Iterative hybrid quantum-classical mechanism for approximating geometric entanglement of multi-qubit pure states with noise mitigation in a quantum device

Abstract

A novel and useful hybrid quantum-classical mechanism for approximating geometric entanglement of multi-qubit pure states with noise mitigation in a quantum device. The mechanism applies the Higher-Order Power Method that approximates solutions to rank-1 tensor approximation and is executed virtually solely on a quantum computer with only angle computation performed on a classical computer. The mechanism iteratively estimates a closest separable state and an estimate of the corresponding entanglement eigenvalue in a quantum system. During each iteration, the mechanism applies a special unitary operator that skips one of the qubits. The state of the skipped qubit is recovered using one-qubit tomography to obtain updated angles used for a subsequent qubit. This procedure is repeated to converge to a final entanglement eigenvalue. A depolarizing noise mitigation procedure is also provided to generate a reduced noise estimate of geometric entanglement.

Description

[0001] ITERATIVE HYBRID QUANTUM-CLASSICAL MECHANISM FOR APPROXIMATING GEOMETRIC ENTANGLEMENT OF MULTI-QUBIT PURE STATES WITH NOISE MITIGATION IN A QUANTUM DEVICE

[0002] REFERENCE TO PRIORITY APPLICATION

[0003] This application claims the benefit of U.S. Provisional Application No. 63 / 555,448, filed February 20, 2024, entitled “Iterative Quantum Algorithm For Approximating Geometric Entanglement Of Multi-Qubit Pure Quantum States” incorporated herein by reference in its entirety.

[0004] FIELD OF THE DISCLOSURE

[0005] The subject matter disclosed herein relates to the field of quantum computing and more particularly relates to an iterative hybrid quantum-classical mechanism for approximating geometric entanglement of multi-qubit pure states with noise mitigation in a quantum device.

[0006] BACKGROUND OF THE INVENTION

[0007] Quantum computers are machines that perform computations using the quantum effects between different types of particles, like electrons, holes, ions, photons, atoms, molecules, etc. Quantum computing utilizes quantum-mechanical phenomena such as superposition and entanglement to perform computation. Quantum computing is fundamentally linked to the superposition and entanglement effects and the processing of the resulting entanglement states. A quantum computer is used to perform such computations which can be implemented theoretically or physically.

[0008] Currently, analog and digital are the two main approaches to physically implementing a quantum computer. Analog approaches are further divided into quantum simulation, quantum annealing, and adiabatic quantum computation. Digital quantum computers use quantum logic gates to do computation. Both approaches use quantum bits referred to as qubits.

[0009] Qubits are fundamental to quantum computing and are somewhat analogous to bits in a classical computer. Qubits can be in a |0> or |1> quantum state but they can also be in a superposition of the |0> and |1> states. When qubits are measured, however, they always yield a |0> or a |1> based on the quantum state they were in.

[0010] Entanglement is the defining property of quantum computation. Entanglement is one of the fundamental properties of a quantum state and is a crucial differentiator between classical and quantum computation. There are many ways to define entanglement and its measure, depending on the problem or application under consideration. Each of these measures may be computed or approximated by multiple methods. Hardly any of these methods, however, can be run on near-term quantum hardware. For a quantum algorithm to provide advantage over a classical alternative, entangled states are a necessary ingredient. The mere presence of entanglement is not sufficient, however. The degree of entanglement in a quantum state is an important property for many applications. If there is not “enough” entanglement, a quantum circuit can be efficiently simulated by classical devices. Some systems, however, with maximally entangled states, such as stabilizer codes, give efficient classical simulations. Also highly entangled states are not useful as computational resources in the measurement based computing paradigm.

[0011] In quantum machine learning applications, entanglement contributes to “barren plateaus” in the cost function landscape that make training a challenge. Entanglement also plays a crucial role in almost all quantum information technologies, permitting quantum teleportation, quantum communication, and quantum cryptography. In condensed matter physics entanglement metrics have been applied to study hard to detect phase transitions. Since inducing entanglement is such a key operation for a quantum computer, demonstrating that entanglement can be created and benchmarked is an essential task when developing new quantum hardware.

[0012] Consequently, there has been an effort to define and quantify entanglement, with different methods being preferred depending on the application. Some definitions, e.g., quantum mutual information or von Neumann entropy, are measures of bipartite entanglement only and are computationally expensive for mixed states. Others, such as concurrence, have no unique definition for higher (>2) dimensional systems.

[0013] The geometric measure of entanglement (EG is a known multi-partite entanglement metric with a clear geometric interpretation, which extends to the n-partite case and to mixed states. Geometric entanglement is, among other applications, useful when defining entanglement witnesses which are used as a positive indicator of entanglement for a subset of quantum states.

[0014] The Groverian measure of entanglement computes EG to assess the probability of success for an initial state in Grover's algorithm. This method was rephrased in terms of eigenvalues and singular value decomposition (SVD) and extended to mixed states. A similar method was formulated in terms of Tucker decomposition using the Higher Order Orthogonal Iteration (HOOI) algorithm. There is thus a need for measuring entanglement on a physical quantum device. The approaches described above are all intended to be executed on classical computing machines. To perform this on a classical computer requires using full-state tomography to reconstruct the quantum state to calculate its entanglement. A problem arises, however, as the number of qubits increases, the number of measurements required for tomography increases exponentially.

[0015] Alternatively, if the quantum state in known advance, the appropriate entanglement witness can be prepared. The latter approach, however, does not allow to obtain the measure of entanglement itself. Variational quantum circuits (VQC) can be used to compute geometric entanglement of pure states on a quantum computer. VQC algorithms, however, suffer from “barren plateaus” which hinders the scalability of this approach.

[0016] SUMMARY OF THE INVENTION

[0017] The present invention is a novel and useful hybrid quantum-classical mechanism for approximating and / or estimating geometric entanglement of multi-qubit pure states in a quantum device. The mechanism provides a quantum adaptation of an iterative higher-order power method for estimating the geometric measure of entanglement of multi-qubit pure states using rank-1 tensor approximation. The mechanism is executable on hybrid quantum-classical hardware and does not depend on quantum memory. The present invention also provides a mechanism to mitigate the effects of noise on the results of the computation.

[0018] In one embodiment, the mechanism applies the Higher-Order Power Method that approximates solutions to rank-1 tensor approximation and is executed virtually solely on a quantum computer with only angle computation performed on a classical computer. The mechanism iteratively estimates a closest separable state and an estimate of the corresponding entanglement eigenvalue in a quantum system. During each iteration, the mechanism applies a special unitary operator that skips one of the qubits. The state of the skipped qubit is recovered using one-qubit tomography to obtain updated angles used for a subsequent qubit. This procedure is repeated to converge to a final entanglement eigenvalue. A depolarizing noise mitigation procedure is then applied to generate a noise mitigated estimate of geometric entanglement.

