Method and system for solving integer or mixed integer programming problems using circuit-based continuously variable quantum optical devices - Patents.com

Abstract

A method for solving an integer or mixed integer programming problem, the method including: (a) obtaining an indication of a quantum Hamiltonian that represents the integer or mixed integer programming problem; (b) implementing the quantum Hamiltonian in a quantum optical device, the quantum optical device comprising a quantum gate; (c) preparing a quantum state of a quantum-mode system using the quantum optical device; (d) performing measurements of the quantum-mode system; and (e) providing a solution to the integer or mixed integer programming problem based on the measurements.

Description

[Technical field]

[0001] cross reference This application claims the benefit of U.S. Provisional Application No. 63 / 255,777, filed October 14, 2021, and U.S. Provisional Application No. 63 / 217,689, filed July 1, 2021, each of which is incorporated by reference herein for all purposes. [Background technology]

[0002] Integer and mixed integer programming problems may be NP-complete. That is, the time used to solve a problem using classical algorithms or computing platforms may increase rapidly (e.g., exponentially) as the size of the problem increases. Integer programming problems may be of various types, such as integer linear programming or binary optimization. Integer and mixed integer programming have many applications in various fields, such as finance, industrial manufacturing, land parceling, and telecommunication networks. Integer and mixed integer programming may be used to solve problems such as budget planning, maximum clique problems, and scheduling. Summary of the Invention

[0003] Classical algorithms for solving mixed integer programming problems may scale poorly with problem size. Heuristic quantum algorithms may address this scaling issue. Qubit-based quantum algorithms may scale polynomially with size, but may incur resource overhead due to the fact that each of the integer and continuous variables of the problem may need to be cast to multiple qubits as binary variables. By utilizing the internal capabilities of quantum optical devices, our approach may be used to mitigate these issues by directly implementing the problem variables in the quantum modes (qumodes) of the quantum optical devices. Since photon numbers are essentially non-negative integers, the integer variables of the problem may be represented by the photon number operators of the corresponding quantum modes. Photon number operators with only integer eigenvalues ​​may represent the integer variables of the problem without the need for ancilla quantum modes (ancilla qumodes). The fundamental feature of quantum optical devices, that the photon number states are discrete, allows the representation of the integer variables of the problem using the internal capabilities of the quantum optical devices. This approach may efficiently utilize the resources available in the quantum optical devices, which may reduce overhead. Combining this technique with the usual technique of expressing continuous variables using orthogonal operators, quantum optical devices may be able to efficiently solve integer or mixed integer programming problems. Improvements in methods for solving NP-hard problems may improve the field of optimization as a whole. Furthermore, improvements in the implementation of hard computational problems on the architecture of quantum optical devices may improve the quantum optical devices themselves by extending their ability to solve hard computational problems. Furthermore, there remains a need to solve NP-hard problems that may be solved inefficiently by classical devices.

[0004] Recognized herein is a need for improved methods and systems that can improve the time used to solve integer or mixed integer programming problems. The methods and systems disclosed herein can improve upon methods and systems that employ classical algorithms or computing platforms.

[0005] The present disclosure provides a method and system for solving integer or mixed integer programming problems. The present disclosure may improve existing optimization solvers in at least some aspects by using non-classical devices, such as quantum and quantum optical devices. In particular, by directly implementing problem variables (e.g., variables of an optimization problem to be optimized) in the quantum mode of a quantum optical device, the unique capabilities of the quantum optical device may be exploited to solve problems that are inefficiently solved by classical computers.

[0006] In an aspect, the present disclosure provides a method for solving an integer or mixed integer programming problem using a quantum optical device, the method including: (a) obtaining an indication of a quantum Hamiltonian representing the integer or mixed integer programming problem, the quantum Hamiltonian including a class of operators corresponding to variables of the integer or mixed integer programming problem, (b) implementing the quantum Hamiltonian in the quantum optical device, the quantum optical device including at least one quantum gate configured to operate on a system of quantum modes and one or more quantum modes in the system of quantum modes, one or more operators in the class of operators corresponding to the one or more quantum modes in the system of quantum modes, and (c) providing a solution to the integer or mixed integer programming problem based at least in part on a measurement of the system of quantum modes.

[0007] In some embodiments, the class of operators includes a photon number operator of the quantum Hamiltonian. In some embodiments, integer variables of the integer programming problem or the mixed integer programming problem correspond to the photon number operator of the quantum Hamiltonian. In some embodiments, the class of operators includes a momentum operator and a position operator of the quantum Hamiltonian. In some embodiments, the quantum Hamiltonian represents the mixed integer programming problem and continuous variables of the mixed integer programming problem correspond to the momentum and position operators of the quantum Hamiltonian.

[0008] In some embodiments, the quantum state of the quantum mode system overlaps with a ground state of the quantum Hamiltonian. In some embodiments, the measurement includes a measurement corresponding to the photon number operator to obtain a photon number value. In some embodiments, the measurement includes a homodyne measurement corresponding to the position operator and the momentum operator to obtain a position and momentum value. In some embodiments, the homodyne measurement includes homodyne detection using a beam splitter and two photodetectors to measure two out-of-phase components of an optical field.

[0009] In some embodiments, (b) further comprises (i) preparing a quantum state of the quantum mode system using the quantum optical device. In some embodiments, the method further comprises simulating a Hamiltonian of quantum adiabatic evolution. In some embodiments, (i) comprises an approximation of a quantum adiabatic evolution from a ground state of a mixed Hamiltonian to the quantum state of the quantum mode system. In some embodiments, the mixed Hamiltonian comprises an operator that does not commute with a photon number operator. In some embodiments, the mixed Hamiltonian comprises a position operator or a momentum operator as a non-commutative operator with a photon number operator. In some embodiments, the approximation of the quantum adiabatic evolution comprises a discretized quantum adiabatic algorithm (dQAA) procedure. In some embodiments, the approximation of the quantum adiabatic evolution comprises a quantum approximate optimization algorithm (QAOA) procedure. In some embodiments, (i) comprises performing one or more gate operations.

[0010] In some embodiments, (c) comprises (ii) performing said measurement of the system of said quantum modes. In some embodiments, said measurement in (ii) comprises photon-number-resolving measurement of said one or more quantum modes. In some embodiments, (b) includes configuring the quantum optical device, wherein the configuring includes setting one or more of a rotation gate, a Kerr gate, a cross-Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeezing gate, a controlled addition gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross-position-rotation gate, a momentum-rotation gate, a quadratic momentum rotation gate, or a cross-momentum-rotation gate.

[0011] In some embodiments, the at least one quantum gate is implemented using at least one of Kerr nonlinearity, quantum dots, Rydberg blockades, or four-wave mixing atomic systems. In some embodiments, the quantum Hamiltonian is quadratic in the photon number operator. In some embodiments, the quantum Hamiltonian is fourth order in the momentum and position operators.

[0012] In some embodiments, (b) includes (i) preparing a quantum state of the quantum mode system using the quantum optical device, and (c) includes (ii) performing a measurement of the quantum mode system, and (i), (ii), and (c) are repeated one or more times. In some embodiments, (i), (ii), and (c) are repeated one or more times until a convergence condition is satisfied, optionally, the convergence condition includes a threshold number of iterations or a threshold change in an objective function. In some embodiments, the indication of a quantum Hamiltonian is obtained from a user. In some embodiments, the indication of a quantum Hamiltonian is obtained from a computer-implemented method for solving the integer programming problem or the mixed integer programming problem.

[0013] In another aspect, the present disclosure provides a method for solving an integer or mixed integer programming problem using a quantum optical device, which may include, in a digital computer operatively connected to the quantum optical device, (a) obtaining an indication of a quantum Hamiltonian representing the integer or mixed integer programming problem, the quantum Hamiltonian including a class of operators corresponding to variables of the integer or mixed integer programming problem, (b) indicating the quantum Hamiltonian to the quantum optical device, the quantum optical device implementing the quantum Hamiltonian, the quantum optical device comprising a system of quantum modes and at least one quantum gate configured to operate on one or more quantum modes in the system of quantum modes, one or more operators in the class of operators corresponding to the one or more quantum modes in the system of quantum modes, and (c) providing a solution to the integer or mixed integer programming problem based at least in part on a measurement of the system of quantum modes on the quantum optical device.

[0014] In some embodiments, the class of operators includes a photon number operator of the quantum Hamiltonian. In some embodiments, integer variables of the integer programming problem or the mixed integer programming problem correspond to the photon number operator of the quantum Hamiltonian. In some embodiments, the class of operators includes a momentum operator and a position operator of the quantum Hamiltonian. In some embodiments, the quantum Hamiltonian represents the mixed integer programming problem and continuous variables of the mixed integer programming problem correspond to the momentum operator and the position operator of the quantum Hamiltonian.

[0015] In some embodiments, the quantum state of the quantum mode system overlaps with a ground state of the quantum Hamiltonian. In some embodiments, the measurement includes a measurement corresponding to the photon number operator to obtain a photon number value. In some embodiments, the measurement includes a homodyne measurement corresponding to the position operator and the momentum operator to obtain a position and momentum value. In some embodiments, the homodyne measurement includes homodyne detection using a beam splitter and two photodetectors to measure two out-of-phase components of an optical field.

