Variational quantum eigensolver with constraints

Abstract

Embodiments of a variational quantum model for solving various types of constrained optimization problems are described. In one example, a method of solving constrained optimization problems using a variational quantum approach can include determining primal optimization variables for a function defining a constrained optimization problem based at least in part on execution of a parameterized variational quantum circuit corresponding to the function in a quantum computing environment. The method can further include determining optimization variables for the function based at least in part on the primal optimization variables and dual variables associated with constraints of the constrained optimization problem. The optimization variables being indicative of a solution to the constrained optimization problem.

Description

Attorney Docket: 222204-2970 VARIATIONAL QUANTUM EIGENSOLVER WITH CONSTRAINTS STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0001] This invention was made with government support under grant number 1751085, awarded by the National Science Foundation. The government has certain rights in the invention. CROSS REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit of and priority to U.S. Provisional Application Serial No. 63 / 597,866, filed November 10, 2023, titled “VARIATIONAL QUANTUM EIGENSOLVER WITH CONSTRAINTS” (“the ’866 Application”) the entire contents of which, including Appendix A1, are hereby incorporated herein by reference. BACKGROUND

[0003] Quantum computing utilizes quantum mechanics to solve complex problems that “classical” computers, including supercomputers, cannot easily solve. Quantum computing devices solve such problems by performing operations on one or more quantum bits (“qubits”), which are used as the basic units of information in quantum computing operations and are analogous to “bits” used in classical computing. However, unlike a classical bit that exists in either one of two states (i.e., 0 or 1), a qubit can exist in a superposition of these two states (i.e., both 0 and 1).

[0004] Quantum computing could be a disruptive technology in dealing with challenging computational tasks. Seminal works have developed quantum computing algorithms to tackle problems, such as integer factorization searching in databases, solving systems of linear equations, and various machine learning tasks, with polynomial or exponential speedups over their classical computing alternatives. However, these algorithms are assumed to operate on fault-tolerant quantum computers, which are projected not to be available in the near future. Recent research and development efforts focus on devising algorithms that are of relevance to practical applications on contemporary, qubit-limited, low-circuit depth, and noisy quantum hardware, often referred to as noisy intermediate-scale quantum (NISQ). Variational quantum approaches (VQAs) exploit parameterized quantum circuits (VQCs) of limited depth and reduced number of qubits and they have become the leading candidates to showcase quantum advantage in the NISQ context.

[0005] Variational quantum eigensolver (VQE) is one of the most well-studied variational quantum approaches. Given a sequence of parameterized quantum gates, VQE aims at seeking theAttorney Docket: 222204-2970 eigenvector corresponding to the minimum eigenvalue (energy) of an exponentially large Hermitian matrix representing a quantum observable. While VQE has been successfully utilized as a heuristic to find near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems, quadratic problems oftentimes come with constraints. SUMMARY

[0006] The present disclosure is directed to solving various types of constrained optimization problems using a variational quantum model referred to herein as variational quantum eigensolver with constraints (VQEC). The VQEC model can compute optimal or near-optimal solutions to various types of constrained optimization problems by implementing a hybrid quantum-classical algorithmic approach. The VQEC model can implement a variational quantum eigensolver optimization approach in a quantum computing environment in one example to model at least one of a cost or constraint function corresponding to a particular optimization problem. The VQEC model can further optimize such a function in a classical computing environment in this example while enforcing one or more constraints by implementing at least one of a primal-dual method or a primal-dual perturbation method that is based at least in part on a parameter shift rule.

[0007] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description or can be learned from the description or through practice of the embodiments. Other aspects and advantages of embodiments of the present disclosure will become better understood with reference to the appended claims and the accompanying drawings, all of which are incorporated in and constitute a part of this specification. The drawings illustrate example embodiments of the present disclosure and, together with the description, serve to explain the related concepts of the present disclosure.

[0008] According to one example embodiment, a method of solving constrained optimization problems using a variational quantum approach can include determining primal optimization variables for a function defining a constrained optimization problem based at least in part on execution of a parameterized variational quantum circuit corresponding to the function in a quantum computing environment. The method can further include determining optimization variables for the function based at least in part on the primal optimization variables and dual variables associated with constraints of the constrained optimization problem. The optimization variables being indicative of a solution to the constrained optimization problem.Attorney Docket: 222204-2970 BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Many aspects of the present disclosure can be better understood with reference to the following figures. The components in the figures are not necessarily to scale, with emphasis instead being placed upon clearly illustrating the concepts of the disclosure. Moreover, repeated use of reference characters or numerals in the figures is intended to represent the same or analogous features, elements, or operations across different figures. Repeated description of such repeated reference characters or numerals is omitted for brevity.

[0010] FIG. 1 illustrates a block diagram of an example environment according to various aspects and embodiments of the present disclosure.

[0011] FIG. 2 illustrates a diagram of an example data flow between example classical and quantum computing devices of the example environment shown in FIG. 1 according to various aspects and embodiments of the present disclosure.

[0012] FIG. 3 illustrates a flow diagram of an example computer-implemented method according to various aspects and embodiments of the present disclosure. DETAILED DESCRIPTION

[0013] Variational quantum approaches have shown great promise in finding near-optimal solutions to computationally challenging tasks. Nonetheless, enforcing constraints in a disciplined fashion has been largely unexplored. To address this gap, embodiments described herein are directed to various aspects of a hybrid quantum-classical algorithmic model termed Variational Quantum Eigensolver with Constraints (VQEC) that extends the celebrated variational quantum eigensolver (VQE) to perform optimization with constraints. As with the standard VQE, the VQEC model can capture a vector of primal optimization variables from a state of a variational quantum circuit (VQC). To deal with constraints, the VQEC model can also optimize a Lagrangian function classically over parameters of the VQC and dual variables associated with the constraints. To comply with the quantum setup, the VQEC model can further update variables using a perturbed primal-dual method leveraging the parameter shift rule. Among a wide gamut of potential applications, the VQEC model can approximately solve quadratically-constrained binary optimization (QCBO) problems, find stochastic binary policies satisfying quadratic constraints on the average and in probability, and solve large-scale linear programs (LP) over the probability simplex. Under an assumption on an error for the VQC to approximate an arbitrary probability mass function (PMF), the VQEC model can use bounds on an optimality gap attained by the VQC in some embodiments. Numerical tests on a quantum simulator in some implementationsAttorney Docket: 222204-2970 investigated effects of various parameters and corroborated the VQEC model’s ability to generate high-quality solutions.

[0014] As noted above, quantum computing could be a disruptive technology in dealing with challenging computational tasks. Seminal works have developed quantum computing algorithms to tackle problems, such as integer factorization searching in databases, solving systems of linear equations, and various machine learning tasks, with polynomial or exponential speedups over their classical computing alternatives. However, these algorithms are assumed to operate on fault- tolerant quantum computers, which are projected not to be available in the near future. Recent research and development efforts focus on devising algorithms that are of relevance to practical applications on contemporary, qubit-limited, low-circuit depth, and noisy quantum hardware, often referred to as noisy intermediate-scale quantum (NISQ). Variational quantum approaches (VQAs) exploit parameterized quantum circuits (VQCs) of limited depth and reduced number of qubits and they have become the leading candidates to showcase quantum advantage in the NISQ context.