[0019] Almost all the steps of the HOPM are implemented in the quantum domain. Entanglement is measured on near-term quantum devices more time efficiently than full-state tomography and more space efficiently (i.e. qubits versus classical memory) than executing HOPM on a classical device.

[0020] An advantage of the mechanism of the present invention is that it provides an ability to measure multi-qubit quantum entanglement on quantum computers that scales linearly in the number of gates as well as in the number of qubits. In addition, the mechanism teaches how to implement a rank-1 tensor decomposition algorithm on a quantum computer. It also provides a means of mitigating the effects of machine noise on the results of the algorithm.

[0021] This, additional, and / or other aspects and / or advantages of the embodiments of the present invention are set forth in the detailed description which follows; possibly inferable from the detailed description; and / or learnable by practice of the embodiments of the present invention.

[0022] There is thus provided in accordance with the invention, a method for use on a quantum computer of estimating geometric entanglement of multi-qubit pure quantum states in a quantum system, the method comprising initializing, on the quantum computer, the quantum system to a known quantum state, for each qubit in the quantum system: implementing, on the quantum computer, an operator received from a user to create a quantum target state, applying, on the quantum computer, a unitary to the quantum state where the qubit is skipped, for the kthiteration, meas uring, on the quantum computer, the state of the skipped qubit, calculating updated angles corresponding to the skipped qubit, using the updated angles to encode the quantum state of the skipped qubit with X and Z rotations for subsequent qubits, and iterating the above steps for each qubit to generate an estimate of geometric entanglement.

[0023] There is also provided in accordance with the invention, an apparatus for approximating geometric entanglement of multi-qubit pure quantum states in a quantum system, comprising a quantum computer having a plurality of qubits forming a quantum system, a circuit coupled to said quantum system and operative to: initialize the quantum system to a known quantum state, for each qubit in the quantum system: implement an operator received from a user to create a quantum target state, apply a unitary to the quantum state where the / " qubit is skipped, measure the state of the skipped ithqubit, calculate updated angles corresponding to the skipped qubit, encode, using the updated angles, the quantum state of the skipped qubit with X and Z rotations for subsequent qubits, and iterate the above steps for each qubit to generate an estimate of geometric entanglement.

[0024] There is further provided in accordance with the invention, a method for use on a quantum computer of estimating geometric entanglement of multi-qubit pure quantum states in a quantum system, the method comprising initializing, on the quantum computer, the quantum system to a desired quantum target state, iteratively estimating a closest separable state and a corresponding entanglement eigenvalue, at each iteration k: applying a unitary to the target state whereby an ithqubit is skipped, recovering a state of the skipped qubit using one-qubit tomography, obtaining angles which are used to calculate for a next (i+l)thqubit, and computing the entanglement eigenvalue in accordance with said angles. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Fig. 1 is a high level block diagram illustrating a first example quantum computer system constructed in accordance with the present invention;

[0026] Fig. 2 is a diagram illustrating an example quantum core incorporating one or more quantum circuits;

[0027] Fig. 3 is a high level block diagram illustrating a second example quantum computer system constructed in accordance with the present invention;

[0028] Fig. 4 is a diagram illustrating the processor of Figure 3 in more detail;

[0029] Fig. 5 is a diagram illustrating example pattern generator and sequence control circuit in more detail;

[0030] Fig. 6 is a diagram illustrating an example signal generation and control path;

[0031] Fig. 7 is a diagram illustrating a circuit representation of a separable state used for the mode qubit update;

[0032] Fig. 8 is a diagram illustrating a circuit representation for the similarity metric measurement without the Hadamard test procedure;

[0033] Fig. 9 is a diagram illustrating an example Hadamard test circuit;

[0034] Fig. 10 is a diagram illustrating an example circuit representation without Hadamard test procedure for computing the next iteration of a qubit in a separable state;

[0035] Fig. 11 is a flow diagram illustrating an example method of estimating geometric entanglement of pure quantum states in accordance with the present invention;

[0036] Fig. 12A is a flow diagram illustrating a first example method of one-qubit tomography used in the method of Figure 11;

[0037] Fig. 12B is a flow diagram illustrating a second example method of one-qubit tomography used in the method of Figure 11;

[0038] Fig. 13 is a flow diagram illustrating an example method of measuring an ancilla qubit used in the method of Figure 11;

[0039] Fig. 14 is a flow diagram illustrating an example method of measuring the entanglement eigenvalue A(fe);

[0040] Fig. 15 is a diagram illustrating an example circuit representation of the main steps of the quantum HOPM for the kthiteration using one-bit tomography;

[0041] Fig. 16 is a diagram illustrating an example circuit representation of the main steps of the quantum HOPM for the kthiteration including measuring A(fe); Fig. 17 is a graph illustrating the results of classical and quantum HOPM for approximating the geometric entanglement EG of GHZ;

[0042] Fig. 18 is a graph illustrating the absolute errorof classical and quantum HOPM for approximating the geometric entanglement EG of GHZ;

[0043] Fig. 19 is a graph illustrating the absolute error approximating EG of random [w] states with no noise;

[0044] Fig. 20 is a graph illustrating simulation results for quantum HOPM with target states GHZ[9];

[0045] Fig. 21 is a graph illustrating simulation results for quantum HOPM with target states GHZ[9] with noise mitigation;

[0046] Fig. 22 is a graph illustrating simulation results for quantum HOPM with target states Random [6];

[0047] Fig. 23 is a graph illustrating simulation results for quantum HOPM with target states Random[6] with noise mitigation;

[0048] Fig. 24 is a first graph illustrating the error of quantum HOPM approximating EG for 100 different samples of Random [ / / ] with unmitigated noise;

[0049] Fig. 25 is a second graph illustrating the error of quantum HOPM approximating EG for 100 different samples of Random [ / / ] with noise mitigation;

[0050] Fig. 26 is a first graph illustrating the errorof quantum HOPM approximating EG for 100 different samples of Random [ / / ] with unmitigated noise; and

[0051] Fig. 27 is a second graph illustrating the errorof quantum HOPM approximating EG for 100 different samples of Random [ / / ] with noise mitigation.

[0052] DETAILED DESCRIPTION

[0053] In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the invention. It will be understood by those skilled in the art, however, that the present invention may be practiced without these specific details. In other instances, well-known methods, procedures, and components have not been described in detail so as not to obscure the present invention.