[0016] In some embodiments, (b) comprises (i) preparing a quantum state of the quantum mode system using the quantum optical device. In some embodiments, the method further comprises simulating a Hamiltonian for quantum adiabatic evolution. In some embodiments, (i) comprises an approximation of quantum adiabatic evolution from a ground state of a mixed Hamiltonian to the quantum state of the quantum mode system. In some embodiments, the mixed Hamiltonian comprises an operator that does not commute with a photon number operator. In some embodiments, the mixed Hamiltonian comprises a position operator or a momentum operator as a non-commutative operator with a photon number operator. In some embodiments, the approximation of the quantum adiabatic evolution comprises a discretized quantum adiabatic algorithm (dQAA) procedure. In some embodiments, the approximation of the quantum adiabatic evolution comprises a quantum approximation optimization algorithm (QAOA) procedure. In some embodiments, (i) comprises performing one or more gate operations.

[0017] In some embodiments, (c) includes (ii) performing the measurement of the system of the quantum modes. In some embodiments, the measurement in (ii) includes photon-number resolved measurement of the one or more quantum modes. In some embodiments, (b) includes configuring the quantum optical device, where the configuring includes setting one or more of a rotation gate, a Kerr gate, a cross Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled summation gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, or a cross momentum rotation gate.

[0018] In some embodiments, the at least one quantum gate is implemented using at least one of a Kerr nonlinearity, a quantum dot, a Rydberg cutoff, or a four-wave mixing atomic system. In some embodiments, the quantum Hamiltonian is quadratic in the photon number operator. In some embodiments, the quantum Hamiltonian is fourth order in the momentum and position operators.

[0019] In some embodiments, (b) includes (i) preparing a quantum state of the quantum mode system using the quantum optical device, and (c) includes (ii) performing a measurement of the quantum mode system, and (i), (ii), and (c) are repeated one or more times. In some embodiments, (i), (ii), and (c) are repeated one or more times until a convergence condition is satisfied, optionally, the convergence condition includes a threshold number of iterations or a threshold change in an objective function. In some embodiments, the indication of the quantum Hamiltonian is obtained from a user. In some embodiments, the indication of the quantum Hamiltonian is obtained from a computer-implemented method for solving the integer programming problem or the mixed integer programming problem.

[0020] In another aspect, the present disclosure provides a quantum optical device comprising a quantum mode system and at least one quantum gate configured to act on one or more quantum modes of the quantum mode system, the at least one quantum gate comprising at least one member of the group including a position rotation gate, a secondary position rotation gate, a cross position rotation gate, a momentum rotation gate, a secondary momentum rotation gate, and a cross momentum rotation gate.

[0021] In another aspect, the present disclosure provides a system for solving integer or mixed integer programming problems using a quantum optical device, which may include: (a) a quantum optical device comprising a control system, a quantum mode system, and at least one quantum gate configured to act on one or more quantum modes of the quantum mode system, the quantum optical device configured to at least (i) implement a quantum Hamiltonian, (ii) prepare a quantum state of the quantum mode system, and (iii) perform a measurement of the quantum mode system, and (b) a classical computer operably connected to the quantum optical device, the classical computer being a digital computer comprising a memory including instructions, the digital computer configured to execute at least (i) obtain an indication of a quantum Hamiltonian representing an integer or mixed integer programming problem, (ii) provide the instructions to the quantum optical device, and (iii) receive results from the quantum optical device.

[0022] In some embodiments, the quantum Hamiltonian includes a class of operators corresponding to variables of the integer programming problem or the mixed integer programming problem, and one or more operators in the class of operators correspond to the one or more quantum modes of the quantum mode system. In some embodiments, the class of operators includes a photon number operator of the quantum Hamiltonian. In some embodiments, integer variables of the integer programming problem or the mixed integer programming problem correspond to the photon number operator of the quantum Hamiltonian. In some embodiments, the class of operators includes a momentum operator and a position operator of the quantum Hamiltonian. In some embodiments, the quantum Hamiltonian represents the mixed integer programming problem, and continuous variables of the mixed integer programming problem correspond to the momentum operator and the position operator of the quantum Hamiltonian. In some embodiments, the quantum state of the quantum mode system overlaps with a ground state of the quantum Hamiltonian. In some embodiments, the measurement includes a measurement corresponding to the photon number operator to obtain a value of a photon number. In some embodiments, the measurement includes a homodyne measurement corresponding to the position operator and the momentum operator to obtain position and momentum values, hi some embodiments, the homodyne measurement includes homodyne detection using a beam splitter and two photodetectors to measure two out-of-phase components of an optical electric field.

[0023] In some embodiments, the quantum optical device is configured to at least simulate a Hamiltonian of quantum adiabatic evolution. In some embodiments, (a)(ii) comprises an approximation of quantum adiabatic evolution from a ground state of a mixed Hamiltonian to the quantum state of the system of the quantum modes. In some embodiments, the mixed Hamiltonian comprises an operator that does not commute with a photon number operator. In some embodiments, the mixed Hamiltonian comprises a position operator or a momentum operator as a non-commutative operator with a photon number operator. In some embodiments, the approximation of the quantum adiabatic evolution comprises a discretized quantum adiabatic algorithm (dQAA) procedure. In some embodiments, the approximation of the quantum adiabatic evolution comprises a quantum approximation optimization algorithm (QAOA) procedure. In some embodiments, (a)(ii) comprises one or more gate operations. In some embodiments, the measurement in (a)(iii) comprises a photon-number resolved measurement of the one or more quantum modes. In some embodiments, (b)(ii) includes configuring the quantum optical device, wherein the configuring includes setting one or more of a rotation gate, a Kerr gate, a cross Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled summation gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, or a cross momentum rotation gate.

[0024] In some embodiments, the at least one quantum gate is implemented using at least one of Kerr nonlinearity, quantum dots, Rydberg cutoff, or four-wave mixing atomic systems. In some embodiments, the quantum Hamiltonian is quadratic in the photon number operator. In some embodiments, the quantum Hamiltonian is fourth order in the momentum and position operators. In some embodiments, (a) is repeated one or more times. In some embodiments, (a) is repeated one or more times until a convergence condition is satisfied, optionally, the convergence condition includes a threshold number of iterations or a threshold change in an objective function. In some embodiments, the indication of a quantum Hamiltonian is obtained from a user. In some embodiments, the indication of a quantum Hamiltonian is obtained from a computer-implemented method for solving the integer programming problem or the mixed integer programming problem.

[0025] In another aspect, the present disclosure provides a system for solving integer or mixed integer programming problems using quantum optical devices, which may include a quantum optical device including a quantum mode and at least one quantum gate configured to act on one or more quantum modes of a system of the quantum modes, the quantum optical device configured to implement a quantum Hamiltonian, the quantum Hamiltonian representing an integer or mixed integer programming problem, the quantum Hamiltonian including a class of operators corresponding to variables of the integer or mixed integer programming problem, and one or more operators in the class of operators corresponding to the one or more quantum modes of the system of the quantum modes.

[0026] In some embodiments, the quantum optical device further comprises a control system configured to receive instructions from a digital computer operably coupled to the quantum optical device, the digital computer comprising a memory containing instructions, the digital computer configured to execute the instructions to at least (i) obtain an indication of the quantum Hamiltonian representing the integer programming problem or the mixed integer programming problem, (ii) implement the quantum Hamiltonian in the quantum optical device, and (iii) receive results from the quantum optical device, the results being based at least in part on measurements of the quantum mode of the system on the quantum optical device.

[0027] In some embodiments, the class of operators includes a photon number operator of the quantum Hamiltonian. In some embodiments, integer variables of the integer programming problem or the mixed integer programming problem correspond to the photon number operator of the quantum Hamiltonian. In some embodiments, the class of operators includes a momentum operator and a position operator of the quantum Hamiltonian. In some embodiments, the quantum Hamiltonian represents the mixed integer programming problem, and continuous variables of the mixed integer programming problem correspond to the momentum operator and the position operator of the quantum Hamiltonian. In some embodiments, the quantum state of the quantum mode system overlaps with a ground state of the quantum Hamiltonian.

[0028] In some embodiments, the measurement includes a measurement corresponding to the photon number operator to obtain a photon number value. In some embodiments, the measurement includes a homodyne measurement corresponding to the position operator and the momentum operator to obtain a position and momentum value. In some embodiments, the homodyne measurement includes homodyne detection using a beam splitter and two photodetectors to measure two out-of-phase components of the optical electric field.

[0029] In some embodiments, the quantum optical device is configured to prepare a quantum state of the system of the quantum mode. In some embodiments, the quantum optical device is configured to simulate a quantum adiabatic evolution Hamiltonian. In some embodiments, the quantum optical device is configured to implement an approximation of a quantum adiabatic evolution from a ground state of a mixed Hamiltonian to the quantum state of the system of the quantum mode. In some embodiments, the mixed Hamiltonian includes an operator that does not commute with a photon number operator. In some embodiments, the mixed Hamiltonian includes a position operator or a momentum operator as a non-commutative operator with a photon number operator. In some embodiments, the approximation of the quantum adiabatic evolution includes a discretized quantum adiabatic algorithm (dQAA) procedure. In some embodiments, the approximation of the quantum adiabatic evolution includes a quantum approximation optimization algorithm (QAOA) procedure.

[0030] In some embodiments, the quantum optical device is configured to perform one or more gate operations. In some embodiments, the quantum optical device is configured to perform the measurement of the system of the quantum modes. In some embodiments, the measurement comprises a photon number resolved measurement of the one or more quantum modes.

[0031] In some embodiments, the control system is configured to configure the quantum optical device, and the configuring includes setting one or more of a rotation gate, a Kerr gate, a cross Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled summation gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, or a cross momentum rotation gate.

[0032] In some embodiments, the at least one quantum gate is implemented using at least one of a Kerr nonlinearity, a quantum dot, a Rydberg cutoff, or a four-wave mixing atomic system. In some embodiments, the quantum Hamiltonian is quadratic in the photon number operator. In some embodiments, the quantum Hamiltonian is fourth order in the momentum and position operators.