[0015] VQE is one of the most well-studied variational quantum approaches. Given a sequence of parameterized quantum gates, VQE aims at seeking the eigenvector corresponding to the minimum eigenvalue (energy) of an exponentially large Hermitian matrix representing a quantum observable. While VQE has been successfully utilized as a heuristic to find near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems, quadratic problems oftentimes come with constraints. A constrained binary problem can be converted to an unconstrained binary problem upon penalizing constraint violations by adding them to an objective function associated with the problem. However, the weights associated with each penalty term are non-trivial to select unless the constraints are of specific forms, such as Boolean functions or linear equalities. Another VQA that has been particularly successful for binary optimization is the quantum approximate optimization algorithm (QAOA). The QAOA is a special case of VQE that uses a problem-dependent VQC or ansatz. To incorporate constraints, some existing approaches adapt the mixer Hamiltonian of the QAOA’s ansatz to ensure that the target quantum state remains within the feasible subspace. This strategy is also studied on quantum annealing, the analog counterpart of QAOA. However, confining the mixer Hamiltonian applies only to a single linear equality constraint and requires a larger number of additional gates.

[0016] One existing approach considers binary quadratic programs with linear constraints again, by combining the quantum adiabatic approach with the classical branch-and-bound method. Another existing approach uses a VQA to minimize an objective expressed as a sum of a quadratic function over binary variables and a convex function over continuous variables. Binary andAttorney Docket: 222204-2970 quadratic variables are set to be equal through linear equality constraints. The problem is solved using the alternating direction method of multipliers (ADMM). Each ADMM iteration entails solving the convex subproblem over the continuous variables using standard convex optimization techniques on a classical computer, and the QUBO subproblem via VQE / QAOA. Although, solving a VQE or QAOA to optimality for each ADMM iteration could be computationally demanding. In another approach, a VQA method is used to improve the chance of acquiring feasible solutions for constrained combinatorial problems through a greedy post-processing method, however it’s only applicable to linear constraints.

[0017] Lagrangian duality offers a more systematic way of handling optimization problems with constraints. In the context of VQA, the vector of VQC parameters corresponds to the primal optimization variables, and the vector of Lagrange multipliers corresponds to dual variables. Albeit the primal problem is non-convex, the dual problem is known to be always convex. Other existing approaches address quadratic constrained binary optimization (QCBO) using dual decomposition, a variation of subgradient descent that aims at solving the dual problem. However, each update of dual variables involves a complete run of quantum annealing, or solving a VQE to optimality, either of which can be computationally formidable.

[0018] In contrast, a primal-dual method used by some approaches to train neural networks under constrained optimization problems is more suitable since each primal step only requires updating the primal variable inexactly. The convergence of this primal-dual method is guaranteed under strict settings, such as the Lagrangian function being strictly convex and strictly concave. However, by updating the primal and dual variables at appropriate perturbed points, one existing approach ensures the sequence of the primal-dual pairs converging to the optimal point without putting strict assumptions on the Lagrangian function. Another variant of the primal-dual method with perturbations is used in an existing approach often referred to as the extragradient method (EGM). EGM has been expanded recently to the stochastic setting. Nonetheless, in the VQC context, EGM increases substantially the number of VQC compilations.

[0019] Although the VQEC model is implemented to solve optimization problems with constraints in many examples herein, it can be expanded to other setups in some cases such as when a VQC is used as a machine learning (ML) model. This idea has attracted sizable research interest recently. A popular choice of the loss function for quantum ML is the minimization of expectations of quantum observables concerning a quantum state prepared by the VQC. Although there is no quantum equivalent to automatic differentiation, some existing approaches derive an analytical formula for the gradient of typical loss functions. However, estimating the gradient through quantum measurements is always subject to noise. Some studies show that estimatingAttorney Docket: 222204-2970 gradients using a finite number of measurements engenders an unbiased estimator and can facilitate stochastic gradient descent. In light of interpreting VQCs as ML models, it is also of crucial importance to explore VQCs from the perspective of learning models for constrained setups.

[0020] Acknowledging the gap in incorporating constraints into VQAs and its relevance to optimization and ML tasks, the contribution of the VQEC model described in embodiments herein is on four fronts. First, the VQEC model provides a novel algorithmic framework for handling optimization problems with constraints via a VQA. This VQEC framework can be applied to problems where cost and constraint functions can be captured as quantum observables over general, exponentially large, Hermitian matrices. As with VQE, rather than solving the problem over the original, exponentially large decision variable, the VQEC model adopts a hybrid quantum-classical approach. In many examples, a quantum circuit parameterized over fewer parameters stored in a vector can be used to measure quantum observables, and a classical computer can be used to update iteratively. To incorporate constraints in a disciplined fashion, the VQEC model can implement a primal-dual method in many examples using a classical computer to update not only , but also the vector of Lagrange multipliers associated with the constraints. For improved convergence properties, the VQEC model can adapt a perturbed variant termed the perturbed primal-dual (PPD) method to the quantum setup in some cases by capitalizing on the parameter shift rule. Interestingly, if cost and constraint observables can be measured simultaneously, implementation of the VQEC model involves approximately the same quantum computations as VQE.

[0021] Second, the VQEC model can be applied to problems with diagonal observables (e.g., observables defined over diagonal Hermitian matrices) as described in examples herein that also show that a wide gamut of optimization tasks can be formulated as such. Examples include problems such as constrained quadratic binary optimization (QCBO), designing stochastic policies over binary-valued vectors that satisfy constraints on the average or in probability as chance constraints, learning large-scale PMFs, and solving large-scale LPs over the probability simplex. Such problems abound in diverse application domains, including reinforcement and machine learning in general, wireless communications, portfolio optimization, and optimal resource allocation.

[0022] Third, the VQEC model provides analytical bounds on the optimality gap experienced when a problem with diagonal observables is solved in its variational form over rather than its original form over an exponentially large variable. Under an assumption resembling the universal approximation theorem for deep neural networks, the optimality gap is shown in some examplesAttorney Docket: 222204-2970 to depend on the approximation within which a VQC can approximate any PMF as well as the sensitivity of the original problem to perturbations in the constraints.

[0023] Fourth, some examples herein describe numerical evaluation of the performance of the VQEC model on problems with diagonal observables. The tests in these examples demonstrate the VQEC model’s convergence, reasonable performance even with a finite number of quantum measurements, sensitivity to VQC depth, and ability to provide relatively high-quality solutions to binary programs with constraints and large-scale LPs over the probability simplex.