[0054] Among those benefits and improvements that have been disclosed, other objects and advantages of this invention will become apparent from the following description taken in conjunction with the accompanying figures. Detailed embodiments of the present invention are disclosed herein; however, it is to be understood that the disclosed embodiments are merely illustrative of the invention that may be embodied in various forms. In addition, each of the examples given in connection with the various embodiments of the invention which are intended to be illustrative, and not restrictive.

[0055] The subject matter regarded as the invention is particularly pointed out and distinctly claimed in the concluding portion of the specification. The invention, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference to the following detailed description when read with the accompanying drawings.

[0056] The figures constitute a part of this specification and include illustrative embodiments of the present invention and illustrate various objects and features thereof. Further, the figures are not necessarily to scale, some features may be exaggerated to show details of particular components. In addition, any measurements, specifications and the like shown in the figures are intended to be illustrative, and not restrictive. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a representative basis for teaching one skilled in the art to variously employ the present invention. Further, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements.

[0057] Because the illustrated embodiments of the present invention may for the most part, be implemented using electronic components and circuits known to those skilled in the art, details will not be explained in any greater extent than that considered necessary, for the understanding and appreciation of the underlying concepts of the present invention and in order not to obfuscate or distract from the teachings of the present invention. Any reference in the specification to a method should be applied mutatis mutandis to a system capable of executing the method. Any reference in the specification to a system should be applied mutatis mutandis to a method that may be executed by the system.

[0058] Throughout the specification and claims, the following terms take the meanings explicitly associated herein, unless the context clearly dictates otherwise. The phrases “in one embodiment,” “in an example embodiment,” and “in some embodiments” as used herein do not necessarily refer to the same embodiment s), though it may. Furthermore, the phrases “in another embodiment,” “in an alternative embodiment,” and “in some other embodiments” as used herein do not necessarily refer to a different embodiment, although it may. Thus, as described below, various embodiments of the invention may be readily combined, without departing from the scope or spirit of the invention.

[0059] In addition, as used herein, the term “or” is an inclusive “or” operator, and is equivalent to the term “and / or,” unless the context clearly dictates otherwise. The term “based on” is not exclusive and allows for being based on additional factors not described, unless the context clearly dictates otherwise. In addition, throughout the specification, the meaning of “a,” “an,” and “the” include plural references. The meaning of “in” includes “in” and “on.”

[0060] The following definitions apply throughout this document.

[0061] A quantum particle is defined as any atomic or subatomic particle suitable for use in achieving the controllable quantum effect. Examples include electrons, holes, ions, photons, atoms, molecules, artificial atoms. A carrier is defined as an electron or a hole in the case of semiconductor electrostatic qubit. Note that a particle may be split and present in multiple quantum dots. Thus, a reference to a particle also includes split particles. Qubits may comprise position based qubits, electrostatic based qubits, and / or charge based qubits. Furthermore, qubits can be hybrid, meaning they can explore a magnetic spin of the quantum particle in addition to its position (i.e. electrostatic or charge nature).

[0062] A qubit or quantum bit is defined as a two state (two level) quantum structure and is the basic unit of quantum information. A qudit is defined as a d-state (d-level) quantum structure. A qubyte is a collection of eight qubits.

[0063] In quantum computing, the qubit is the basic unit of quantum information, i.e. the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two state quantum mechanical system in which the states can be in a superposition. Examples include (1) the spin of the particle (e.g., electron, hole) in which the two levels can be taken as spin up and spin down; (2) the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization; and (3) the position of the particle (e.g., electron) in a structure of two quantum dots or qdots, in which the two states correspond to the particle being in one qdot or the other. In a classical system, a bit is in either one state or the other. Quantum mechanics, however, allows the qubit to be in a coherent superposition of both states simultaneously, a property fundamental to quantum mechanics and quantum computing. Multiple qubits can be further entangled with each other.

[0064] A quantum dot or qdot (also referred to in literature as QD) is a nanometer-scale structure where an addition or removal of a particle changes its properties is some ways. In one embodiment, quantum dots are constructed in silicon semiconductor material having typical dimension in nanometers. The position of a particle in a qdot can attain several states. Qdots are used to form qubits and qudits where multiple qubits or qudits are used as a basis to implement quantum processors and computers.

[0065] A quantum interaction gate is defined as a basic quantum logic circuit operating on a small number of qubits or qudits. They are the building blocks of quantum circuits, just like the classical logic gates are for conventional digital circuits.

[0066] A quantum structure or circuit is a plurality of quantum interaction gates. A quantum computing core is a plurality of quantum structures. A quantum computer is a circuit having one or more computing cores. A quantum fabric is a collection of quantum structures, circuits, or interaction gates arranged in a grid like matrix where any desired signal path can be configured by appropriate configuration of access control gates placed in access paths between qdots and structures that make up the fabric.

[0067] Throughout this document, a representation of the state of the one-qubit quantum state in spherical coordinates includes two angles θ and φ . The state of the one-qubit system is completely described by a normalized vector ψ in the two dimensional complex Hilbert space. Taking into account that the global phase of a quantum state is not measurable, the vector T can be described in spherical coordinates with two angles 0 and <p. The angle 0 is between the vector ψ and the z-axis and the angle φ is the angle between the projection of the vector on the XY plane and the x-axis. Thus, any position on the sphere is described by these two angles 0 and φ . Note that for one qubit the spherical representation is two dimensional. For multiple qubits the dimensionality will grow nonlinearly and two angles will not be enough.

[0068] Quantum Computer Architecture

[0069] A high-level block diagram illustrating a first example quantum computer system constructed in accordance with the present invention is shown in Figure 1. The quantum computer, generally referenced 10, comprises a conventional (i.e. not a quantum circuit) external support unit 12, software unit 20, cryostat unit 36, quantum processing unit 38, clock generation units 33, 35, and one or more communication busses between the blocks. The external support unit 12 comprises operating system (OS) 18 coupled to communication network 76 such as LAN, WAN, PAN, etc., decision logic 16, and calibration block 14. Software unit 20 comprises control block 22 and digital signal processor (DSP) 24 blocks in communication with the OS 18, calibration engine / data block 26, and application programming interface (API) 28.