[0033] In some embodiments, the quantum optical device is configured to (i) prepare a quantum state of the quantum mode system, and (ii) perform a measurement of the quantum mode system, and the digital computer is configured to (iii) provide the solution of the integer programming problem or the mixed integer programming problem based at least in part on the measurement of the quantum mode system, and (i), (ii), and (iii) are repeated one or more times. In some embodiments, (i), (ii), and (iii) are repeated one or more times until a convergence condition is satisfied, optionally, the convergence condition includes a threshold number of iterations or a threshold change in an objective function. In some embodiments, the indication of a quantum Hamiltonian is obtained from a user. In some embodiments, the indication of a quantum Hamiltonian is obtained from a computer-implemented method for solving the integer programming problem or the mixed integer programming problem.

[0034] In another aspect, the present disclosure provides a system for solving integer or mixed integer programming problems using a quantum optical device, which may include a digital computer operably connected to the quantum optical device, the digital computer comprising a memory including instructions, the digital computer configured to execute the instructions to at least (i) obtain an indication of a quantum Hamiltonian representing the integer or mixed integer programming problem, (ii) indicate the quantum Hamiltonian to a quantum optical device, and (iii) receive a result from the quantum optical device, the quantum optical device implementing the quantum Hamiltonian, the quantum optical device comprising a system of quantum modes and at least one quantum gate configured to act on one or more quantum modes in the system of quantum modes, one or more operators in the class of operators correspond to the one or more quantum modes in the system of quantum modes, and the result is based at least in part on a measurement of the system of quantum modes on the quantum optical device.

[0035] In some embodiments, the system further comprises a quantum optical device comprising a quantum mode system and at least one quantum gate configured to act on one or more quantum modes of the quantum mode system. In some embodiments, the quantum optical device further comprises a control system, the control system configured to receive the instructions from the digital computer operably coupled to the quantum optical device. In some embodiments, the class of operators comprises a photon number operator of the quantum Hamiltonian. In some embodiments, integer variables of the integer programming problem or the mixed integer programming problem correspond to the photon number operator of the quantum Hamiltonian. In some embodiments, the class of operators comprises a momentum operator and a position operator of the quantum Hamiltonian. In some embodiments, the quantum Hamiltonian represents the mixed integer programming problem and continuous variables of the mixed integer programming problem correspond to the momentum operator and the position operator of the quantum Hamiltonian.

[0036] In some embodiments, the quantum state of the quantum mode system overlaps with a ground state of the quantum Hamiltonian. In some embodiments, the measurement includes a measurement corresponding to the photon number operator to obtain a photon number value. In some embodiments, the measurement includes a homodyne measurement corresponding to the position operator and the momentum operator to obtain a position and momentum value. In some embodiments, the homodyne measurement includes homodyne detection using a beam splitter and two photodetectors to measure two out-of-phase components of an optical field.

[0037] In some embodiments, the quantum optical device prepares a quantum state of the system of the quantum mode. In some embodiments, the quantum optical device simulates a quantum adiabatic evolution Hamiltonian. In some embodiments, the quantum optical device implements an approximation of a quantum adiabatic evolution from a ground state of a mixed Hamiltonian to the quantum state of the system of the quantum mode. In some embodiments, the mixed Hamiltonian includes an operator that does not commute with a photon number operator. In some embodiments, the mixed Hamiltonian includes a position operator or a momentum operator as a non-commutative operator with a photon number operator. In some embodiments, the approximation of the quantum adiabatic evolution includes a discretized quantum adiabatic algorithm (dQAA) procedure. In some embodiments, the approximation of the quantum adiabatic evolution includes a quantum approximation optimization algorithm (QAOA) procedure.

[0038] In some embodiments, the quantum optical device performs one or more gate operations. In some embodiments, the quantum optical device performs the measurement of the system of the quantum modes. In some embodiments, the measurement includes a photon number resolved measurement of the one or more quantum modes. In some embodiments, the digital computer is configured to direct a control system to configure the quantum optical device, the configuring including setting one or more of a rotation gate, a Kerr gate, a cross Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled summation gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, or a cross momentum rotation gate.

[0039] In some embodiments, the at least one quantum gate is implemented using at least one of Kerr nonlinearity, quantum dots, Rydberg cutoff, or four-wave mixing atomic systems. In some embodiments, the quantum Hamiltonian is quadratic in the photon number operator. In some embodiments, the quantum Hamiltonian is fourth order in the momentum and position operators. In some embodiments, the quantum optical device is configured to (i) prepare a quantum state of the quantum mode system and (ii) perform a measurement of the quantum mode system, and the digital computer (iii) provide a solution to the integer programming problem or the mixed integer programming problem based at least in part on the measurement of the quantum mode system. In some embodiments, (i), (ii), and (iii) are repeated one or more times. In some embodiments, (i), (ii), and (iii) are repeated one or more times until a convergence condition is satisfied, optionally the convergence condition includes a threshold number of iterations or a threshold change in an objective function. In some embodiments, the indication of the quantum Hamiltonian is obtained from a user. In some embodiments, the indication of a quantum Hamiltonian is obtained from a computer-implemented method for solving the integer programming problem or the mixed integer programming problem.

[0040] Another aspect of the present disclosure provides a system comprising one or more computer processors and a computer memory coupled thereto, the computer memory comprising machine executable code that, when executed by the one or more computer processors, performs any of the methods disclosed elsewhere herein.

[0041] Further aspects and advantages of the present disclosure will become readily apparent to those skilled in the art from the following detailed description, in which only exemplary embodiments of the present disclosure are shown and described. As will be understood, the present disclosure can be realized in other different embodiments, and its several details can be modified in various obvious respects, all without departing from the present disclosure. Thus, the drawings and description should be regarded as illustrative in nature, and not as restrictive.

[0042] Incorporation by Reference All publications, patents, and patent applications mentioned herein are incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent that the publications and patents or patent applications incorporated by reference conflict with the disclosure contained herein, the present specification is intended to supersede and / or take precedence over any such conflicting material. [Brief description of the drawings]

[0043] The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also referred to herein as "Figure" and "FIG"), in which: [Figure 1] FIG. 1 is a schematic diagram of an example of a system for solving integer or mixed integer programming problems using quantum optical devices according to some embodiments disclosed herein. [Diagram 2] 1 is a flowchart of an example of a method for solving integer programming problems using quantum optical devices according to some embodiments disclosed herein. [Diagram 3] 1 is a flowchart of an example of a method for solving a mixed integer programming problem using a quantum optical device according to some embodiments disclosed herein. Detailed Description of the Invention

[0044] While various embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It is understood that various alternatives to the embodiments of the present invention described herein may be employed.

[0045] Unless otherwise defined, all technical terms used herein have the same meaning as commonly understood by those skilled in the art to which this invention belongs. As used in this specification and the appended claims, the singular forms "a", "an" and "the" include plural references unless the context clearly dictates otherwise. Any reference to "or" in this specification is intended to include "and / or" unless specifically stated otherwise.

[0046] The term "plurality" generally refers to "two or more" unless specifically stated otherwise.

[0047] The term "eg" and similar terms mean "for example," and thus do not limit the term or phrase they describe. For example, in the sentence "a computer transmits data (e.g., instructions, data structures) over the Internet," the term "eg" explains that "instructions" are an example of "data" that a computer can transmit over the Internet, and also explains that "data structures" are an example of "data" that a computer can transmit over the Internet. However, both "instructions" and "data structures" are merely examples of "data," and things other than "instructions" and "data structures" may be "data."

[0048] Whenever the terms "at least," "greater than," or "greater than or equal to" precede a first number in a series of two or more numerical values, the terms "at least," "greater than," or "greater than or equal to" apply to each and every number in the series. For example, 1, 2, or 3 or more is equivalent to 1 or more, 2 or more, or 3 or more.

[0049] Whenever the term "no more than," "less than," or "less than or equal to" precedes a first number in a series of two or more numbers, the term "no more than," "less than," or "less than or equal to" applies to each and every number in the series. For example, 3, 2, or 1 or less is equivalent to 3 or less, 2 or less, or 1 or less.

[0050] Certain inventive embodiments herein contemplate numerical ranges. When a range exists, the range includes the end points of the range. Furthermore, all subranges and values ​​within the range exist as if explicitly written out. The term "about" or "approximately" can mean within an acceptable error range for a particular value, which depends in part on how the value is measured or determined, e.g., on the limitations of the measurement system. For example, "about" can mean within one or more standard deviations, in accordance with the practice in the art. Alternatively, "about" can mean within a range of up to 20%, up to 10%, up to 5%, or up to 1% of a given value.

[0051] In the following detailed description, reference is made to the accompanying drawings, which form a part of this specification. In the drawings, similar symbols typically identify similar components unless the context dictates otherwise. The exemplary embodiments described in the detailed description, drawings, and claims are not meant to be limiting. Other embodiments may be used and other changes may be made without departing from the scope of the subject matter presented herein. It will be readily understood that the aspects of the present disclosure, as generally described herein and illustrated in the drawings, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations, all of which are expressly contemplated herein.

[0052] "Qubit," an abbreviation for "quantum bit," commonly refers to the basic unit of quantum information.

[0053] As used herein, the term "quantum gate" generally refers to one of a set of operations in a gate model quantum computer. The term "quantum gate" may also refer to a logical operator comprising one or more quantum bits that may be used to perform a logical operation. A quantum gate may be a physical device that transforms the quantum state of its input according to a unitary transformation that describes the specific operation of the quantum gate.