[0024] For context, FIG. 1 illustrates a block diagram of an example environment 100 according to various aspects and embodiments of the present disclosure. The environment 100 can be a computing environment in which classical and quantum computing operations can be performed, among other operations. For instance, the environment 100 can be embodied as and include classical and quantum computing environments in which classical and quantum computing operations can be performed, respectively. The environment 100 is illustrated as a representative example, and the VQEC framework concepts described herein are not limited to use with any particular type of computing environment.

[0025] In the example illustrated in FIG.1, the environment 100 includes a computing device 102, one or more remote computing devices 104 (or “remote computing devices 104”), a quantum computing device 106, and a signal generator 108, among other components. In this example, the computing device 102, the remote computing devices 104, the quantum computing device 106, and the signal generator 108 are coupled to one another by way of one or more networks 110 (or, “networks 110”). In some examples, the computing device 102 can be directly coupled to the signal generator 108, which can be directly coupled to the quantum computing device 106. Such direct coupling can be achieved by way of a wired connection or another connection that can allow for at least one of a communicative, electrical, operative, or optical coupling of the computing device 102 to the signal generator 108 and coupling of the signal generator 108 to the quantum computing device 106. In one example, such direct coupling can be achieved by way of one or more coaxial cables.

[0026] The computing device 102 and any or all of the remote computing devices 104 can each be embodied or implemented as, for example, at least one of a server computing device, a client computing device, a general-purpose computer, a special-purpose computer, a virtual machine, a supercomputer, a laptop, a tablet, a smartphone, or another type of computing device that can be configured and operable to perform various operations described herein. A detailed description of the computing device 102 and the operations it can perform is provided below.Attorney Docket: 222204-2970

[0027] The quantum computing device 106 can be embodied or implemented as, for example, a qubit-based quantum computing device that can be configured and operable to perform quantum operations involving one or more qubits. For instance, the quantum computing device 106 can be embodied or implemented as at least one of a superconducting qubit device, a quantum dot spin qubit device, a superconducting transmons device, a trapped ions device, or another qubit device that can be configured and operable to perform quantum operations involving one or more qubits. Examples of such quantum operations can include, but are not limited to, at least one of a quantum or qubit evolution, a quantum or qubit operation, a quantum or qubit gate operation, a single-qubit gate operation, a multi-qubit gate operation, a sequence of qubit gate operations (e.g., a quantum circuit), or another quantum operation.

[0028] The signal generator 108 can be embodied or implemented as, for example, a pulse generator that can be configured and operable to generate various signals or pulses that can be used to perform various operations associated with the quantum computing device 106. For instance, the signal generator 108 can generate control signals or pulses that can be used to drive quantum operations performed by the quantum computing device 106. The signal generator 108 can be embodied and implemented as a microwave pulse generator configured to generate resonant microwave pulses in one example. The resonant microwave pulses can drive quantum operations performed by the quantum computing device 106.

[0029] The networks 110 can include, for instance, the Internet, intranets, extranets, wide area networks (WANs), local area networks (LANs), wired networks, wireless networks (e.g., cellular, WiFi®), cable networks, satellite networks, other suitable networks, or any combinations thereof. The computing device 102, the remote computing devices 104, the quantum computing device 106, and the signal generator 108 can communicate data with one another over the networks 110 using any suitable systems interconnect models and / or protocols. Example interconnect models and protocols include hypertext transfer protocol (HTTP), simple object access protocol (SOAP), representational state transfer (REST), real-time transport protocol (RTP), real-time streaming protocol (RTSP), real-time messaging protocol (RTMP), user datagram protocol (UDP), internet protocol (IP), transmission control protocol (TCP), and / or other protocols for communicating data over the networks 110, without limitation. Although not illustrated, the networks 110 can also include connections to any number of other network hosts, such as website servers, file servers, networked computing resources, databases, data stores, or other network or computing architectures in some cases.

[0030] Among other types of operations, the computing device 102 can be configured to provide optimal or near-optimal solutions to various types of constrained optimization problemsAttorney Docket: 222204-2970 using the VQEC model of the present disclosure. To provide such solutions the computing device 102 can use the VQEC model to implement a method of solving a constrained optimization problem using a variational quantum approach. In one particular example, the computing device 102 can provide such solutions by implementing the methodologies, equations, theorems, and VQEC paradigm described in the paper titled “SOLVING CONSTRAINED OPTIMIZATION PROBLEMS VIA THE VARIATIONAL QUANTUM EIGENSOLVER WITH CONSTRAINTS,” by Thinh Viet Le and Vassilis Kekatos, Phys. Rev. A 110, 022430 – Published August 20, 2024, https: / / doi.org / 10.1103 / PhysRevA.110.022430 (“the Kekatos paper”), the entire contents of which is hereby incorporated herein by reference.

[0031] To solve one or more constrained optimization problems such as those described in examples herein, the computing device 102 can include at least one processing and memory system. In the example depicted in FIG. 1, the computing device 102 includes at least one processor 112 and at least one memory 114, both of which are communicatively coupled, operatively coupled, or both, to a local interface 116. The memory 114 includes a data store 118, a variational quantum eigensolver with constraints (VQEC) module 120 (“VQEC module 120”), a quantum simulator module 122, a signal generator control module 124, and a communications stack 126 in the example shown. The computing device 102 is coupled to the networks 110 by way of the local interface 116 in this example. In some cases, the computing device 102 can be coupled to the signal generator 108, in addition to or in place of the networks 110, by way of the local interface 116. The computing device 102 can also include other components that are not illustrated in FIG.1.

[0032] The processor 112 can be embodied as or include any processing device (e.g., a processor core, a microprocessor, an application specific integrated circuit (ASIC), a field- programmable gate array (FPGA), a controller, a microcontroller, or a quantum processor) and can include one or multiple processors that can be operatively connected. In some examples, the processor 112 can include one or more complex instruction set computing (CISC) microprocessors, one or more reduced instruction set computing (RISC) microprocessors, one or more very long instruction word (VLIW) microprocessors, or one or more processors that are configured to implement other instruction sets.

[0033] The memory 114 can be embodied as one or more memory devices and can store data and software or executable-code components executable by the processor 112. For example, the memory 114 can store executable-code components associated with the VQEC module 120, the quantum simulator module 122, the signal generator control module 124, and the communications stack 126 for execution by the processor 112. The memory 114 can also store data such as the dataAttorney Docket: 222204-2970 described below that can be stored in the data store 118, among other data. For instance, the memory 114 can also store data indicative of quantum related aspects (e.g., quantum dynamics, features, or properties) associated with various quantum computing devices or quantum operations to be performed, data indicative of constrained optimization problems, data indicative of constraints associated with constrained optimization problems, data indicative of functions (e.g., cost functions, constraint functions, Lagrangian functions) defining constrained optimization problems, data indicative of parameterized variational quantum circuits corresponding to constrained optimization problems, data indicative of quantum observables corresponding to quantum states resulting from execution of parameterized variational quantum circuits (e.g., readouts or measurements of observed quantum states or properties), data indicative of vectors of optimization variables or optimization variables associated with functions defining constrained optimization problems, data indicative of solutions to constrained optimization problems solved according to embodiments described herein, or other data indicative of other classical or quantum related computing aspects of the embodiments herein.