[0070] Quantum processing unit 38 comprises a plurality of quantum core circuits 60 supporting, inter alia, the execution of the quantum portion of the hybrid quantum-classical mechanism of estimating geometric entanglement of a quantum system, classical computer coprocessor 41 for supporting, inter alia, the execution of the classical portion of the hybrid quantum-classical mechanism of estimating geometric entanglement of a quantum system, high speed interface 58, detectors / samplers / output buffers 62, quantum error correction (QEC) 64, digital block 66, analog block 68, correlated data sampler (CDS) 70 coupled to one or more analog to digital converters (ADCs) 74 as well as one or more digital to analog converters (DACs, not shown), clock / divider / pulse generator circuit 42 coupled to the output of clock generator 35 which comprises high frequency (HF) generator 34. The quantum processing unit 38 further comprises serial peripheral interface (SPI) low speed interface 44, cryostat software block 46, microcode 48, command decoder 50, software stack 52, memory 54, and pattern generator 56. The clock generator 33 comprises low frequency (LF) generator 30 and power amplifier (PA) 32, the output of which is input to the quantum processing unit (QPU) 38. Clock generator 33 also functions to aid in controlling the spin of the quantum particles in the quantum cores 60.

[0071] The cryostat unit 36 is the mechanical system that cools the QPU down to cryogenic temperatures. Typically, it is made from metal and it can be fashioned to function as a cavity resonator 72. It is controlled by cooling unit control 40 via the external support unit 12. The cooling unit control 40 functions to set and regulate the temperature of the cryostat unit 36. By configuring the metal cavity appropriately, it is made to resonate at a desired frequency. A clock is then driven via a power amplifier which is used to drive the resonator which creates a magnetic field. This magnetic field can function as an auxiliary magnetic field to aid in controlling one or more quantum structures in the quantum core.

[0072] The external support unit / software units may comprise any suitable computing device or platform such as an FPGA / SoC board. In one embodiment, it comprises one or more general purpose CPU cores and optionally one or more special purpose cores (e.g., DSP core, floating point, etc.) that that interact with the software stack that drives the hardware, i.e. the QPU. The one or more general purpose cores execute general purpose opcodes while the special purpose cores execute functions specific to their purpose. Main memory comprises dynamic random access memory (DRAM) or extended data out (EDO) memory, or other types of memory such as ROM, static RAM, flash, and non-volatile static random access memory (NV SRAM), bubble memory, etc. The OS may comprise any suitable OS capable of running on the external support unit and software units, e.g., Windows, MacOS, Linux, QNX, NetBSD, etc. The software stack includes the API, the calibration and management of the data, and all the necessary controls to operate the external support unit itself.

[0073] The clock generated by the high frequency clock generator 35 is input to the clock divider 42 that functions to generate the signals that drive the QPU. Low frequency clock signals are also input to and used by the QPU. A slow serial / parallel interface (SPI) 44 functions to handle the control signals to configure the quantum operation in the QPU. The high speed interface 58 is used to pump data from the classic computer, i.e. the external support unit, to the QPU. The data that the QPU operates on is provided by the external support unit.

[0074] Non-volatile memory may include various removable / non-removable, volatile / nonvolatile computer storage media, such as hard disk drives that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive that reads from or writes to a removable, nonvolatile magnetic disk, an optical disk drive that reads from or writes to a removable, nonvolatile optical disk such as a CD ROM or other optical media. Other removable / non-removable, volatile / nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like.

[0075] The computer may operate in a networked environment via connections to one or more remote computers. The remote computer may comprise a personal computer (PC), server, router, network PC, peer device or other common network node, or another quantum computer, and typically includes many or all of the elements described supra. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.

[0076] When used in a LAN networking environment, the computer is connected to the LAN via network interface 76. When used in a WAN networking environment, the computer includes a modem or other means for establishing communications over the WAN, such as the Internet. The modem, which may be internal or external, is connected to the system bus via user input interface, or other appropriate mechanism.

[0077] Computer program code for carrying out operations of the present invention may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Smalltalk, C++, C# or the like, conventional procedural programming languages, such as the “C” programming language, and functional programming languages such as Python, Hotlab, Prolog and Lisp, machine code, assembler or any other suitable programming languages.

[0078] Also shown in Figure 1 is the optional data feedback loop between the quantum processing unit 38 and the external support unit 12 provided by the partial quantum data read out. The quantum state is stored in the qubits of the one or more quantum cores 60. The detectors 62 function to measure / collapse / detect some of the qubits and provide a measured signal through appropriate buffering to the output ADC block 74. The resulting digitized signal is sent to the decision logic block 16 of the external support unit 12 which functions to reinject the read out data back into the quantum state through the high speed interface 58 and quantum initialization circuits. In an alternative embodiment, the output of the ADC is fed back to the input of the QPU.

[0079] In one embodiment, quantum error correction (QEC) is performed via QEC block 64 to ensure no errors corrupt the read out data that is reinjected into the overall quantum state. Errors may occur in quantum circuits due to noise or inaccuracies similarly to classic circuits. Periodic partial reading of the quantum state functions to refresh all the qubits in time such that they maintain their accuracy for relatively long time intervals and allow the complex computations required by a quantum computing machine.

[0080] It is appreciated that the architecture disclosed herein can be implemented in numerous types of quantum computing machines. Examples include semiconductor quantum computers, superconducting quantum computers, magnetic resonance quantum computers, optical quantum computers, etc. Further, the qubits used by the quantum computers can have any nature, including charge qubits, spin qubits, hybrid spin-charge qubits, etc.

[0081] In one embodiment, the quantum structure disclosed herein is operative to process a single particle at a time. In this case, the particle can be in a state of quantum superposition, i.e. distributed between two or more locations or charge qdots. In an alternative embodiment, the quantum structure processes two or more particles at the same time that have related spins. In such a structure, the entanglement between two or more particles could be realized. Complex quantum computations can be realized with such a quantum interaction gate / structure or circuit. In alternative embodiments, the quantum structure processes (1) two or more particles at the same time having opposite spin, or (2) two or more particles having opposite spins but in different or alternate operation cycles at different times. In the latter embodiment, detection is performed for each spin type separately.

[0082] A diagram illustrating an example quantum core incorporating one or more quantum circuits is shown in Figure 2. The quantum core, generally referenced 80, comprises one or more quantum interaction gates, circuits, or cores 84 each comprising one or more quantum wells. The quantum interaction gates / circuits / cores or fabric have corresponding control circuitry including reset circuitry 82 for placing the quantum circuit into a known state, injector circuitry 88 for injecting one or more particles (e.g., electron or holes) into the circuit, imposer circuitry 90 for controlling the quantum operation, and detector circuitry 86 for detecting the presence or absence of particles. Together these classic electronic interface circuits electronically control the operation of the semiconductor quantum interaction gates / circuits / cores.