[0054] As used herein, the term "quantum computing" generally refers to a computational method that utilizes the concepts of quantum superposition and quantum entanglement to manipulate information instead of the binary bits of 0 and 1 in classical computers. Quantum entanglement generally refers to the phenomenon whereby when multiple quantum bits interact with each other, their quantum states become "entangled" and can no longer be represented individually. Quantum superposition generally refers to the principle that the quantum state of a quantum bit can be represented by adding two or more different quantum states, each associated with a probability. In some cases, the probabilities of all states add up to 1. Quantum circuits, consisting of one or more quantum gates, can be designed to perform quantum computations, such as large-scale prime factorization, which may be infeasible or highly inefficient for classical computers.

[0055] As used herein, the term "classical" as used in the context of computing or computation generally refers to computations performed using binary values ​​using discrete bits without the use of quantum mechanical superposition and quantum mechanical entanglement. A classical computer may be a digital computer, such as, for example, a computer that employs discrete bits (e.g., 0 and 1) without the use of quantum mechanical superposition and quantum mechanical entanglement.

[0056] As used herein, the term "non-classical" when used in the context of computing or computation generally refers to any method or system for performing computational procedures outside the paradigm of classical computing.

[0057] As used herein, the term "quantum mode" generally refers to a quantum state expressed as a quantum mode. A quantum mode can be an optical mode expressed as a superposition of a set of possible quantum photon number states. An optical mode can be a single optical mode of a quantum optical device as described herein. A superposition can be an infinite superposition of all possible quantum photon number states. A quantum mode can be an alternative to representing quantum information in quantum bits or qudits. A quantum bit can be a discrete packet of information that represents a binary value, e.g., a superposition of 0 and 1.

[0058] As used herein, the term "quantum optical device" generally refers to a non-classical computer that is an optical computing device. An optical computing device may use light for data processing, data storage, or data communication. For example, in an optical computing device, photons may be used as information carriers. A quantum optical device may be a device that includes optical elements that can act as quantum gates for photons. In some cases, the optical elements may maintain quantum coherence between different quantum states. In some cases, the inputs and outputs of a quantum optical device are quantum modes. In some cases, the quantum optical device is an optical quantum computer, such as, for example, a linear optical quantum computer or a nonlinear optical quantum computer.

[0059] As used herein, the term "integer programming problem" generally refers to a mathematical problem in which the values ​​of variables are integers. Integer programming problems may include variables that are constrained to be integers.

[0060] As used herein, the term "mixed-integer programming problem" generally refers to a mathematical problem in which at least one variable is an integer and the rest of the variables can take on continuous values. A mixed-integer programming problem may include at least one variable that is constrained to be an integer.

[0061] An optimization problem may be a problem of finding an improved solution from a set of feasible solutions. In some cases, the improved solution may be a "best" or "optimum" solution, such as a minimum or maximum. The problem of finding an improved solution may involve minimizing or maximizing an objective function that is a function of a set of variables. The minimum may be a local minimum or a global minimum. The maximum may be a local maximum or a global maximum. However, in many cases, a solution that reasonably approximates the "best" or "optimum" solution may be a sufficient solution. In some cases, what constitutes a reasonable approximation may be contingent on conditions set by a user. For example, an optimization problem may be solved subject to conditions that may set bounds on whether a solution is sufficient, or a reasonable approximation of an "optimal" solution. In some cases, an optimization problem may employ a heuristic. The solution to the heuristic may be a reasonable approximation rather than the "best" of all possible solutions. Heuristics may be used alone or in combination with formally "exact" optimization algorithms.

[0062] As used herein, the term "integer optimization problem" generally refers to a class of integer programming problems in which the mathematical problem is an optimization problem. Integer and mixed integer programming problems can be expressed by objective functions that are functions of integer or continuous variables or both. In some cases, the point at which the objective function is minimized refers to the solution of the problem.

[0063] As used herein, the term "mixed-integer optimization problem" generally refers to a class of mixed-integer programming problems in which the mathematical problem is an optimization problem.

[0064] There are various algorithms and methods for solving integer or mixed integer programming problems. Systems and methods for solving integer or mixed integer programming problems can be exact. Systems and methods for solving integer or mixed integer programming problems can include various heuristics that may not be exact. Even medium-sized cases can be intractable for exact solvers. Some heuristic methods may be efficient for solving some optimization problems, while they may exploit a particular structure of the class of problems being solved, and therefore may have limited application. Further details can be found in "Integer programming" by Conforti et al., vol. 271, Berlin: Springer, 2014, or "Integer programming with a fixed number of variables" by Lenstra Jr. et al., Mathematics of operations research 8, no. 4: pp. 538-548, 1983, or "Tabu search - part II" by Glover et al., ORSA Journal on computing 2, no. 1: pp. 4-32, 1990, each of which is incorporated herein by reference for all purposes.

[0065] As used herein, the term “photon-number operators” generally refers to quantum operators that act on photon quantum states whose eigenstates may be photon-number quantum states and whose eigenvalues ​​may give the number of photons in a given quantum number state.

[0066] Neither the title nor the Abstract should be construed as limiting the scope of the disclosed invention in any way. The title of this application and the section headings provided in this application are for convenience only and should not be construed as limiting the disclosure in any way.

[0067] An advantage of some examples of the methods and systems disclosed herein is that the solutions provided by the quantum optical devices are integers because the photon numbers are integers, which may eliminate the need to round the solutions to the nearest integer, which may introduce additional errors and inaccuracies.

[0068] Another advantage of some examples of the methods and systems disclosed herein is that quantum optical devices can heuristically find solutions to quantum computational methods in polynomial time. For many of the problems studied, the number of gate operations can scale polynomially as the problem size increases. Polynomial scaling can be advantageous over other methods in that integer programming problems are generally NP-complete and scale exponentially.

[0069] Another advantage of some examples of the methods and systems disclosed herein is that the solutions provided by the quantum optical device may include both integers and continuous numbers, which may allow for solving mixed-integer programming problems without the need for rounding or approximations.

[0070] The methods and systems disclosed herein utilize the internal functionality of quantum optical devices to solve integer or mixed integer programming problems. The internal functionality refers to quantum optical devices that are essentially bosonic systems, which are represented by an infinite-dimensional Hilbert space with photon number states as a basis. Photons in bosonic quantum optical devices follow Bose-Einstein statistics, which means that multiple photons can exist in the same quantum state simultaneously. The bosonic nature of quantum optical devices can be represented by writing the quantum states as a superposition of all possible photon number states, infinitely, as described elsewhere herein. Since photon numbers are essentially non-negative integers, integer variables of the problem can be represented by photon number operators of the corresponding quantum modes. Photon number operators with only integer eigenvalues ​​can represent integer variables of the problem without the need for auxiliary quantum modes. The fundamental feature of quantum optical devices, that photon number states are discrete, allows for the representation of integer variables of the problem using the internal functionality of quantum optical devices. This approach may efficiently utilize resources available in quantum optical devices, which may reduce overhead. By combining this with the conventional approach of representing continuous variables using orthogonal operators, quantum optical devices may be able to efficiently solve integer or mixed integer programming problems.

[0071] Discrete Quantum Adiabatic Algorithm (dQAA) and Quantum Approximate Optimization Algorithm (QAOA) The ground state of the quantum Hamiltonian (treated as the target Hamiltonian) can be found using a discretized quantum adiabatic algorithm (dQAA) or a quantum approximate optimization algorithm (QAOA), which are approximation methods of quantum adiabatic (QA) evolution, in which the Hamiltonian of the system can be adiabatically evolved from a mixed Hamiltonian to a target Hamiltonian, which may represent an integer or mixed integer programming problem.

[0072] The discretized quantum adiabatic algorithm may be a version of the quantum adiabatic evolution that is discretized in time. Changes in the system Hamiltonian may occur in steps where the Hamiltonian may remain constant. The time evolution of the system Hamiltonian may approach the actual quantum adiabatic evolution by increasing the number of steps.

[0073] According to the quantum adiabatic theorem, by initializing the system in the ground state of the mixed Hamiltonian and evolving it according to the QA method, the ground state of the target Hamiltonian can be reached with high probability. In practice, a quantum state that overlaps with at least one of the ground states of the target Hamiltonian can be reached using the quantum adiabatic evolution. In some examples, the degree of overlap can be at least 80% or 90%. In some cases, the overlap can involve two or more ground states. In some cases, the overlap can involve three or more ground states. The existence of overlap means that the quantum state of the target Hamiltonian can be a superposition of multiple eigenstates, but with a large amplitude relative to at least one of the ground states of the target Hamiltonian. By performing Trotterization of the quantum adiabatic evolution (i.e., evolving the quantum state, alternating between the mixed Hamiltonian and the target Hamiltonian, with the evolution duration being determined according to some scheduling), a discretized quantum adiabatic algorithm (dQAA) can be obtained. The performance of the discretized quantum adiabatic algorithm (dQAA) can reach that of the QA method in the limit of infinite discretization. If the number of discretizations is small (e.g., only a few discretizations are performed) and classical algorithms are used to optimize the scheduling of the evolution duration of the mixed Hamiltonian and the target Hamiltonian, the quantum approximate optimization algorithm (QAOA) can be obtained. The discretized quantum adiabatic algorithm (dQAA) and the quantum approximate optimization algorithm (QAOA) can be implemented on a circuit model quantum device.Alternatively, the QA method can be approximated by Hamiltonian simulation techniques, such as Taylor series expansions or qubitization methods (see, e.g., Berry et al., “Simulating Hamiltonian dynamics with a truncated Taylor series,” Physical Review Letters 114, no. 9:090502, 2015, and Low et al., “Hamiltonian simulation by qubitization,” Quantum 3:163, 2019, each of which is incorporated herein by reference for all purposes).