[0034] The memory 114 can store other executable-code components for execution by the processor 112. For example, an operating system can be stored in the memory 114 for execution by the processor 112. Where any component discussed herein is implemented in the form of software, any one of a number of programming languages can be employed such as, for example, C, C++, C#, Objective C, JAVA®, JAVASCRIPT®, Perl, PHP, VISUAL BASIC®, PYTHON®, RUBY, FLASH®, or other programming languages.

[0035] As discussed above, the memory 114 can store software for execution by the processor 112. In this respect, the terms “executable” or “for execution” refer to software forms that can ultimately be run or executed by the processor 112, whether in source, object, machine, or other form. Examples of executable programs include, for instance, a compiled program that can be translated into a machine code format and loaded into a random access portion of the memory 114 and executed by the processor 112, source code that can be expressed in an object code format and loaded into a random access portion of the memory 114 and executed by the processor 112, source code that can be interpreted by another executable program to generate instructions in a random access portion of the memory 114 and executed by the processor 112, or other executable programs or code.

[0036] The local interface 116 can be embodied as a data bus with an accompanying address / control bus or other addressing, control, and / or command lines. In part, the local interface 116 can be embodied as, for instance, an on-board diagnostics (OBD) bus, a controller areaAttorney Docket: 222204-2970 network (CAN) bus, a local interconnect network (LIN) bus, a media oriented systems transport (MOST) bus, ethernet, or another network interface.

[0037] The data store 118 can include data for the computing device 102 such as, for instance, one or more unique identifiers for the computing device 102, digital certificates, encryption keys, session keys and session parameters for communications, and other data for reference and processing. The data store 118 can also store computer-readable instructions for execution by the computing device 102 via the processor 112, including instructions for the VQEC module 120, the quantum simulator module 122, the signal generator control module 124, and the communications stack 126.

[0038] In some cases, the data store 118 can also store data indicative of quantum related aspects (e.g., quantum dynamics, features, or properties) associated with various quantum computing devices or quantum operations to be performed, data indicative of constrained optimization problems, data indicative of constraints associated with constrained optimization problems, data indicative of functions (e.g., cost functions, constraint functions, Lagrangian functions) defining constrained optimization problems, data indicative of parameterized variational quantum circuits corresponding to constrained optimization problems, data indicative of quantum observables corresponding to quantum states resulting from execution of parameterized variational quantum circuits (e.g., readouts or measurements of observed quantum states or properties), data indicative of vectors of optimization variables or optimization variables associated with functions defining constrained optimization problems, data indicative of solutions to constrained optimization problems solved according to embodiments described herein, or other data indicative of other classical or quantum related computing aspects of the embodiments herein.

[0039] The VQEC module 120 can be embodied as one or more software applications or services executing on the computing device 102. The VQEC module 120 can be executed by the processor 112 to provide optimal or near-optimal solutions to various types of constrained optimization problems using a variational quantum model. To provide such solutions the VQEC module 120 can implement a method of solving a constrained optimization problem using a variational quantum approach. In one particular example, the VQEC module 120 can provide such solutions by implementing the methodologies, equations, remarks, theorems, assumptions, lemmas, and VQEC paradigm described in the Kekatos paper and Appendix A1 of the ’866 Application. For instance, the VQEC module 120 can provide such solutions by implementing the methodology described in the Kekatos paper and Appendix A1 of the ’866 Application in connection with Equations (1) to (28) in Sections 2 and 3 titled “Proposed Algorithm” and “Applications,” respectively. In some examples, the VQEC module 120 can further implement theAttorney Docket: 222204-2970 methodology described in the Kekatos paper and Appendix A1 of the ’866 Application in connection with Equations (29) to (41) in Sections 4 and 5 titled “Performance Analysis” and “Numerical Tests,” respectively, to analyze performance associated with implementing the methodology described in Sections 2 and 3 of each of the Kekatos paper and Appendix A1 of the ’866 Application.

[0040] As noted above, the VQEC module 120 can provide optimal or near-optimal solutions to various types of constrained optimization problems by implementing the methodology described in connection with Equations (1) to (41) in Sections 2, 3, 4, and 5 of the Kekatos paper and Appendix A1 of the ’866 Application. Examples of constrained optimization problems for which the VQEC module 120 can provide such solutions according to various embodiments herein include, but are not limited to, a quadratically-constrained binary optimization problem, a problem of designing stochastic binary policies satisfying quadratic constraints on average or in probability, a problem of learning large-scale probability mass functions, and a problem of solving large-scale linear programs over a probability simplex. In some embodiments, the computing device 102 can perform at least one operation based at least in part on computing an optimal or near-optimal solution to a constrained optimization problem. Example operations that can be performed by the computing device 102 based at least in part on computing such a solution to one or more constrained optimization problems described herein can include, but are not limited to, a machine learning or a reinforcement machine learning operation, a wireless communication operation, an asset portfolio optimization operation, and an optimal resource allocation operation.

[0041] To provide optimal or near-optimal solutions to a constrained optimization problem according to one specific embodiment, the VQEC module 120 can implement the methodology described in connection with Equations (1) to (28) in Sections 2 and 3 of the Kekatos paper and Appendix A1 of the ’866 Application to perform a method of solving a constrained optimization problem using a variational quantum approach. In this example, the VQEC module 120 can determine a vector of primal optimization variables from quantum observables measured for a quantum state created or resulting at least partly from a parameterized VQC executed in a quantum computing environment. The parameterized VQC can correspond to a function such as at least one of a cost function or a constraint function defining a constrained optimization problem in some cases. The VQEC module 120 can further determine optimization variables for the function based at least in part on the vector of primal optimization variables and dual variables associated with constraints of the constrained optimization problem. The optimization variables in many examples can be indicative of a solution to the constrained optimization problem.Attorney Docket: 222204-2970

[0042] To determine the vector of primal optimization variables in one example, the VQEC module 120 can implement the methodology described in connection with the aforementioned Equations and Sections of the Kekatos paper and Appendix A1 of the ’866 Application. For instance, the VQEC module 120 can define a sequence of parameterized quantum gates that is indicative of the parameterized variational quantum circuit. The sequence of parameterized quantum gates can be executed by a quantum computing device such as the quantum computing device 106 in a quantum computing environment. The VQEC module 120 can employ at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 in many cases to implement the sequence of parameterized quantum gates in a quantum computing environment such as a simulated quantum computing environment or a quantum computing environment included in or generated by the quantum computing device 106.

[0043] The VQEC module 120 can further employ at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 in many examples to measure quantum observables corresponding to a quantum state created or resulting at least partly from implementation of the sequence of parameterized quantum gates in the quantum computing environment. The quantum observables can be indicative of a function that defines a constrained optimization problem in some examples such as at least one of a cost function or a constraint function associated with the constrained optimization problem. The quantum observables can include cost quantum observables and constraint quantum observables defined by at least one Hermitian matrix in one example. In another example, the quantum observables can include diagonal quantum observables defined over at least one diagonal Hermitian matrix. In some examples, the VQEC module 120 can employ at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 to simultaneously measure cost quantum observables and constraint quantum observables based at least in part on implementation of the sequence of parameterized quantum gates in the quantum computing environment described above.