[0083] A high level block diagram illustrating a second example quantum computer system constructed in accordance with the present invention is shown in Figure 3. The quantum computer, generally referenced 170, comprises an external support unit (ESU) 174, server computer 176, communications network 178, client computer 180, cooling unit control 172, communications bus 188, power management 182, quantum processing unit (QPU) 184 and I / O circuits 186. At the core of the system is the quantum processing unit 184 which comprises the quantum interaction gates, circuits, and cores and includes digital and analog sections, initialization and control section, and measurement section.

[0084] The server 176 may comprise a suitably programmed conventional computer that is linked to one or more conventional client computers 180 via a communications network 178 such as the internet using well known remote communications protocols such as secure shell (SSH). Thus clients are able to perform cloud computing using the quantum computer. The server computer interfaces with the external support unit 174 which may comprise any suitable computing device or platform such as a system on chip / field programmable gate array (SoC / FPGA) board such as the Zynq ARM / FPGA SoC development board manufactured by Xilinx, Inc., San Jose, California, USA. The ESU FPGA board comprises an SD card boot loader 192, processor 190, Ethernet port 194, and PCIe port 196. The function of the FPGA board is to interface to the hardware in the system including the quantum processing unit 184 and the server 176. In addition, the ESU FPGA board generates the signals for the serial interface, parallel interface, high speed interface, low speed interface, etc. The board typically runs an operating system (OS) such as Linux but can be any suitable OS.

[0085] A diagram illustrating the processor of Figure 3 in more detail is shown in Figure 4. The processor 190 comprises a calibration engine and data block 200, control block 202, DSP block 204, and software API stack 206. The software stack and API controls the activities of the ESU FPGA board which also includes software for performing calibration, data operations, and all required digital signal processing.

[0086] The ESU FPGA board also interfaces with the cooling system via the cooling unit control block 172 accessible over the communications bus. The cooling unit controls the temperature the quantum processing unit operates at. In one embodiment, the quantum processing unit operates at 4K. The quantum computer system also comprises a power management unit 182 that functions to control the different supply voltages required by the quantum processing unit integrated circuit (IC) chip. The system also includes control I / O 186 directly from the ESU FPGA board as well as the signal path I / Os which are generally higher frequency I / O. It is through the I / Os that data is injected into the quantum core 184 and data is detected at the output.

[0087] The quantum processing unit (QPU) 184 comprises the quantum interaction gates, circuits, and / or cores, referred herein simply as the quantum core or quantum fabric. The quantum core(s) may comprise any number of quantum structures, e.g., qubits, qudits, etc., arranged in any desired fashion, e.g., linear array, two dimensional array, circular, etc.

[0088] A diagram illustrating example pattern generator and sequence control circuit in more detail is shown in Figure 5. The circuit, generally referenced 210, comprises pattern generator 216, heater 221, serial / parallel interface (SPI) 212, low voltage differential signaling (LVDS) interface 214, multiplexer 234, address decoding and sequence control block 222, and QPU 226. The QPU 226 comprises quantum core / fabric 225, amplitude generator 228, pulse generator 230, and heater 232.

[0089] The quantum core / fabric comprises a plurality of quantum cells that include a plurality of qdots, qubits, and control lines. The signals that control the quantum cell have both amplitude and pulse width characteristics. In addition, the pulse width also has an element of time as each pulse begins and ends at a certain point in time and has a certain pulse width. These two characteristics of the control signals must be controlled. The function of the pattern generator is to provide the sequence of signals which ultimately are used to control the quantum core. Such control signals include, for example, reset, injectors, various imposers, and detector reference and detector signals, sampling of the output of the detector, etc. These signal pulses are generated in an appropriate sequence to properly operate the quantum core and perform quantum operations. The pattern generator comprises an array of sequences 218, i.e. sequence 1, sequence 2, . . . sequence N, each having an address 220 in memory. It is appreciated that the present invention is not limited to the pattern generator presented herein as numerous other mechanisms may be used to generate the appropriate signal sequences, including those stored in a memory.

[0090] A quantum command pointer (i.e. analogous to a program counter) is maintained and program execution or flow is established. Address decoding and sequence control block 222 functions to convert each sequence into an action, counter, data value, and address. The sequencing circuit is capable of sequential command execution, branching, delays, looping, and any other desired operations. An example sequence execution 224 is provided starting at address 0 with sequence 1. Sequence 2 is then executed which causes a branch to address+9 with sequence 10. A loopback to address 0 and sequence 1 occurs and the sequence execution repeats.

[0091] A diagram illustrating an example signal generation and control path is shown in Figure 6. The signal path, generally referenced 240, comprises programmable pattern generator 242, pulse generator 246, amplitude generation 248, quantum core 252, detector 254, correlated double sampling (CDS) 256, and buffer / amplifier 258. Initially, the pattern generator creates some pattern as represented by a series of sequences from commands received from the external support unit (ESU). In one embodiment, each sequence comprises an ordered array of zeros and ones. Pattern data from the sequences is provided to the pulse generator that functions to convert sequences of zeros and ones into digital pulses. The amplitude generator, via DACs 250, converts the digital signals to analog levels, e.g., tens or hundreds of millivolts. The analog levels generated are those required by the quantum core.

[0092] Thus, the programmable pattern generator creates the digital sequence while the pulse generator and amplitude generator create the implementation of the pulse sequence with analog circuits. The amplitude control controls the actual amplitude of the signals output to the quantum cell / core 252. By having the pulse generation cascaded with amplitude generation, both the amplitude and pulse width of the signal is controlled. The signal is then input to the quantum cell which has the four functions including reset, inject, impose, and measure / detect. Detectors 254 at the output convert the quantum state back into classic circuit signals. After detection, a sampling CDS is performed. After sampling and amplification circuitry, the signal is converted to digital via analog to digital converters (ADCs) where the output can then drive digital circuits. Note that alternatively, the sequences to be executed may be generated from memory rather than the ESU and pattern generator. In this embodiment, the sequences are retrieved from memory and then decoded and executed to control the quantum core.

[0093] Approximating / Estimating Geometric Entanglement

[0094] The geometric measure of entanglement is one example entanglement metric with a clear geometric interpretation which naturally extends to the n-partite case. Geometric entanglement has many applications including defining entanglement witnesses which are used as a positive indicator of entanglement for a subset of quantum states.

[0095] The notion of entanglement can be defined in different ways, especially in the multiqubit setting. It is simpler, however, to define the absence of entanglement. Each (pure) state of a qubit (and of quantum systems in general) can be presented as a vector. We define a separable (i.e. non-entangled) state as the “composition” of states of individual qubits in the sense that changes in a state of any qubit do not affect states of other qubits. If the state of a system cannot be presented in this way, we call this state entangled. This manner of constructing the state precludes the presence of entanglement.