[0074] Quantum Optical Devices The computational tasks discussed herein may be performed by a quantum optical device. The device may prepare initial photonic quantum states, known as quantum modes, and perform a series of quantum gates on these photonic quantum states. These gates may use quantum optical elements that may perform appropriate quantum unitary transformations. The quantum optical device may further perform quantum measurements, such as photon number resolved measurements and homodyne detection, on the quantum modes to determine the final quantum state. This determination may be performed by statistically determining a probability distribution of the final quantum state in the quantum number basis, which may be determined by repeated measurements. Further details may be found in US2019 / 0325589, and in "Applications of near-term photonic quantum computers: software and algorithms" by Bromley et al., Quantum Science and Technology 5, no. 3:034010, 2020, each of which is incorporated herein by reference for all purposes. In Xanadu quantum optical devices, for example, the initial states can be squeezed vacuum states prepared by squeezing on-chip vacuum states using a highly second-order nonlinear optical crystal. An example of such a crystal can be lithium niobate, which is used in quantum optical devices. These squeezed vacuum states can then be traversed by quantum optical gates that can perform predetermined operations. Phase shifters can be implemented, for example, using materials with a temperature-dependent refractive index, which can be manipulated using voltage-controlled heating plates. In some examples, quantum optical gates such as on-chip and fiber optic directional couplers or beam splitters can be used, which can generate quantum superposition between quantum modes.In some instances, the superposition may be between two or more quantum modes.

[0075] Measurements performed on quantum modes may include quantum photon number resolved measurements and homodyne measurements. By measuring the intensity of the optical field and comparing its phase to a reference phase, the results of homodyne measurements can be used to infer the results of performing quantum measurements in the position or momentum basis. Homodyne measurements are continuous measurements that can be performed with high precision, while quantum photon number resolved measurements may be technically more demanding to perform due to the difficulty of detecting single photons. Examples of photon number detectors may include, but are not limited to, single-photon avalanche detectors (SPADs), which can detect whether the photon number is zero or non-zero, and photon-number-resolving detectors (PNRDs), which can report the number of detected photons down to a sub-number of photons. In some examples, a PNRD may output at least 2, at least 3, at least 4, at least 5, at least 6, at least 7, at least 8, at least 9, or at least 10 photons. The control voltage signals for the squeezer and quantum gates, as well as the results of the quantum measurements, can be controlled and collected by a classical computational device such as a field-programmable gate array (FPGA) and converted into appropriate voltages for the optical gates and detectors, among values ​​that make sense in terms of the mathematical description of the computational task.

[0076] Classical Computers In some cases, the systems, media, networks, and methods described herein include a classical computer, or the use thereof. In some cases, the classical computer may include a digital computer. In some cases, the classical computer includes one or more hardware central processing devices (CPU, also referred to herein as "processor" and "computer processor") that perform the functions of the classical computer. This may be a single-core or multi-core processor, or multiple processors for parallel processing. In some cases, the classical computer further includes an operating system (OS) configured to execute executable instructions. The instructions may be directed to the CPU, which may then program or otherwise configure the CPU to perform the methods of the present disclosure. Examples of operations performed by the CPU include fetch, decode, execute, and writeback.

[0077] In some cases, the classical computer is connected to a computer network. In some cases, the classical computer is connected to the Internet to access the World Wide Web. In some cases, the classical computer is connected to one or more computer servers, which may enable distributed computing, such as a cloud computing infrastructure. In some cases, the classical computer is connected to an intranet and / or extranet, or an intranet and / or extranet that communicates with the Internet. In some cases, the classical computer is connected to a data storage device. In some cases, the network is a telecommunications and / or data network. In some cases, the network is a peer-to-peer network, which may enable devices coupled to a computer system to behave as clients or servers.

[0078] In accordance with the description herein, suitable classical computers may include, by way of non-limiting examples, server computers, desktop computers, laptop computers, notebook computers, sub-notebook computers, netbook computers, netpad computers, set-top computers, media streaming devices, handheld computers, Internet appliances, mobile smart phones, tablet computers, personal digital assistants, video game consoles, and vehicles. Smart phones may be suitable for use with the methods and systems described herein. Selected televisions, video players, and digital music players, if they have computer network connectivity, may be suitable for use in the systems and methods described herein. Suitable tablet computers may include those having booklet, slate, and convertible configurations.

[0079] In some cases, classical computers include an operating system configured to execute executable instructions. An operating system may be software, for example, including programs and data, that manages the device's hardware and provides services for the execution of applications. Suitable server operating systems include, by way of non-limiting example, FreeBSD, OpenBSD, NetBSD, Linux, Apple® Mac OS X Server®, Oracle® Solaris®, Windows Server®, and Novell® NetWare®. Suitable personal computer operating systems may include, by way of non-limiting example, UNIX-like operating systems, such as Microsoft® Windows®, Apple® Mac OS X®, Apple® macOS®, UNIX®, and GNU / Linux®. In some cases, the operating system is provided by cloud computing. Suitable mobile smartphone operating systems include, by way of non-limiting example, Nokia® Symbian® OS, Apple® iOS®, Research In Motion® BlackBerry OS®, Google® Android®, Microsoft® Windows Phone® OS, Microsoft® Windows Mobile® OS, Linux®, and Palm® WebOS®.Suitable media streaming device operating systems may include, by way of non-limiting example, Apple TV®, Roku®, Boxee®, Google TV®, Google Chromecast®, Amazon Fire®, and Samsung® HomeSync®. Suitable video game console operating systems may include, by way of non-limiting example, Sony® PS3®, Sony® PS4®, Microsoft® Xbox 360®, Microsoft® Xbox One®, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®.

[0080] In some cases, a classical computer includes a storage device and / or a memory device. In some cases, a storage device and / or a memory device is one or more physical devices used to temporarily or permanently store data or programs. In some cases, a storage device and / or a memory device may have one or more additional data storage units located outside the classical computer, for example, on a remote server in communication with the classical computer through an intranet or the Internet. In some cases, the device is a volatile memory and requires power to maintain the stored information. In some cases, the device is a non-volatile memory and retains the stored information when the classical computer is not powered. In some cases, the non-volatile memory includes a flash memory. In some cases, the non-volatile memory includes a dynamic random access memory (DRAM). In some cases, the non-volatile memory includes a ferroelectric random access memory (FRAM). In some cases, the non-volatile memory includes a phase change random access memory (PRAM). In other embodiments, the device is a storage device, including, by way of non-limiting examples, CD-ROMs, DVDs, flash memory devices, magnetic disk drives, magnetic tape drives, optical disk drives, and cloud computing-based storage. In some cases, the storage device and / or memory device is a combination of devices such as those disclosed herein.

[0081] In some cases, classical computers include a display for transmitting visual information to a user. In some cases, the display is a cathode ray tube (CRT). In some cases, the display is a liquid crystal display (LCD). In some cases, the display is a thin film transistor liquid crystal display (TFT-LCD). In some cases, the display is an organic light emitting diode (OLED) display. In some cases, the OLED display is a passive matrix OLED (PMOLED) display or an active matrix OLED (AMOLED) display. In some cases, the display is a plasma display. In other embodiments, the display is a video projector. In some cases, the display is a combination of devices such as those disclosed herein.

[0082] In some cases, classical computers include input devices for receiving information from a user. In some cases, the input device is a keyboard. In some cases, the input device is a pointing device, including, by way of non-limiting examples, a mouse, a trackball, a trackpad, a joystick, a game controller, or a stylus. In some cases, the input device is a touch screen or a multi-touch screen. In some cases, the input device is a microphone for capturing voice or other sound input. In some cases, the input device is a video camera or other sensor for capturing motion or visual input. In some cases, the input device is a Kinect, Leap Motion, or the like. In some cases, the input device is a combination of devices, such as those disclosed herein.

[0083] 1, there is shown a schematic diagram of an example of a system for solving integer or mixed integer programming problems using a quantum optical device. The system comprises i) a classical computer (8), which in this embodiment is a digital computer, and ii) a quantum optical device (10). The classical computer (8) can be any classical computer disclosed elsewhere herein.

[0084] The quantum optical device 10 includes a quantum optical processor 12 and a control system (e.g., a readout control system) 14. The quantum optical device 10 may be any quantum optical device disclosed elsewhere herein.

[0085] The classical computer (8) comprises a processing device (20), a display device (24), an input device (26), a communication port (28), and a memory (22). The processing device (20), the display device (24), the input device (26), the communication port (28), and the memory (22) can be of various types, such as any of the types disclosed elsewhere herein. The memory (22) contains a computer program executable by the processing device (20). The communication port (28) communicates with the quantum optical device (10) via the readout control system (14).

[0086] 2, a flow chart of an example of a method for solving integer programming problems using quantum optical devices is shown, the example including quantum gates that may correspond to photon number operators.

[0087] According to processing operations (202), an indication of a quantum Hamiltonian representing the integer programming problem may be obtained. The quantum Hamiltonian may include a class of operators that correspond to variables of the integer programming problem. The class of operators may be a photon number operator of the quantum Hamiltonian. The integer variables of the integer programming problem may correspond to the photon number operator of the quantum Hamiltonian. The indication of the quantum Hamiltonian may be of various types. In some examples, the indication of the quantum Hamiltonian is a mathematical operator that represents an energy operator. In some examples, the quantum Hamiltonian may be a polynomial (e.g., quadratic) in the photon number operator. A ground state of the quantum Hamiltonian may correspond to a solution of the integer programming problem.