[0044] To enforce constraints associated with a constrained optimization problem, the VQEC module 120 can implement the methodology described in connection with the aforementioned Equations and Sections of the Kekatos paper and Appendix A1 of the ’866 Application. can update at least one of a vector of primal optimization variables or a vector of dual variables described herein using at least one of a classical optimizer, a first order method, a second order method, or another optimization method in many cases. For instance, the VQEC module 120 can update atAttorney Docket: 222204-2970 least one of a vector of primal optimization variables or a vector of dual variables described herein using a primal-dual (PD) method or a perturbed primal-dual (PPD) method (or “primal-dual perturbation method”) in many cases. For example, after each iteration of a parameterized variational quantum circuit described herein, the VQEC module 120 can update at least one of a vector of primal optimization variables or a vector of dual variables using a PD or PPD method to ultimately identify a saddle point of a function defining a constrained optimization problem. The saddle point being indicative of a pair of primal-dual vectors that corresponds to an optimal value of the function. A primal vector can denote a vector of primal optimization variables and a dual vector can denote constraints of a constrained optimization problem in some cases.

[0045] In an example where a Lagrangian function defines a constrained optimization problem, the VQEC module 120 can implement a PD method to identify a saddle point of the Lagrangian function over dual variables corresponding to constraints of the problem and primal variables corresponding to a vector of primal optimization variables. The saddle point being indicative of a pair of primal-dual vectors that corresponds to an optimal value of the Lagrangian function in this example. The dual variables can include Lagrange multipliers associated with an inequality or equality constraint of the constrained optimization problem in one example.

[0046] The VQEC module 120 can update at least one of a vector of primal optimization variables or a vector of dual variables described herein using a PD or PPD method that is based at least in part on a parameter shift rule in some examples. For instance, the VQEC module 120 can update perturbed primal variables and perturbed dual variables corresponding to a vector of primal optimization variables and dual variables, respectively. The VQEC module 120 can iteratively update the perturbed primal variables and perturbed dual variables in many cases using a PD or PPD method that is based at least in part on a parameter shift rule. For example, after each iteration of a parameterized variational quantum circuit described herein, the VQEC module 120 can update such perturbed primal and dual variables using a PD or PPD method that is based at least in part on a parameter shift rule.

[0047] As noted above, the VQEC module 120 can implement the methodology described in connection with Equations (29) to (41) in Sections 4 and 5 of the Kekatos paper and Appendix A1 of the ’866 Application to analyze performance associated with implementing the methodology described in Sections 2 and 3 of the Kekatos paper and Appendix A1 of the ’866 Application. In one example, the VQEC module 120 can implement such methodology in connection with the aforementioned Equations and Sections of the Kekatos paper and Appendix A1 of the ’866 Application to bound an optimality gap attained by implementation of the parameterized variational quantum circuit (e.g., by the quantum computing device 106). The optimality gap inAttorney Docket: 222204-2970 this example being associated with a distance between a pair of primal-dual vectors that is indicative of a saddle point of function such as a Lagrangian function.

[0048] The quantum simulator module 122 can be embodied as one or more software applications or services executing on the computing device 102. The quantum simulator module 122 can be executed by the processor 112 to simulate or cause an external device to simulate execution of a parameterized variational quantum circuit in the form of a sequence of parameterized quantum gates. The sequence of parameterized quantum gates being indicative of at least one of a cost function, a constraint function, or a Lagrangian function that defines a particular constrained optimization problem in some examples. Based at least in part on such simulation, the quantum simulator module 122 can generate or obtain simulated performance data or numerical tests data described in connection with Equations (29) to (41) in Sections 4 and 5 of the Kekatos paper and Appendix A1 of the ’866 Application.

[0049] The signal generator control module 124 can be embodied as one or more software applications or services executing on the computing device 102. The signal generator control module 124 can be executed by the processor 112 to control or otherwise cause the signal generator 108 to generate and transmit various signals to the quantum computing device 106. For example, the signal generator control module 124 can cause the signal generator 108 to generate and transmit microwave and low-frequency electrical signals (e.g., pulses) to the quantum computing device 106. The signal generator control module 124 can cause the signal generator 108 to generate and transmit a resonant microwave pulse in some cases to drive a quantum operation or a quantum circuit executed by the quantum computing device 106. The signal generator control module 124 can cause the signal generator 108 to generate and transmit a probing pulse in other examples to retrieve data indicative of the results observed from performing such an operation.

[0050] In one example, the signal generator control module 124 can cause the signal generator 108 to generate and transmit to the quantum computing device 106 one or more signals that are indicative of or allow for execution of the aforementioned parameterized variational quantum circuit. For instance, the signal generator control module 124 can cause the signal generator 108 to generate and transmit to the quantum computing device 106 one or more signals indicative of or allow for execution of the aforementioned sequence of parameterized quantum gates. The sequence of parameterized quantum gates being indicative of at least one of a cost function, a constraint function, a Lagrangian function, or another function that defines in part or is otherwise associated with a particular constrained optimization problem. Once the quantum computing device 106 executes the sequence of parameterized quantum gates using the provided signals, the signal generator control module 124 can then cause the signal generator 108 to generate andAttorney Docket: 222204-2970 transmit a probing signal (e.g., pulse) to the quantum computing device 106 to retrieve data indicative of the results observed from executing the sequence of parameterized quantum gates. Based at least in part on such results data, the computing device 102 can generate performance data or numerical tests data described in connection with Equations (29) to (41) in Sections 4 and 5 of the Kekatos paper and Appendix A1 of the ’866 Application.

[0051] The communications stack 126 can include software and hardware layers to implement data communications such as, for instance, Bluetooth®, Bluetooth® Low Energy (BLE), WiFi®, cellular data communications interfaces, or a combination thereof. Thus, the communications stack 126 can be relied upon by the computing device 102 to establish cellular, Bluetooth®, WiFi®, and other communications channels with the networks 110 and with at least one of the remote computing devices 104 or the quantum computing device 106.

[0052] The communications stack 126 can include the software and hardware to implement Bluetooth®, BLE, and related networking interfaces, which provide for a variety of different network configurations and flexible networking protocols for short-range, low-power wireless communications. The communications stack 126 can also include the software and hardware to implement WiFi® communication, and cellular communication, which also offers a variety of different network configurations and flexible networking protocols for mid-range, long-range, wireless, and cellular communications. The communications stack 126 can also incorporate the software and hardware to implement other communications interfaces, such as X10®, ZigBee®, Z-Wave®, and others.

