element, the four waveguides comprising two sets of two waveguides, the transformer radical-2 qubit quantum circuit element operating on two qubits, the two qubits comprising a first radical-2 qubit and a second radical-2 qubit, a first of the two sets acting as a medium for the first radical-2 qubit and a second of the two sets acting as a medium for the second radical-2 qubit. Transradical quantum circuits.

2. 10. The trans-radix quantum circuit of claim 1, further comprising one or more radix-2 quantum circuit elements coupled to the quantum circuit element, wherein the trans-radix quantum circuit is adapted to operate as a control quantum circuit.

3. 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled S quantum circuit element.

4. 4. The trans-radix quantum circuit of claim 3, wherein the radix-2 controlled S quantum circuit element is applied to the first radix-2 qubit before the first radix-2 qubit is provided to the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element.

5. 5. The trans-radix quantum circuit of claim 4, wherein the radix-2 controlled S quantum circuit element is applied to the first radix-2 qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the first radix-2 qubit.

6. 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled-Z quantum circuit element.

7. 7. The trans-radix quantum circuit of claim 6, wherein the radix-2 controlled-Z quantum circuit element is applied to the first radix-2 qubit before the first radix-2 qubit is provided to the quantum circuit element implementing the Chrestenson transformation matrix that is adapted to operate as the trans-radix-2 qubit quantum circuit element for the first time.

8. 8. The trans-radix quantum circuit of claim 7, wherein the base-2 controlled-Z quantum circuit element is applied to the first base-2 qubit after the quantum circuit element adapted to operate as the transformer base-2 qubit quantum circuit element is applied to the first base-2 qubit for the first time and before the quantum circuit element adapted to operate as the transformer base-2 qubit quantum circuit element is applied to the first base-2 qubit for a second time.

9. 9. The trans-radix quantum circuit of claim 8, wherein the radio-2 controlled-Z quantum circuit element is applied to the second radio-2 qubit before the second radio-2 qubit is provided a second time to the radio-2 quantum circuit element implementing the Chrestenson transformation matrix that is adapted to operate as the trans-radix-2 qubit quantum circuit element.

10. 10. The trans-radix quantum circuit of claim 9, wherein the base-2 controlled-Z quantum circuit element is applied to the second base-2 qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the second base-2 qubit a second time.

11. 3. The trans radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the trans radix quantum circuit is adapted to operate as a radix-2 controlled-X quantum circuit element.

12. 12. The trans-radix quantum circuit of claim 11 , wherein the base-2 controlled-X quantum circuit element is applied to the first base-2 qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the first base-2 qubit for the first time and before the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the first base-2 qubit for a second time, and wherein the base-2 controlled-X quantum circuit element is applied to the second base-2 qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the second base-2 qubit for the first time and before the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the second base-2 qubit for the second time.

13. 13. The trans-radix quantum circuit of claim 12, wherein the base-2 controlled-X quantum circuit element is applied to the first base-2 qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the first base-2 qubit a second time, and the base-2 controlled-X quantum circuit element is applied to the second base-2 qubit after the quantum circuit element adapted to operate as the trans-radix-2 qubit quantum circuit element is applied to the second base-2 qubit a second time.

14. 10. The trans-radix quantum circuit of claim 1, wherein the quantum circuit elements implementing the Chrestenson transformation matrix are implemented in photonics.

15. 15. The trans-radix quantum circuit of claim 14, wherein the quantum circuit element is an optical four-port directional coupler.

16. 16. The trans-radix quantum circuit of claim 15, wherein applying the quantum circuit element to the first radical-2 qubit and the second radical-2 qubit comprises providing the first radical-2 qubit to a first face or a second face of the optical four-port directional coupler, and providing the second radical-2 qubit to a third face or a fourth face of the optical four-port directional coupler.

17. 3. The transformer radix quantum circuit of claim 2, wherein the one or more radix-2 quantum circuit elements include a radix-2 quantum circuit element implementing a Hadamard transform matrix, and the transformer radix quantum circuit is adapted to operate as a radix-2 controlled S quantum circuit element.

18. 1. A method comprising: A quantum information processing system comprising: operating an optical four-port directional coupler as a transformer radix-2 qubit QIP circuit element, the optical four-port directional coupler comprising a first port, a second port, a third port, and a fourth port; the optical four-port directional coupler configured to perform a characteristic transformation matrix operation utilizing the first, second, third, and fourth ports; and the operating includes: the quantum information processing system providing a first radix-2 qubit at the first port or the second port of the optical four-port directional coupler; the quantum information processing system providing a second radix-2 qubit at the third port or the fourth port of the optical four-port directional coupler; The method is achieved by

19. 20. The method of claim 18, wherein the first base-2 qubit and the second base-2 qubit are time-aligned at the first port, second port, third port, or fourth port.

20. A transradix quantum circuit, comprising: Quantum circuit elements configured to perform radix-2 quantum operations Equipped with A trans-radix quantum circuit, wherein the quantum circuit elements include one or more radix-r quantum circuit elements, where r>2.

21. 21. The trans-radix quantum circuit of claim 20, further comprising one or more radix-2 quantum circuit elements, wherein an input or output port of the one or more radix-r quantum circuit elements supports an input or output of a radix-2 qubit transmission channel.

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