[0088] For example, a target Hamiltonian for solving the maximum clique problem using the Motzkin-Straus formulation is described in Motzkin et al., "Maxima for graphs and a new proof of a theorem of Turan," Canadian Journal of Mathematics 17:533-540, 1965, which is incorporated by reference for all purposes. This can be implemented using integer variables,

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[0093] An indication of the quantum Hamiltonian can be obtained following various examples.

[0094] In some examples, the instructions for the quantum Hamiltonian may be obtained using a digital computer. The digital computer may be of various types, such as any digital computer disclosed elsewhere herein. In some examples, the digital computer may be the digital computer (8) disclosed herein with respect to FIG. 1. The instructions for the quantum Hamiltonian may be stored in a storage (not shown) or memory disclosed herein. In some examples, the memory device may be memory (22) of the digital computer (8).

[0095] In some examples, the instructions for the quantum Hamiltonian may be provided by a user interacting with a digital computer (8).

[0096] In some examples, the instructions for the quantum Hamiltonian may be obtained from a remote processing unit (not shown) operably coupled to the digital computer (8). The remote processing unit may be operably coupled to the digital computer (8) according to various embodiments. In some examples, the remote processing unit may be coupled to the digital computer (8) via a network as disclosed elsewhere herein. In some examples, the network may be a data network. The data network may be selected from the group consisting of a local area network (LAN), a metropolitan area network (MAN), and a wide area network (WAN). In one example, the data network includes the Internet.

[0097] In some examples, an indication of the quantum Hamiltonian may be obtained from a computer-implemented method for solving an integer programming problem.

[0098] Continuing to refer to FIG. 2, according to processing operation (204), the quantum Hamiltonian is implemented on a quantum optical device. The quantum optical device may comprise a system of quantum modes and at least one quantum gate configured to act on the quantum modes. The operators in the operator class may correspond to the quantum modes in the system of quantum modes. The photon number operator of the quantum Hamiltonian may correspond to the quantum modes of the quantum optical device. The quantum optical device may be of various types, such as any quantum optical device disclosed elsewhere herein. In some examples, the quantum optical device may be the quantum optical device (10) disclosed herein with respect to FIG. 1. Implementing the quantum Hamiltonian on the quantum optical device may include configuring the quantum optical device. In some examples, configuring may include setting at least one member of the group consisting of a rotation gate, a Kerr gate, a cross Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled summation gate, a controlled phase gate, and a two-mode squeeze gate.Quantum gates can be based on Kerr nonlinearities (see, e.g., Chuang, Isaac L. and Yamamoto, Yoshihisa, "Simple Quantum Computer", Physical Review A 52.5 (1995): 3489), quantum dots (see, e.g., Fushman, Ilya et al., "Controlled Phase Shifts with a Single Quantum Dot", Science 320.5877 (2008) 769-772), Rydberg cutoff (see, e.g., Tiarks, Daniel et al., "A Photon-Photon Quantum Gate Based on Rydberg Interactions", Nature Physics 15.2 (2019): 124-126), or four-wave mixing using atomic systems (see, e.g., Sagona-Stophel, Steven et al., "Conditional π-phase Shift of Single-Photon-Level Pulses at Room Temperature", Physical Review Letters 125.24(2020):243601), each of which is incorporated by reference in its entirety.

[0099] For example, the target Hamiltonian defined in the description of the processing operation (202) can be implemented using Kerr, cross-Kerr, and rotation gates and combinations with corresponding parameters. These gates are implemented using the unitary transformation

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[0102] According to the processing operation (206), the quantum optical device may be used to prepare a quantum state of the quantum mode system such that the state overlaps (e.g., overlaps by at least 80%) with at least one of the ground states of the quantum Hamiltonian. In some cases, the overlap may be with two or more ground states. In some cases, the overlap may be with three or more ground states. In some examples, preparing the quantum state of the quantum mode system may include simulating a quantum evolution Hamiltonian. In some examples, preparing the quantum state of the quantum mode system may include performing a quantum evolution such as an approximation of a quantum adiabatic evolution from a ground state of a mixed Hamiltonian to a quantum state of the quantum mode system. In some examples, the mixed Hamiltonian may include a quantum operator that does not commute with the photon number operator. In some examples, the evolution of the quantum optical device may be performed using a discretized quantum adiabatic algorithm (dQAA). This procedure may be performed using a scheduling parameter γ j and η j This can be implemented by evolving the quantum state while alternating between the mixed Hamiltonian and the target Hamiltonian (e.g., as described in process step (202)) for the duration of the evolution according to

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[0106] In this specification, the operator

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[0118] In some examples, the evolution of quantum optical devices can be performed using a quantum approximate optimization algorithm (QAOA). This procedure is similar to the discretized quantum adiabatic algorithm (dQAA) procedure described herein, but uses a classical optimization protocol to find the value η j and γ j In some cases, we can find j and γ j Certain values ​​of P may yield optimal solutions. In some instances, because it may be difficult to run the quantum approximate optimization algorithm (QAOA) for large values ​​of P, the quantum approximate optimization algorithm (QAOA) may be deployed for small values ​​of P compared to other methods such as the discretized quantum adiabatic algorithm (dQAA). Reducing the value of P reduces the vector η = (η1, η2, ..., η P ) and γ = (γ1, γ2, ..., γ P ) can be used to reduce the parameter space of the optimization defined by Quantum Approximate Optimization Algorithms (QAOA) with small values ​​of P can be used for devices where it is not physically possible to have a large number of layers.

[0119] In some examples, the development of quantum optical devices may be performed using gate operations. The quantum optical devices may be evolved P times. In some cases, the quantum optical devices may be evolved by feeding the output state back to the same quantum optical device as the input and adjusting the parameters of the gate. In some cases, the output state may be evolved by feeding it to another device with a different set of parameters. The input parameters for the corresponding gates of each layer of the quantum approximate optimization algorithm (QAOA) corresponding to each quantum optical device are the scheduling parameters η j and γ j In some cases, other coefficients related to the mixed Hamiltonian and the target Hamiltonian may also be used. For example, for the problem defined in the description of process step (202), the input values ​​for gates K(κ), CK(κ), R(θ), P(α), and X(x) for each layer j of the quantum optical device may be determined by η j , γ j, W, p0, S, and the adjacency matrix A. The output state is obtained after deployment of the final layer P of the quantum optical device. Measurements may be performed on the optical device to infer the quantum state of the output state for one run of the procedure, as described herein. The process of measuring a single run of the procedure may be repeated to achieve a probability distribution from which the state with the highest probability may be inferred. The number of times this process may be repeated may depend on the accuracy of the probability required. In some examples, repeating this process 1000 times may give the probability of the output quantum states within an accuracy of 0.001. The state with the highest probability may represent a solution to the problem. The iterations of the process may also be performed by multiple quantum optical devices. In some examples, these iterations may be performed in parallel by multiple quantum optical devices. In some examples, these devices may be similarly programmed quantum optical devices. In some examples, two or more devices may be used to perform these iterations. In some examples, three or more devices may be used to perform these iterations.

[0120] Continuing to refer to FIG. 2, according to processing operation (208), a measurement of the system of quantum modes may be performed. The measurement may be a quantum measurement of the number of photons corresponding to the photon number operator. In some examples, the quantum measurement includes a photon-number-resolved measurement of the quantum modes. The photon-number-resolved measurement may be performed by sending the output quantum modes to photon detectors. In some examples, these detectors are photon-number-resolved detectors (PNRDs). These detectors are capable of determining the number of photons that may be present in each quantum mode in a particular instance. These detectors may output a series of integers corresponding to the number of photons in each quantum mode.

[0121] Continuing to refer to FIG. 2, according to processing operation (210), photon counts may be provided as integer solutions to an integer programming problem. By generating a histogram of these counts over one or more runs of the experiment, a probability distribution of quantum states in the photon count basis may be inferred.

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[0123] The processing operations (206, 208, and 210) may be repeated one or more times, which may improve the quality of the solution. In some examples, due to potential inaccuracies of the photon number resolving detector (PNRD) in outputting the actual number of photons in each quantum mode, increasing the number of iterations may help determine a more accurate probability distribution for the output quantum states. The number of iterations may depend on the accuracy required to achieve the final probability distribution for a given problem. In some examples, the accuracy of the probabilities may decrease as the problem size increases, so a greater number of iterations may be performed for larger problem sizes. This number of iterations for a given problem may be empirically determined for each problem type and problem size.

[0124] Referring now to FIG. 3, a flow chart of an example of a method for solving mixed integer programming problems using quantum optical devices is shown.

[0125] According to processing operation (302), an instruction of a quantum Hamiltonian representing a mixed-integer programming problem is obtained. The mixed-integer programming problem includes integer variables and continuous variables. The quantum Hamiltonian may include a class of operators corresponding to the variables. The class of operators may be a photon number operator of the quantum Hamiltonian, as well as a momentum operator and a position operator of the quantum Hamiltonian. The integer variables of the integer programming problem correspond to the photon number operator of the quantum Hamiltonian, and the continuous variables of the mixed-integer programming problem correspond to the momentum operator or the position operator of the quantum Hamiltonian. The instructions of the quantum Hamiltonian can be of various types. In some examples, the instruction of the quantum Hamiltonian is a mathematical operator representing an energy operator. In one or more embodiments, the quantum Hamiltonian can be quadratic in the photon number operator. In some examples, the quantum Hamiltonian can be quartic in the momentum operator or the position operator.

[0126] An example of a mixed-integer programming problem is a sparse optimization problem, and the goal is to solve an optimization problem subject to the constraint that a certain number of components of the solution are maximally non-zero. An example of an N-variable sparse optimization problem is

[0127] [Number] is.