[0053] The communications stack 126 can be configured to communicate various data or information amongst the computing device 102, the remote computing devices 104, and the quantum computing device 106. Examples of such data or information can include, but is not limited to, at least one of data indicative of quantum related aspects (e.g., quantum dynamics, features, or properties) associated with various quantum computing devices or quantum operations to be performed, data indicative of constrained optimization problems, data indicative of constraints associated with constrained optimization problems, data indicative of functions (e.g., cost functions, constraint functions, Lagrangian functions) defining constrained optimization problems, data indicative of parameterized variational quantum circuits corresponding to constrained optimization problems, data indicative of quantum observables corresponding to quantum states resulting from execution of parameterized variational quantum circuits (e.g., readouts or measurements of observed quantum states or properties), data indicative of vectors of optimization variables or optimization variables associated with functions defining constrained optimization problems, data indicative of solutions to constrained optimization problems solvedAttorney Docket: 222204-2970 according to embodiments described herein, data indicative of empirical or simulated test results obtained from simulating a parameterized variational quantum circuit described herein, or other data indicative of other classical or quantum related computing aspects of the embodiments herein.

[0054] In some cases, the computing device 102 can implement the VQEC framework described herein as a service. For instance, any of the remote computing devices 104 can send a request (e.g., via the networks 110) to the computing device 102 requesting the computing device 102 to provide an optimal or near-optimal solution to a particular constrained optimization problem as described in examples herein.

[0055] FIG. 2 illustrates a flow diagram of an example data flow 200 between example classical and quantum computing devices of the example environment 100 shown in FIG. 1 according to various aspects and embodiments of the present disclosure. In the example shown, the data flow 200 is between the computing device 102 and the quantum computing device 106.

[0056] In several examples, the computing device 102 can implement the VQEC module 120 to perform the data flow 200 iteratively until optimization variables indicative of a solution to a constrained optimization problem are determined. The VQEC module 120 can implement the data flow 200 while running a primal-dual (PD) method or perturbed primal-dual (PPD) method described in examples herein. The VQEC module 120 can implement the data flow 200 while running the PD or PPD method to solve a constrained optimization problem through a variational quantum approach as described in embodiments herein.

[0057] At VQC implementation 210, the VQEC module 120 can encode a VQC 212 with aunitary matrix S( ) in the example shown. The VQEC module 120 can also employ the quantumcomputing device 106 to implement the encoded VQC 212 in a quantum computing environment at VQC implementation 210.

[0058] At measurement 220, the VQEC module 120 can instruct the quantum computing device 106 to measure quantum observables of a quantum state generated at least in part from implementing the encoded VQC 212. The VQEC module 120 can also instruct the quantum computing device 106 to provide the computing device 102 with data indicative of such quantum observables at measurement 220.

[0059] At optimization 230, the VQEC module 120 can update primal and dual variables 234 using the quantum observables data measured at measurement 220. For instance, the VQEC module 120 can update the primal and dual variables 234 using a gradient of a Lagrangian function 232 with respect to the primal and dual variables 234. The VQEC module 120 can update the primal and dual variables 234 individually or simultaneously in many cases using at least one of a PD method, a PPD method, or a PD or PPD method based on a parameter shift rule. The VQECAttorney Docket: 222204-2970 module 120 can perform the data flow 200 iteratively until the primal and dual variables 234 are determined to be indicative of a solution to a constrained optimization problem.

[0060] The computing device 102 can implement the VQEC module 120 to provide optimal or near-optimal solutions to various constraint optimization problems with minimal computational overhead compared to existing, unconstrained VQE approaches. For instance, when the VQEC module 120 facilitates simultaneous measurement or estimation of all involved quantum observablesthe computational overhead is insignificant and may be confined only to the computing device 102.

[0061] FIG. 3 illustrates a flow diagram of an example computer-implemented method 300 (or “method 300”) according to various aspects and embodiments of the present disclosure. The method 300 can be implemented by the computing device 102 using the VQEC module 120 and at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 in one example. The computing device 102 can implement the method 300 iteratively to ultimately determine optimization variables indicative of a solution to a constrained optimization problem.

[0062] At 302, the method 300 includes implementing a parameterized VQC in a quantum computing environment where the parameterized VQC is indicative of and corresponds to a function such as a Lagrangian function defining a constrained optimization problem. For example, at 302 the computing device 102 can implement the VQEC module 120 to define a sequence of parameterized quantum gates that represents and is indicative of such a parameterized VQC where the sequence of parameterized quantum gates is executable by a quantum computing device in a quantum computing environment. The computing device 102 can further employ the VQEC module 120 to implement the sequence of parameterized quantum gates in a quantum computing environment to model the constrained optimization problem in such an environment. For instance, the VQEC module 120 can employ at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 to implement the sequence of parameterized quantum gates in a quantum computing environment such as a simulated quantum computing environment or a quantum computing environment included in or generated by the quantum computing device 106. The parameterized VQC in some cases can be encoded with or include data indicative of at least one of primary optimization or dual variables or parameters of a constrained optimization problem with the dual variables or parameters corresponding to constraints of the problem.

[0063] At 304, the method 300 includes measuring quantum observables defining a quantum state generated from execution of the parameterized VQC. For example, at 304 the computingAttorney Docket: 222204-2970 device 102 can implement the VQEC module 120 to measure quantum observables corresponding to and defining a quantum state created or resulting at least partly from implementation of the sequence of parameterized quantum gates in the quantum computing environment at 302. The VQEC module 120 can employ at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 in many cases to capture such quantum observable measurements from a simulated quantum computing environment or a quantum computing environment included in or generated by the quantum computing device 106.

[0064] Similar to the parameterized VQC, the quantum observables resulting from the implementation of the parameterized VQC can also be indicative of and correspond to the aforementioned function that defines a constrained optimization problem. In some examples, the function can be at least one of a cost function or a constraint function associated with the constrained optimization problem. The quantum observables can include cost quantum observables and constraint quantum observables defined by at least one Hermitian matrix in one example. In another example, the quantum observables can include diagonal quantum observables defined over at least one diagonal Hermitian matrix. In some examples, the VQEC module 120 can employ at least one of the quantum simulator module 122, the signal generator control module 124, the signal generator 108, or the quantum computing device 106 to simultaneously measure cost quantum observables and constraint quantum observables based at least in part on implementation of the sequence of parameterized quantum gates in the quantum computing environment described above.

[0065] At 306, the method 300 includes performing a primal-dual method to update primal optimization variables and dual variables. For example, the computing device 102 can implement the VQEC module 120 at 306 to update at least one of a vector of primal optimization variables or a vector of dual variables by performing calculations using data indicative of quantum observables measured at 304. Vectors of primal optimization variables can correspond to and be extracted or calculated from primary optimization or cost quantum observables in many cases. Vectors of dual variables can correspond to and be extracted or calculated from cost or constraint quantum observables in some examples. Primal and dual variables corresponding to constraints of a constrained optimization problem can be updated at each iteration of 306 for the next iteration of the method 300 by performing a primal-dual method in many cases. For example, the computing device 102 can implement the VQEC module 120 at 306 to perform a primal-dual method such as a primal-dual perturbation method based on a parameter shift rule. The VQEC module 120 can perform such a PD or PPD method to update a vector of the primal optimization variables and aAttorney Docket: 222204-2970 vector of the dual variables in many examples. For an initial iteration in some examples, a vectorof the primal variables can be drawn uniformly within 0, 2 , and a vector of the dual variablescan be initialized at 0. Such a PD or PPD method can be performed at 306 to identify a saddle point of the function (e.g., Lagrangian function) that defines the constrained optimization problem and to estimate updated values for the primal optimization and dual variables if the saddle point is not identified.