[0096] Several definitions are provided below. Let be an / / -qubit Hilbert space and A pure n-partite state is fully separable if and only if it is a product state of 1 -qubit states

[0097] We say an n-partite pure state is entangled if it is not fully separable. Let denote the set of fully separable states.

[0098] Let the geometric measure of entanglement of a pure state be the following where called the entanglement eigenvalue.

[0099] The scalar product can be understood as a similarity metric where if the states are identical it will be 1 and if they are orthogonal it will be 0. The intuition of geometric entanglement is that we are looking to find the most similar separable state to the target state is not entangled, then there exists a separable state that is identical to itself, and the similarity metric will be 1, and is entangled then no separable state will be identical to but their similarity will be limited by the degree of entanglement, then Note that the closest separable state is formally states as a minimization problem over the objective function being the distance between as follows where is the reference distance with being the Frobenius norm. Note further that the minimizer of Equation 3 exists since the set of fully separable states Snis the classical Segre variety which is closed in Zariski and Euclidean topology when defined over the complex numbers.

[0100] Note that in one embodiment, the target hardware for the mechanism of the present invention is a quantum computer with a classical co-processor such as described supra which is capable of generating and executing quantum programs and processing measurements from the quantum device.

[0101] In accordance with the present invention, the classical high-order power method (HOPM) algorithm for estimating entanglement is implemented on a quantum computer. The algorithm is provided below as follows.

[0102] The notations used herein include: is a two-dimensional complex vector space) is a tensor representation of in the basis with components vector of components of the one-qubit states defined in Equation 1, in the same basis is the non-normalized components of in the corresponding basis); the symbol represents the complex conjugate; the symbolxi denotes an n-mode vector product over all modes except the ith one, which is a contraction of a tensor on the left with the tuple of vectors on the right, skipping the ith index (i.e. mode) of the tensor (when the subscript is not given, none of the modes are skipped);

[0103] The input for the mechanism is a quantum state encoded as a set of instructions typically defined by a user that implement the operator that creates a quantum target state on an n-qubit device. After the state is initialized, the initial guess for the closest separable state is chosen. The mechanism is iterative in that after k iterations we have an estimate of the closest separable state and the corresponding entanglement eigenvalue the next estimate at k+ improves over the current estimate k. During each iteration, the algorithm applies to the target state a special unitary that skips one of the qubits, i.e. the qubit. The resulting state of the skipped qubit is then recovered using the one- qubit state tomography by measuring the expectation values of X, Y and Z on the skipped qubit with all other qubits being projected onto the state Note that in one embodiment, this is achieved using either post-selection or the Hadamard test procedure, described in more detail infra. These values are processed by an expression to obtain the angles that will be used in +i for the next qubit, i.e. the (z+l)thqubit. The above procedure is then repeated for each qubit within each iteration.

[0104] A flow diagram illustrating an example method of estimating geometric entanglement of pure quantum states in accordance with the present invention is shown in Figure 11. The method which is applicable to any type of qubit includes an inner loop i for each qubit in the quantum system and an outer loop k for improving the accuracy of the results. A flow diagram illustrating a first example method of one-qubit tomography is shown in Figure 12A. A flow diagram illustrating a second example method of one-qubit tomography is shown in Figure 12B. A flow diagram illustrating an example method of measuring an ancilla qubit is shown in Figure 13. A flow diagram illustrating an example method of measuring the entanglement eigenvalue A(fc)is shown in Figure 14. With reference to Figures 11, 12A, 12B, 13, and 14, the entanglement estimation mechanism will now be described. Quantum Algorithm and Inputs and Outputs

[0105] Note that the number of iterations k required depends on the target state I and the desired absolute accuracy c (step 300) as well as an initial guess of random angles the solution (step 302). In this step, the outer loop variable k is initialized to The inner loop i is initialized to 1 (step 304) and the quantum system is placed in the state

[0106] The input for the algorithm is a user defined set of instructions for a quantum machine, e.g., an encoding of a quantum circuit, that implements some unitary operator controlled by the ancilla qubit which on application to state creates the quantum state This can be expressed as Optional input arguments may include the error tolerance e, the minimum number of iterations, and / or the maximum number of iterations required.

[0107] Next, operator is performed controlled by the ancilla qubit (step 310) as described in more detail infra. One-qubit tomography of the ith qubit is then carried out (step 312). The angles are updated (step 314) and i is incremented (step 324) and the inner loop repeats for the next qubit. Once all of the n qubits have been processed, an estimate of the entanglement eigenvalue A^ is calculated (step 318). The outer loop index k is incremented (step 326) and the outer loop steps are repeated until the eigenvalue stops improving with respect to a predefined absolute accuracy c is achieved (step 320). Alternatively, the number of iterations can be fixed, e.g., 10 iterations, if it is known that the algorithm is likely to converge within that number of iterations.

[0108] Once the algorithm is considered to have converged, the final output values of the algorithm on the kth iteration are the entanglement eigenvalue and a set of angles that are used to encode the current separable state (step 322). Note that the separable state encoded as a tensor product of one-qubit x and z rotations acting on

[0109] Note that in the example embodiment described herein, the updating of the angles in step 314 is performed on the classical computer co-processor portion of the quantum computer.

[0110] Encoding the Separable State

[0111] Any one-qubit stat up to a global phase, can be encoded using two angles for Rxand Rzrotations as follows: This encoding is used to encode the closest separable state: side.

[0112] An initial separable state chosen by randomly choosing the angles as a starting point for the algorithm. At each iteration, these angles are updated qubit by qubit. A diagram illustrating a circuit representation of a separable state used for the ith mode qubit update is shown in Figure 7. The circuit, generally referenced 290, comprises n qubits input to a circuit 292 which is the vector part of the / / -mode vector product for line 6 of Algorithm 1 shown supra.

[0113] A diagram illustrating a circuit representation for the similarity metric measurement without the Hadamard test procedure is shown in Figure 8. The circuit, generally referenced 390, comprises operation Z and X rotations 394, and measurement 396.