[0128] Here, when x i is non-zero, n i = 1, and when x i = 0, n i = 0. The constant B is the number of non-zero variables, and in order to underdetermine the optimization problem, it is necessary to satisfy the condition B < N. The constant μ i represents the minimum value of the objective function (excluding constraints). The first constraint is a simplex condition expressing that the sum of non-zero variables must be equal to S. To solve this problem, N quantum modes are defined to represent the continuous variable x i , and the integer variable n iAn additional N quantum modes are defined to represent the target Hamiltonian of the sparse optimization problem:

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[0130] The first term implements the objective function f(x), the second term implements the first condition, the third term implements the second condition, and the last term implements n i can only be 0 or 1. A total of 2N quantum modes are needed to solve this problem. The quantum operator for the first N quantum modes

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[0137] An indication of the quantum Hamiltonian can be obtained by various examples.

[0138] In some examples, the instructions for the quantum Hamiltonian may be obtained using a digital computer. The digital computer may be of various types, such as any digital computer disclosed herein. In some examples, the digital computer may be the digital computer (8) disclosed herein with respect to FIG. 1. The instructions for the quantum Hamiltonian may be stored in a storage and / or memory device disclosed herein. In some examples, the memory device may be memory (22) of the digital computer (8).

[0139] In some examples, the instructions for the quantum Hamiltonian may be provided by a user interacting with a digital computer (8).

[0140] In some examples, the instructions of the quantum Hamiltonian may be obtained from a remote processing unit (not shown) operably coupled to the digital computer (8). The remote processing unit may be operably coupled to the digital computer (8) according to various embodiments. In some examples, the remote processing unit may be coupled to the digital computer (8) via a network as disclosed herein. In some cases, the network may be a data network. The data network may be selected from the group consisting of a local area network (LAN), a metropolitan area network (MAN), and a wide area network (WAN). In one example, the data network includes the Internet.

[0141] In some examples, an indication of the quantum Hamiltonian may be obtained from a computer-implemented method for solving a mixed integer programming problem.

[0142] Continuing to refer to FIG. 3, according to processing operation (304), a quantum Hamiltonian may be implemented on a quantum optical device. The quantum optical device may comprise a system of quantum modes and at least one quantum gate configured to act on the quantum modes. A class of operators may correspond to the system of quantum modes. Photon number, position, and momentum operators of the quantum Hamiltonian may correspond to the quantum modes of the quantum optical device.

[0143] The quantum optical device may be of various types, such as any quantum optical device disclosed herein. In some examples, the quantum optical device may be the quantum optical device (10) disclosed herein with respect to FIG. 1. Implementing the quantum Hamiltonian on the quantum optical device may include configuring the quantum optical device. In some examples, configuring may include setting at least one member of the group consisting of a rotation gate, a Kerr gate, a cross Kerr gate, a P-gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled summation gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, and a cross momentum rotation gate. Some of the quantum gates may be implemented using Kerr nonlinearity, quantum dots, Rydberg cutoff, or four-wave mixing atomic systems. The quantum Hamiltonian in the processing operation (302) may be a position rotation gate.

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[0147] Some of the quantum gates disclosed elsewhere herein, such as the position rotation, second order position rotation, momentum rotation, second order momentum rotation, cross position rotation, and cross momentum rotation gates, may be required to implement the methods disclosed elsewhere herein for solving mixed integer programming problems on quantum optical devices.

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[0150] The methods and systems disclosed herein utilize the internal capabilities of quantum optical devices to solve integer or mixed integer programming problems. This internal capability refers to the quantum optical device being essentially a bosonic system, which is represented by an infinite-dimensional Hilbert space with photon number states as a basis. Photons in a quantum optical device, which is a bosonic system, follow Bose-Einstein statistics, which means that multiple photons can exist in the same quantum state simultaneously. The bosonic nature of a quantum optical device can be represented by writing the quantum state as a superposition of all possible photon number states, to infinity, as described elsewhere herein. Since photon numbers are essentially non-negative integers, integer variables of the problem can be represented by photon number operators of the corresponding quantum modes. Photon number operators with only integer eigenvalues ​​can represent integer variables of the problem without the need for auxiliary quantum modes. The fundamental feature of quantum optical devices, in which photon number states are discrete, allows the representation of integer variables of the problem using the internal capabilities of the quantum optical device. This approach can efficiently utilize resources available in quantum optical devices, which can reduce overhead. By combining this with the usual approach of representing continuous variables using orthogonal operators, quantum optical devices may be able to efficiently solve integer or mixed integer programming problems.

[0151] According to the processing operation (306), the quantum optical device may be used to prepare a quantum state of the quantum mode system to have an overlap (e.g., at least 80% overlap) with at least one of the ground states of the quantum Hamiltonian. In some cases, the overlap may be with two or more ground states. In some cases, the overlap may be with three or more ground states. In some examples, preparing the quantum state of the quantum mode system may include simulating a quantum evolution Hamiltonian. In some examples, preparing the quantum state of the quantum mode system may include performing a quantum evolution, such as an approximation of a quantum adiabatic evolution from a ground state of the mixed Hamiltonian to a quantum state of the quantum mode system. In some examples, the mixed Hamiltonian may include a quantum operator that does not commute with the photon number operator. In some examples, the mixed Hamiltonian may include a position operator or a momentum operator as an operator that does not commute with the photon number operator.

[0152] In some examples, the evolution of quantum optical devices can be performed using the discretized quantum adiabatic algorithm (dQAA). This procedure is implemented by iterating the scheduling parameters γ j and η j This can be implemented by evolving the quantum state using alternating mixed and target Hamiltonians (e.g., as described in process step (302)) for an evolution duration according to:

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[0156] This mixed Hamiltonian does not commute with the target Hamiltonian presented in the description of the processing operation (302). The eigenstates of this mixed Hamiltonian are

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[0162] In some examples, the evolution of quantum optical devices can be performed using a quantum approximate optimization algorithm (QAOA). This procedure is similar to the discretized quantum adiabatic algorithm (dQAA) described herein, but uses a classical optimization protocol to find the value η j and γ j In some cases, we can find η j and γ jCertain values ​​of P may yield optimal solutions. In some instances, because it may be difficult to run the quantum approximate optimization algorithm (QAOA) for larger values ​​of P, the quantum approximate optimization algorithm (QAOA) may be deployed for smaller values ​​of P compared to other methods such as the discretized quantum adiabatic algorithm (dQAA). Reducing the value of P reduces the vector η = (η1, η2, ..., η P ) and γ = (γ1, γ2, ..., γ P ) can be used to reduce the parameter space of the optimization defined by Quantum Approximate Optimization Algorithms (QAOA) with small values ​​of P can be used for devices where it is not physically possible to have a large number of layers.

[0163] In one or more embodiments, the evolution of the quantum optical devices may be performed using gate operations. The quantum optical devices may be evolved P times. In some cases, the quantum optical devices may be evolved by feeding back the output state to the same quantum optical device as the input and adjusting the parameters of the gate. In some cases, the output state may be evolved by feeding it back to another device with a different set of parameters. The input parameters for the corresponding gates of each layer of the quantum approximate optimization algorithm (QAOA) corresponding to each quantum optical device are the scheduling parameters η j and γ j In some cases, other coefficients related to the mixed Hamiltonian and the target Hamiltonian may also be used. For example, for the problem defined in the description of process step (302), the input values ​​for gates XN(α,m), XN2(α,m), CXN(α,m), R(θ), K(κ), CK(κ), P(α), and X(x) for each layer j of the optical device may be determined by the variables η j , γ j, λ1, λ2, λ3, p0, S, μ1, μ2, μ3, and B, with m=2. The output state is obtained after deployment of the final layer P of the optical device. Measurements may be performed on the optical device to infer the quantum state of the output state for one run of the procedure, as described herein. The process of measuring a single run of the procedure may be repeated to achieve a probability distribution from which the state with the highest probability may be inferred. The number of times the process may be repeated may depend on the precision of the probability required. In some examples, repeating the process 1000 times may give a probability of the output quantum state with a precision of 0.001. The state with the highest probability may represent a solution to the problem. The iterations of the process may also be performed by multiple quantum optical devices. In some examples, these iterations may be performed in parallel by multiple quantum optical devices. In some examples, these devices may be similarly programmed quantum optical devices. In some examples, two or more devices may be used to perform these iterations. In some examples, three or more devices may be used to perform these iterations.

[0164] Continuing to refer to FIG. 3 , a measurement of the system of quantum modes may be performed according to processing operation (308). The measurement may be a quantum measurement of a photon number corresponding to a photon number operator. In some examples, the quantum measurement includes a photon-number-resolved measurement of the quantum modes. The photon-number-resolved measurement may be performed by sending the output quantum modes to photon detectors. In some examples, these detectors are photon-number-resolved detectors (PNRDs). These detectors are capable of resolving the number of photons that may be present in each quantum mode in a particular case. These detectors may output a series of integers corresponding to the number of photons in each quantum mode. In some examples, the measurement may include performing a homodyne measurement corresponding to a position operator and a momentum operator to obtain position and momentum values ​​of the quantum modes. The homodyne measurement may include using a beam splitter and two photodetectors to measure two out-of-phase components of the optical field. The two measured components of the optical field are resolved into a position operator and a momentum operator.