[0066] The saddle point in many cases is indicative of a pair of primal-dual vectors that corresponds to an optimal value of the function defining the constrained optimization problem. The optimal value of the function can include optimization variables of the function. For instance, the optimal value can include optimal values for the primal optimization and dual variables. The optimal value and optimization variables of the function can both be indicative of a solution for the function and the constrained optimization problem in many cases.

[0067] At 308, the method 300 includes determining whether a saddle point of the function (e.g., Lagrangian function) that defines the constrained optimization problem was identified when performing the primal-dual method at 306. If it is determined that a saddle point was identified at 306, then at 310 the VQEC module 120 can output optimization variables that are indicative of a solution for the function and the constrained optimization problem. For instance, the optimization variables can include optimal values for the primal optimization and dual variables. In some cases, the VQEC module 120 can perform one or more operations based at least in part on the optimization variables or solution to the constrained optimization problem output at 306. Example operations include, but are not limited to, at least one of a machine learning or a reinforcement machine learning operation, a wireless communication operation, an asset portfolio optimization operation, an optimal resource allocation operation, or another operation.

[0068] If it is determined that a saddle point was not identified at 306, then at 312 the VQEC module 120 can update parameters of the parameterized VQC and repeat steps 302 to 308 of the method 300 using the updated parameterized VQC. For instance, at 312 the VQEC module 120 can estimate values for the primal optimization and dual variables based at least in part on performing the PD or PPD method at 306. The VQEC module 120 can then use such estimated values to define a new or updated sequence of parameterized quantum gates that represents and is indicative of a new or updated parameterized VQC. The VQEC module 120 can then use such an updated parameterized VQC to repeat steps 302 to 308 of the method 300 until a saddle point of the function is identified at 306.

[0069] Referring now to FIG. 1, an executable program can be stored in any portion or component of the memory 114. The memory 114 can be embodied as, for example, a randomAttorney Docket: 222204-2970 access memory (RAM), read-only memory (ROM), magnetic or other hard disk drive, solid-state, semiconductor, universal serial bus (USB) flash drive, memory card, optical disc (e.g., compact disc (CD) or digital versatile disc (DVD)), floppy disk, magnetic tape, or other types of memory devices.

[0070] The memory 114 can include both volatile and nonvolatile memory and data storage components. Volatile components are those that do not retain data values upon loss of power. Nonvolatile components are those that retain data upon a loss of power. Thus, the memory 114 can include, for example, a RAM, ROM, magnetic or other hard disk drive, solid-state, semiconductor, or similar drive, USB flash drive, memory card accessed via a memory card reader, floppy disk accessed via an associated floppy disk drive, optical disc accessed via an optical disc drive, magnetic tape accessed via an appropriate tape drive, and / or other memory component, or any combination thereof. In addition, the RAM can include, for example, a static random-access memory (SRAM), dynamic random-access memory (DRAM), or magnetic random-access memory (MRAM), and / or other similar memory device. The ROM can include, for example, a programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or other similar memory devices.

[0071] As discussed above, the VQEC module 120, the quantum simulator module 122, the signal generator control module 124, and the communications stack 126 can each be embodied, at least in part, by software or executable-code components for execution by general purpose hardware. Alternatively, the same can be embodied in dedicated hardware or a combination of software, general, specific, and / or dedicated purpose hardware. If embodied in such hardware, each can be implemented as a circuit or state machine, for example, that employs any one of or a combination of a number of technologies. These technologies can include, but are not limited to, discrete logic circuits having logic gates for implementing various logic functions upon an application of one or more data signals, application specific integrated circuits (ASICs) having appropriate logic gates, field-programmable gate arrays (FPGAs), or other components.

[0072] Referring now to FIGS. 2 and 3, each flow diagram shown in FIGS. 2 and 3 is representative of certain processes, functionality, and operations of the embodiments discussed herein. Each block can represent one or a combination of steps or executions in a process. Alternatively, or additionally, each block can represent a module, segment, or portion of code that includes program instructions to implement the specified logical function(s). The program instructions can be embodied in the form of source code that includes human-readable statements written in a programming language or machine code that includes numerical instructionsAttorney Docket: 222204-2970 recognizable by a suitable execution system such as the processor 112 and / or a quantum processor of the quantum computing device 106. The machine code can be converted from the source code. Further, each block can represent, or be connected with, a circuit or a number of interconnected circuits to implement a certain logical function or process step.

[0073] Although the flow diagrams shown in FIGS.2 and 3 each illustrate a specific order, it is understood that the order can differ from that which is depicted. For example, an order of execution of two or more blocks can be scrambled relative to the order shown. Also, two or more blocks shown in succession can be executed concurrently or with partial concurrence. Further, in some embodiments, one or more of the blocks can be skipped or omitted. In addition, any number of counters, state variables, warning semaphores, or messages might be added to the logical flow described herein, for purposes of enhanced utility, accounting, performance measurement, or providing troubleshooting aids. Such variations, as understood for implementing the process consistent with the concepts described herein, are within the scope of the embodiments.

[0074] Also, any logic or application described herein, including the VQEC module 120, the quantum simulator module 122, the signal generator control module 124, and the communications stack 126 can be embodied, at least in part, by software or executable-code components, can be embodied or stored in any tangible or non-transitory computer-readable medium or device for execution by an instruction execution system such as a general-purpose processor. In this sense, the logic can be embodied as, for example, software or executable-code components that can be fetched from the computer-readable medium and executed by the instruction execution system. Thus, the instruction execution system can be directed by execution of the instructions to perform certain processes such as those illustrated in FIGS.2 and 3. In the context of the present disclosure, a non-transitory computer-readable medium can be any tangible medium that can contain, store, or maintain any logic, application, software, or executable-code component described herein for use by or in connection with an instruction execution system.

[0075] The computer-readable medium can include any physical media such as, for example, magnetic, optical, or semiconductor media. More specific examples of suitable computer-readable media include, but are not limited to, magnetic tapes, magnetic floppy diskettes, magnetic hard drives, memory cards, solid-state drives, USB flash drives, or optical discs. Also, the computer- readable medium can include a RAM including, for example, an SRAM, DRAM, or MRAM. In addition, the computer-readable medium can include a ROM, a PROM, an EPROM, an EEPROM, or other similar memory device.