[0114] Measurement Using Hadamard Test

[0115] In one embodiment, the Hadamard test is used to measure the probability of a state to obtain a scalar product of the form by performing measurements on a single ancilla qubit specially entangled with the quantum system. For example, to determine the probability of a state where U is any operation, being in state 1°) after the measurement, which is the quantity the Hadamard test circuit shown in Figure 9 can be used. The circuit, generally referenced 330, comprises operation U 332, identity gate J 334, H gate 336, and measurement 338. For example, to measure the real component then gate Jis the identity gate. To measure the imaginary component then gate J

[0116] Considering the quantum circuit QC (step 370), the unitary operation U is controlled by the ancilla qubit initialized in the 1+) state and followed by the H gate to measure the real part of to measure the imaginary part. In particular, the Hadamard gate H is applied to the ancilla qubit (step 372). To obtain the probability, the real component of the ancilla qubit is measured first to yield probabilities The real part s then calculated (step 376). The quantum circuit QC is then re-initialized to its input state (step 378). The gate is then applied to the ancilla qubit (step 380). The imaginary component of the ancilla qubit is measured second to yield probabilities 382). The imaginary part is then calculated (step 384) and the value returned (step 386). Note that in one embodiment, this procedure is used in all probability measurements.

[0117] Computing the Entanglement Eigenvalue

[0118] The entanglement eigenvalue is expressed as and computed in line 8 of Algorithm 1 shown supra. On a quantum computer, the measurement of needed. Operation however, is provided by the user. The operation V is defined supra in Equation 5. With the current list of angles provided by a classical co-processor that is part of the quantum device. This circuit can be visualized as shown in Figure 8.

[0119] To measure A^, the quantum system is first initialized to the state Operation is then performed controlled by the ancilla qubit (step 392). Operation is performed controlled by the ancilla qubit (step 394). The ancilla qubit is measured using the method of Figure 13 described supra (step 396). The entanglement eigenvalue is then calculated in accordance with the result of the measurement (step 398).

[0120] Updating the Current Separable State

[0121] Part of the estimation mechanism is to update the current separable state. This step comprises lines 6 and 7 of Algorithm 1 provided supra which is a similar operation to the computation of the entanglement eigenvalue with the difference being that the angles for one of the qubits is skipped. To perform the update procedure, for each qubit a state created where

[0122] The state of qubit i, having all other qubits projected to is an updated state | for the qubit in the separable state In the hybrid cl as si cal -quantum implementation this state is recreated many times for the measurement protocol to build for the other qubits. To recreate the state are extracted using one-qubit tomography which are then used to further encode it using Equation 4, i.e. A diagram illustrating an example circuit representation without Hadamard test procedure for computing the next iteration of a qubit in a separable state is shown in Figure 10. The circuit, generally referenced 520, comprises operation 522, Z rotation 524, X rotation 536, operation Us528, and measurement 529. Note that this circuit representation is without Hadamard test procedure. It can be used to compute the next iteration of a qubit in a separable state. In this example, a three qubit state is provided and the updated values for the second qubit is to be computed. The operation Usis used to modify the state and is used by the one- qubit tomography procedure described in more detail infra.

[0123] One-Qubit Tomography

[0124] To obtain the angle for the one-qubit state qubit at an arbitrary iteration k of the quantum HOPM, the following one-qubit tomography procedure is used. The input to the procedure includes the quantum circuit QC and the qubit number z (i.e. the inner loop index) (step 350). The procedure loops through all possible values of 5 in The operator qubit controlleidthby the ancilla qubit (step 354). The ancilla qubit is then measured using the method of Figure 13 described supra (step 356). The quantum circuit QC is then re-initialized to its input state (step 358). The probability is then calculated using the following (step 360):

[0125] The probability is then used to solve the following set of linear equations (step 362):

[0126] The solution is then returned (step 364).

[0127] Let which for some W returns the angles such that for and any one-qubit basis state where is an orthogonal state to In the example implementation provided herein, solves (using a classical computing device) the following system of equations: is a probability of the qubit being in the state while other qubits are in the state The probabilities are obtained by querying the quantum system and are equal to probabilities pi(V) in the system of Equations 8 supra.

[0128] One way to obtain ^) is by direct measurement of The number of shots (i.e. implementation of identical experiments) for a given accuracy in this approach grows exponentially with the number of qubits. In the example implementation presented herein, a Hadamard test based procedure is used where the number of shots depends only on the accuracy and not the number of qubits.

[0129] A flow diagram illustrating a second example method of one-qubit tomography is shown in Figure 12B. Initially, the coefficients defined in Equation 9 are classically reconstructed for the state in some basis The real and imaginary parts of are measured for each basis state the Hadamard test procedure (step 342). An ancilla qubit is introduced and initialized to the state 1+) (step 344). A unitary operation jsthen performed on the data qubits controlled by the ancilla qubit (step 346). A unitary operation is then performed that transforms state on the data qubit controlled by the ancilla qubit (step 348). The are measured on the ancilla qubit (a= A ® / ")

[0130] (step 350). And finally, the calculated where i is an imaginary unit (step

[0131] 352). Depolarizing Noise Mitigation

[0132] To mitigate the effects of noise on the results of the estimation mechanism, the depolarizing noise (DN) model is assumed in the calculations. The DN channel for a system of n qubits applied to all the qubits is defined as follows: where / is a two-dimensional identity matrix, and is a parameter of the DN model which is referred to as the noise rate. When p = 1 the channel is fully depolarizing. Whenechannel is equivalent to random Pauli errors applied uniformly on each qubit with equal probability.

[0133] Note that the following simplification is assumed. The DN channel is applied d times at a constant rate p to all qubits after each layer of gates, where d is the circuit depth, i.e. the maximum number of operations performed on a qubit in the circuit. The DN channel commutes with any unitary operation as follows: which enables the application of the channel d times at the end of the circuit: where q = (1 — p), and P' is a density matrix of a pure state with no noise. The number d can also be understood as the number of times that the DN channel affects the state with the error rate p.

[0134] It is shown that given the DN channel defined by Equation 13, in terms of the Hadamard test procedure and a sufficient number of shots, the measured for some arbitrary an be expressed as follows: where are the noise free values of respectively; dots in Equation 16 correspond to the terms of the second order and higher in n; is the phase of the noise free scalar product which is recovered from:

[0135] If all the p terms in Equation 16 are negligible, i.e. the oneOqubit tomography procedure Tt will return angles with errors of similar magnitude for any circuit depth. If these errors can be neglected, the quantum implementation will still converge but with a different convergence rate. In the example presented herein the largest error rate considered was p = 0.05 which yields Thus, considering changes at most by approximately 0.05. This change does not affect the convergence as it is seen from simulations performed. This effect can be neglected and only the changes of A(fc)need be taken into account which even for relatively shallow may be significant. With these assumptions, the entanglement eigenvalue of the non-noisy pure state is approximated as follows: where is obtained from Equation 18. The denominator of the upper bound s always less or equal to one, which means that within the assumptions which corresponds to the observed simulation results.