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[0166] 3, in accordance with process operation (310), the photon number, and position and / or momentum values ​​may be provided as feasible integer and continuous solutions to a mixed integer programming problem. The photon number is provided as values ​​of integer variables of the mixed integer programming problem, and the position and momentum values ​​are provided as values ​​of continuous variables of the mixed integer programming problem. For example, in the sparse optimization problem presented in the description of process operation (302), performing homodyne measurements on the first N quantum modes in a position basis provides a feasible solution to the continuous variable x i While performing photon-number resolved measurements on the second N quantum modes can reveal the solution value of i It is possible to clarify the solution value of n i The value of can, with high probability, be either 0 or 1. For a given quantum mode, n i = 0, the corresponding value x i should be ignored, n i = 1, then the corresponding value x i may represent a solution to a sparse optimization problem. In other words, the measured values ​​x i is the corresponding measured photon number value n i A solution to a mixed integer programming problem can be if and only if is equal to 1.

[0167] The processing operations (306, 308, and 310) may be repeated one or more times, which may improve the quality of the solution. In some examples, due to the inaccuracy of the photon-number-resolved detector (PNRD) in outputting the actual number of photons in each quantum mode, increasing the number of iterations may help determine a more accurate probability distribution for the output quantum states. For homodyne measurements of quantum modes that represent continuous variables of a mixed integer programming problem, the solution for the continuous variables may be found by averaging the results obtained over repeated homodyne measurements. The more measurements, the more accurate the average value of the continuous variables may be. The number of iterations for both the photon-number-resolved and homodyne measurements may depend on the desired accuracy of the final probability distribution or the desired accuracy of the averaged continuous variables for a given problem. In some examples, more iterations may be performed for larger problem sizes, since these accuracies may decrease as the problem size increases. This number of iterations for a given problem may be empirically determined for each problem type and problem size.

[0168] While preferred embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that such embodiments are provided by way of example only. The present invention is not intended to be limited by the specific examples provided herein. Although the present invention has been described with reference to the foregoing specification, the description and illustration of the embodiments herein are not intended to be construed in a limiting sense. Numerous variations, changes, and substitutions will occur to those skilled in the art without departing from the present invention. Furthermore, it should be understood that all aspects of the present invention are not limited to the specific depictions, configurations, or relative proportions described herein, which depend upon a variety of conditions and variables. It should be understood that various alternatives to the embodiments of the present invention described herein may be employed in carrying out the present invention. It is therefore contemplated that the present invention is also intended to encompass any such alternatives, modifications, variations, or equivalents. It is intended that the following claims define the scope of the present invention, and that methods and structures within the scope of these claims and their equivalents are covered thereby.

Claims

1. A method for solving an integer programming problem or a mixed-integer programming problem using a quantum optical device, the method comprising: (a) obtaining an instruction of a quantum Hamiltonian representing the integer programming problem or the mixed-integer programming problem, the quantum Hamiltonian including a class of operators corresponding to variables of the integer programming problem or the mixed-integer programming problem; (b) implementing the quantum Hamiltonian on the quantum optical device, the quantum optical device comprising a system of quantum modes and at least one quantum gate configured to act on one or more quantum modes in the system of quantum modes, wherein one or more operators in the class of operators correspond to the one or more quantum modes in the system of quantum modes; (c) providing a solution to the integer programming problem or the mixed-integer programming problem based at least in part on a measurement of the system of quantum modes. A method as described above.

2. The method according to claim 1, wherein the class of operators includes one or more photon number operators, momentum operators, and position operators of the quantum Hamiltonian.

3. The method according to claim 2, wherein the integer variables of the integer programming problem or the mixed-integer programming problem correspond to the photon number operators of the quantum Hamiltonian, and the continuous variables of the mixed-integer programming problem correspond to the momentum operators and the position operators of the quantum Hamiltonian.

4. The method according to claim 1, wherein the quantum state of the system of quantum modes overlaps with the ground state of the quantum Hamiltonian.

5. The measurement includes one or more of (A) a measurement corresponding to the photon number operator for obtaining a value of the photon number, (B) a homodyne measurement corresponding to the position operator and the momentum operator for obtaining values of position and momentum, as described in claim 2.

6. The measurement includes one or more of (A) a measurement corresponding to the photon number operator for obtaining a value of the photon number, (B) a homodyne measurement corresponding to the position operator and the momentum operator for obtaining values of position and momentum, the homodyne measurement including homodyne detection using a beam splitter and two photodetectors for measuring two phase-shifted components of an optical electric field, as described in claim 2. **Claim 7** The method according to claim 2, wherein the measurement is (A) a measurement corresponding to the photon number operator for obtaining a value of the number of photons, including a photon number resolved measurement of the one or more quantum modes, and (B) one or more of homodyne measurements corresponding to the position operator and the momentum operator for obtaining values of position and momentum. **Claim 8** The method according to claim 2, wherein the measurement is (A) a measurement corresponding to the photon number operator for obtaining a value of the number of photons, including a photon number resolved measurement of the one or more quantum modes, and (B) one or more of homodyne measurements corresponding to the position operator and the momentum operator for obtaining values of position and momentum, including homodyne detection using a beam splitter and two photodetectors for measuring two phase-shifted components of the optical electric field. **Claim 9** The method according to claim 1, wherein (b) further includes (i) preparing a quantum state of the system of the quantum modes using the quantum optical device, including performing one or more gate operations. **Claim 10** The method according to claim 9, further including a step of simulating a Hamiltonian of quantum adiabatic evolution. **Claim 11** The method according to claim 9, wherein (i) includes an approximation of quantum adiabatic evolution from the ground state of the hybrid Hamiltonian to the quantum state of the system of the quantum modes. **Claim 12** The method according to claim 11, wherein the hybrid Hamiltonian includes an operator that does not commute with the photon number operator. **Claim 13** The method according to claim 11, wherein the hybrid Hamiltonian includes a position operator or a momentum operator as a non-commuting operator with the photon number operator. **Claim 14** The method according to claim 11, wherein the approximation of the quantum adiabatic evolution includes one or more of (A) a discrete quantum adiabatic algorithm (dQAA) procedure and (B) a quantum approximate optimization algorithm (QAOA) procedure. **Claim 15** The method according to claim 1, wherein (c) further includes (ii) performing the measurement of the system of the quantum modes. Claim 16. The method according to claim 1, wherein (b) includes configuring the quantum optical device, and the configuring includes setting one or more of a rotation gate, a Kerr gate, a cross-Kerr gate, a P gate, a quadratic phase gate, a displacement gate, a displacement momentum gate, a displacement position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled addition gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross-position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, or a cross-momentum rotation gate. Claim 17. The method according to claim 1, wherein the at least one quantum gate is implemented using at least one of a Kerr nonlinearity, a quantum dot, a Rydberg blockade, or a four-wave mixing atomic system. Claim 18. The method according to claim 1, wherein the quantum Hamiltonian is quadratic in the photon number operator and quartic in the momentum operator and the position operator. Claim 19. The method according to claim 1, wherein (b) includes (i) preparing the quantum state of the system of the quantum modes using the quantum optical device, and (c) includes (ii) performing a measurement of the system of the quantum modes, and (i), (ii), and (c) are repeated one or more times. Claim 20. The method according to claim 19, wherein (i), (ii), and (c) are repeated one or more times until a convergence condition is satisfied. Claim 21. The method according to claim 19, wherein (i), (ii), and (c) are repeated one or more times until a convergence condition is satisfied, and the convergence condition includes a threshold number of iterations or a threshold change in an objective function. Claim 22. The method according to claim 1, wherein the indication of the quantum Hamiltonian is obtained from a user or from a computer-implemented method for solving the integer programming problem or the mixed integer programming problem. **Claim 23**: A quantum optical device comprising a quantum mode system and at least one quantum gate configured to act on one or more quantum modes in the quantum mode system, wherein the at least one quantum gate includes at least one member of the group consisting of a rotation gate, a Kerr gate, a cross-Kerr gate, a P gate, a quadratic phase gate, a displacement gate, a displacement-momentum gate, a displacement-position gate, a Fourier gate, a beam splitter, a squeeze gate, a controlled addition gate, a controlled phase gate, a two-mode squeeze gate, a position rotation gate, a quadratic position rotation gate, a cross-position rotation gate, a momentum rotation gate, a quadratic momentum rotation gate, or a cross-momentum rotation gate. **Claim 24**: The quantum optical device of claim 23, wherein the quantum optical device is configured to at least (i) implement a quantum Hamiltonian, (ii) prepare a quantum state of the quantum mode system, and (iii) perform a measurement of the quantum mode system, the quantum optical device is operably coupled to a digital computer, the digital computer comprises a memory containing instructions, the digital computer is configured to at least (i) obtain an indication of a quantum Hamiltonian representing an integer programming problem or a mixed integer programming problem, (ii) provide the instructions to the quantum optical device, and (iii) execute the instructions to receive a result from the quantum optical device, and the quantum optical device further comprises a control system configured to receive instructions from a digital computer operably coupled to the quantum optical device. **Claim 25**: A method for solving an integer programming problem or a mixed integer programming problem using a quantum optical device, the method comprising, in a digital computer operably connected to the quantum optical device, (a) obtaining an indication of a quantum Hamiltonian representing the integer programming problem or the mixed integer programming problem, the quantum Hamiltonian including a class of operators corresponding to variables of the integer programming problem or the mixed integer programming problem. (b) A step of instructing the quantum Hamiltonian to the quantum optical device, wherein the quantum optical device implements the quantum Hamiltonian, the quantum optical device comprising a system of quantum modes and at least one quantum gate configured to act on one or more of the quantum modes in the system of quantum modes, and one or more operators in the class of operators corresponding to the one or more quantum modes in the system of quantum modes; (c) Providing a solution to the integer programming problem or the mixed integer programming problem based at least in part on a measurement of the system of quantum modes on the quantum optical device; A method comprising.

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