[0076] Disjunctive language, such as the phrase “at least one of X, Y, or Z,” unless specifically stated otherwise, is to be understood with the context as used in general to present thatAttorney Docket: 222204-2970 an item, term, or the like, can be either X, Y, or Z, or any combination thereof (e.g., X, Y, and / or Z). Thus, such disjunctive language is not generally intended to, and should not, imply that certain embodiments require at least one of X, at least one of Y, or at least one of Z to be each present. As referenced herein in the context of quantity, the terms “a” or “an” are intended to mean “at least one” and are not intended to imply “one and only one.”

[0077] As referred to herein, the terms “includes” and “including” are intended to be inclusive in a manner similar to the term “comprising.” As referenced herein, the terms “or” and “and / or” are generally intended to be inclusive, that is (i.e.), “A or B” or “A and / or B” are each intended to mean “A or B or both.” As referred to herein, the terms “first,” “second,” “third,” and so on, can be used interchangeably to distinguish one component or entity from another and are not intended to signify location, functionality, or importance of the individual components or entities. As referenced herein, the terms “couple,” “couples,” “coupled,” and / or “coupling” refer to chemical coupling (e.g., chemical bonding), communicative coupling, electrical and / or electromagnetic coupling (e.g., capacitive coupling, inductive coupling, direct and / or connected coupling), mechanical coupling, operative coupling, optical coupling, and / or physical coupling.

[0078] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications can be made to the above- described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.

Claims

Attorney Docket: 222204-2970 CLAIMS Therefore, at least the following is claimed:

1. A method of solving constrained optimization problems using a variational quantum approach, the method comprising: determining, by a computing device, primal optimization variables for a function defining a constrained optimization problem based at least in part on execution of a parameterized variational quantum circuit corresponding to the function in a quantum computing environment; and determining, by the computing device, optimization variables for the function based at least in part on the primal optimization variables and dual variables associated with constraints of the constrained optimization problem, the optimization variables being indicative of a solution to the constrained optimization problem.

2. The method of claim 1, wherein determining the primal optimization variables comprises: defining, by the computing device, a sequence of parameterized quantum gates that is indicative of the parameterized variational quantum circuit, the sequence of parameterized quantum gates being executable by a quantum computing device in the quantum computing environment; instructing, by the computing device, the quantum computing device to implement the sequence of parameterized quantum gates in the quantum computing environment; and obtaining, by the computing device, quantum observables corresponding to a quantum state prepared based at least in part on implementation of the sequence of parameterized quantum gates in the quantum computing environment, the quantum observables being indicative of the function and the function comprising at least one of a cost function or a constraint function associated with the constrained optimization problem.

3. The method of claim 2, wherein the quantum observables comprise cost quantum observables and constraint quantum observables defined by at least one Hermitian matrix.

4. The method of claim 2, wherein the quantum observables comprise diagonal quantum observables defined over at least one diagonal Hermitian matrix.Attorney Docket: 222204-2970 5. The method of claim 2, wherein the quantum observables comprise cost quantum observables and constraint quantum observables, and wherein instructing the quantum computing device to measure the quantum observables comprises: instructing, by the computing device, the quantum computing device to simultaneously measure the cost quantum observables and the constraint quantum observables based at least in part on implementation of the sequence of parameterized quantum gates in the quantum computing environment.

6. The method of claim 1, further comprising: updating, by the computing device, the primal optimization variables and the dual variables using a primal-dual method.

7. The method of claim 6, wherein the function comprises a Lagrangian function and the primal-dual method comprises: identifying, by the computing device, a saddle point of the Lagrangian function over the dual variables and the primal optimization variables, the saddle point being indicative of a pair of primal-dual vectors that corresponds to an optimal value of the Lagrangian function.

8. The method of claim 1, further comprising: updating, by the computing device, the primal optimization variables and the dual variables iteratively using a primal-dual perturbation method that is based at least in part on a parameter shift rule.

9. The method of claim 8, wherein the primal-dual perturbation method comprises: updating, by the computing device, perturbed primal and dual variables corresponding to the primal optimization variables and the dual variables, respectively.

10. The method of claim 9, wherein the primal-dual perturbation method further comprises: updating, by the computing device, the perturbed primal and dual variables iteratively using a primal-dual method and the parameter shift rule.

11. The method of claim 1, further comprising:Attorney Docket: 222204-2970 bounding, by the computing device, an optimality gap attained by implementation of the parameterized variational quantum circuit, the optimality gap being associated with a distance between a pair of primal-dual vectors that is indicative of a saddle point of the function, the function comprising a Lagrangian function.

12. The method of claim 1, wherein: the function comprises at least one of Lagrangian function, a cost function, or a constraint function associated with the constrained optimization problem; and the dual variables comprise Lagrange multipliers associated with an inequality or equality constraint of the constrained optimization problem.

13. The method of claim 1, wherein the constrained optimization problem comprises a quadratically-constrained binary optimization problem, a problem of designing stochastic binary policies satisfying quadratic constraints on average or in probability, a problem of learning large- scale probability mass functions, or a problem of solving large-scale linear programs over a probability simplex.

14. The method of claim 1, further comprising: performing, by the computing device, at least one operation based at least in part on the solution to the constrained optimization problem, the at least one operation comprising one or more of a machine learning or a reinforcement machine learning operation, a wireless communication operation, an asset portfolio optimization operation, and an optimal resource allocation operation.

15. A computing device, comprising: a memory device to store computer-readable instructions thereon; and at least one processing device configured through execution of the computer-readable instructions to: determine primal optimization variables for a function defining a constrained optimization problem based at least in part on execution of a parameterized variational quantum circuit corresponding to the function in a quantum computing environment; and determine optimization variables for the function based at least in part on the primal optimization variables and dual variables associated with constraints of the constrainedAttorney Docket: 222204-2970 optimization problem, the optimization variables being indicative of a solution to the constrained optimization problem.

16. The computing device of claim 15, wherein the at least one processing device is further configured to: update the primal optimization variables and the dual variables using a primal-dual method.

17. The computing device of claim 16, wherein the at least one processing device is further configured to: identify a saddle point of the function over the dual variables and primal variables corresponding to the primal optimization variables, the saddle point being indicative of a pair of primal-dual vectors that corresponds to an optimal value of the function, and the function comprising a Lagrangian function.

18. The computing device of claim 15, wherein the at least one processing device is further configured to: update the primal optimization variables and the dual variables iteratively using a primal- dual perturbation method that is based at least in part on a parameter shift rule.

19. The computing device of claim 18, wherein the at least one processing device is further configured to: update perturbed primal and dual variables corresponding to the primal optimization variables and the dual variables, respectively.

20. A computing system, comprising: a quantum computing device configured to execute a parameterized variational quantum circuit in a quantum computing environment, the parameterized variational quantum circuit corresponding to a function defining a constrained optimization problem; and a computing device configured to: determine primal optimization variables for the function based at least in part on execution of the parameterized variational quantum circuit in the quantum computing environment; and determine optimization variables for the function based at least in part on the primal optimization variables and dual variables associated with constraints of the constrainedAttorney Docket: 222204-2970 optimization problem, the optimization variables being indicative of a solution to the constrained optimization problem.

Images