[0136] To mitigate the error in the estimation mechanism for arbitrary noise models on hardware or simulations (i.e. the depolarizing noise parameter) where the noise model is usually unknown, in one embodiment the following procedure can be used. The estimation algorithm is run on a reference state with a known value of the entanglement eigenvalue A and the noisy value is measured. In the example provided, is chosen to be a reference state since the true value for the entanglement eigenvalue consistent for any number of qubits. Using this value with Equation 17 the value p is estimated as follows:

[0137] The mitigation procedure then proceeds as outlined in the depolarizing noise case described supra.

[0138] Iterative Quantum Implementation

[0139] In another embodiment, by combining the Algorithm 1 with the quantum operations described supra and making use of one-qubit tomography, the most memory intensive operations of Algorithm 1 (i.e. contractions of a tensor of size 2") can be executed in the quantum domain using n qubits using Algorithm 2 shown below.

[0140] A diagram illustrating an example circuit representation of the main steps of the quantum HOPM for a quantum system of n qubits for the ktbiteration using one-bit tomography for measuring s shown in Figure 15. The circuit representation, generally referenced 400, comprises unitary operator 406, and measurement 408. Note that the measurement 408 label shows the operators A, Y to measure.

[0141] Full Quantum Implementation

[0142] As described supra, the estimation algorithm is a hybrid quantum-classical mechanism for estimating geometric entanglement of pure states in a quantum system. In an alternative embodiment, a full quantum mechanism can be used. In this case, the classical HOPM in Algorithm 1 shown supra remains valid even if line 7, i.e. the normalization step, is taken outside the inner loop. In addition, at a fixed number of iterations k < HOPM is operative even if the normalization is performed just before obtaining the final result for A(K). The numerical errors in this case, however, may grow the larger K since line 6 of Algorithm 1 effectively shrinks the updated vectors This allows for small K to potentially fully implement it in a quantum device in one step. To achieve this, however, consider the following:

[0143] (1) the number of qubits needed grows exponentially with the increase in K (2) the quantum device preferably is able to facilitate long range entanglement patterns; (3) since the vector shrink, the number of shots will increase according to the Chernoff bound to estimate the values needed up to a predefined accuracy; and (4) additional measurements may be required to estimate n norms of the vectors comprising the separable state to be able to obtain an estimate for

[0144] Simulation Results

[0145] The estimation mechanism described supra has been simulated to evaluate its robustness to noise using the IBM Qiskit platform. The mitigation of the effects of noise on the algorithm are also provided. Target states are referred to by the notation Statef / z] for n qubits, for example GHZ[9] is the GHZ state with nine qubits and Random[3] is a particular random state with three qubits.

[0146] When using HOPM the minimum EG is chosen from a sample of initial separable states. that the choice of five final iterations is arbitrary.

[0147] First it is verified that the quantum implementation matches the classical algorithm and the effect on convergence of the depolarizing noise (DN) model with noise rate p = 0.01 is

[0148]

[0149] With reference to Figures 20, 21, 22, and 23, simulations with P G {0.001, 0.01, 0.05} representing acceptable, bad and terrible levels of noise, respectively, were run to explore the relationship between DN error rate p and the divergence of quantum HOPM EG value from HOPM. Figures 20 and 22 show simulation results for quantum HOPM with input target states GHZ[9] and Random[6], Figures 21 and 23 show the results of the noise mitigation procedure applied to the results in Figures 20 and 22, respectively. The solid traces represent DN with p = 0.001, the dash traces represent DN with p = 0.01, the dash-dot traces represent DN with p = 0.05, and the dotted traces represent classical HOPM.

[0150]

[0151] Since GHZ[9] is calibrated against itself, to demonstrate the noise mitigation technique on realistic noise models only the state Random[6] is shown with p estimated with GHZ[6], Figures 24, 25, 26, and 27 show the median absolute error of the quantum HOPM results for Random[6] state within FakeSherbrooke (Figure 24) and FakeLima (Figure 26) backends, and their noise mitigation results Figures 25 and 27, respectively. Those skilled in the art will recognize that the boundaries between logic and circuit blocks are merely illustrative and that alternative embodiments may merge logic blocks or circuit elements or impose an alternate decomposition of functionality upon various logic blocks or circuit elements. Thus, it is to be understood that the architectures depicted herein are merely exemplary, and that in fact many other architectures may be implemented which achieve the same functionality.

[0152] Any arrangement of components to achieve the same functionality is effectively “associated” such that the desired functionality is achieved. Hence, any two components herein combined to achieve a particular functionality may be seen as “associated with” each other such that the desired functionality is achieved, irrespective of architectures or intermediary components. Likewise, any two components so associated can also be viewed as being “operably connected,” or “operably coupled,” to each other to achieve the desired functionality.

[0153] Furthermore, those skilled in the art will recognize that boundaries between the above described operations are merely illustrative. The multiple operations may be combined into a single operation, a single operation may be distributed in additional operations and operations may be executed at least partially overlapping in time. Moreover, alternative embodiments may include multiple instances of a particular operation, and the order of operations may be altered in various other embodiments.

[0154] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and / or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0155] In the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. The use of introductory phrases such as “at least one” and “one or more” in the claims should not be construed to imply that the introduction of another claim element by the indefinite articles “a” or “an” limits any particular claim containing such introduced claim element to inventions containing only one such element, even when the same claim includes the introductory phrases “one or more” or “at least one” and indefinite articles such as “a” or “an.” The same holds true for the use of definite articles. Unless stated otherwise, terms such as “first,” “second,” etc. are used to arbitrarily distinguish between the elements such terms describe. Thus, these terms are not necessarily intended to indicate temporal or other prioritization of such elements. The mere fact that certain measures are recited in mutually different claims does not indicate that a combination of these measures cannot be used to advantage. The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. As numerous modifications and changes will readily occur to those skilled in the art, it is intended that the invention not be limited to the limited number of embodiments described herein. Accordingly, it will be appreciated that all suitable variations, modifications and equivalents may be resorted to, falling within the spirit and scope of the present invention. The embodiments were chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.

Claims

CLAIMS1. A method for use on a quantum computer of estimating geometric entanglement of multi-qubit pure quantum states in a quantum system, the method comprising: initializing, on the quantum computer, the quantum system to a known quantum state; for each qubit in the quantum system:

6. The method according to claim 1, wherein said step of measuring comprises a one-qubit tomography procedure including:

Images