Method and system for preparing eigenstates of a target Hamiltonian on a quantum computer

Abstract

A method and system for preparing eigenstates of a target Hamiltonian using a nonclassical computer is disclosed. The method may include obtaining a reflection path between an initial Hamiltonian and a target Hamiltonian, using one or more target eigenstates to obtain a sequence of reflections along the reflection path, and using the nonclassical computer to execute the sequence of reflections along the reflection path. The system may comprise a quantum computer, a digital computer, and a communication interface for providing instructions to the quantum computer and for obtaining quantum measurement results from the quantum computer.

Description

[Technical field]

[0001] cross reference This application claims the benefit of U.S. Provisional Application No. 63 / 249,804, filed September 29, 2021, which is incorporated by reference in its entirety herein. [Background technology]

[0002] Quantum computers typically utilize quantum mechanical phenomena such as superposition and entanglement to perform operations on quantum systems that represent data. The Hamiltonian of a quantum system is an operator that corresponds to the total energy of the system. The Hamiltonian has eigenstates that correspond to total energy levels. In order to find solutions to problems to be solved using a quantum computer, it may be advantageous to prepare eigenstates accurately and efficiently. However, the preparation of eigenstates for both classical and quantum Hamiltonians may be useful in a variety of fields, including applications such as solving NP-hard optimization problems with classical or non-classical objective functions, and quantum simulation of molecular electronic structures in chemistry and materials science. Summary of the Invention

[0003] The present disclosure provides methods and systems for preparing eigenstates of a target Hamiltonian on a quantum computer, which may improve upon existing methods for eigenstate preparation in at least some aspects by advantageously using quantum devices.

[0004] The systems and methods of the present disclosure provide at least some of the following advantages: In some cases, the methods and systems disclosed herein may be used in circuit-based quantum computing and may use quantum error correction, which may allow for improved scalability. In some cases, the methods and systems disclosed herein may avoid intermediate projection measurements, which may be more erroneous and take longer than those for quantum gates. In some cases, the methods and systems disclosed herein may impose fewer constraints on the type of problem. For example, the need to ensure non-degeneracy or detailed balance conditions may be reduced. In some cases, the methods and systems disclosed herein may not require significant overlap between the starting state of the system and the destination state to be prepared. For example, the overlap need only be non-zero. In some cases, the methods and systems disclosed herein may be flexible in their configuration, such that multiple tools can be integrated in similar contexts, such as qubitization. The methods and systems disclosed herein may allow for easy and efficient heuristic implementation of algorithms. In some cases, the methods and systems disclosed herein have the additional advantage that reflections may be deterministic, as opposed to the projective measurements used in some of the existing methods. In some cases, the methods and systems disclosed herein have the additional advantage that they may be able to exploit the structure of the problem. In some cases, the methods and systems disclosed herein have the additional advantage that the probability of success may increase as the number of reflections increases. For example, the probability of success may not be periodic, as opposed to, for example, Grover's algorithm. The average number of reflections required to solve an NP-hard problem may be reduced compared to existing methods.

[0005] In an aspect, the present disclosure provides a method for preparing eigenstates of a target Hamiltonian using a non-classical computer, which may include (a) obtaining a reflection path between an initial Hamiltonian and a target Hamiltonian, (b) using one or more target eigenstates to obtain a sequence of reflections along the reflection path, and (c) using a non-classical computer to execute the sequence of reflections along the reflection path.

[0006] In some embodiments, the non-classical computer is a quantum computer. In some embodiments, prior to (a), the method includes preparing an eigenstate of the initial Hamiltonian on the quantum computer, the eigenstate being non-orthogonal to the one or more target eigenstates. In some embodiments, following (c), the method includes performing a measurement in an eigenbasis of the target Hamiltonian on the quantum computer, optionally performing the measurement in the eigenbasis of the target Hamiltonian to confirm that the one or more target eigenstates have been achieved. In some embodiments, the measurement is a quantum measurement. In some embodiments, the method further includes obtaining a representation of a superposition of the one or more target eigenstates. In some embodiments, the representation of the one or more target eigenstates includes at least one of an energy interval, an integer representing a plurality of eigenstates having a lowest energy, an integer representing a plurality of eigenstates having a highest energy, a label, and a two-variable function indicating a target eigenstate.

[0007] In some embodiments, (c) includes performing the sequence of reflections with a plurality of gating operations at the quantum computer. In some embodiments, the plurality of gating operations include phase kickback. In some embodiments, the plurality of gating operations include energy comparison. In some embodiments, (c) includes performing quantum phase estimation without energy measurement at the quantum computer. In some embodiments, (c) includes performing at least one of qubitization, quantum signal processing, and partial energy measurement at the quantum computer. In some embodiments, the quantum measurement includes performing at least one of qubitization, quantum signal processing, and partial energy measurement.

[0008] In some embodiments, the quantum computer comprises at least one member of the group consisting of a circuit-based quantum computer, a superconducting quantum computer, a trapped ion quantum computer, a quantum dot computer, an optical quantum computer, a nuclear magnetic resonance (NMR) quantum computer, a solid-state NMR Kane quantum computer, an electron-on-helium quantum computer, a cavity quantum electrodynamics-based quantum computer, a molecular magnet-based quantum computer, a fullerene-based ESR quantum computer, a diamond-based quantum computer, a Bose-Einstein condensate-based quantum computer, a transistor-based quantum computer, a rare earth metal ion doped inorganic crystal-based quantum computer, and a metallic carbon nanosphere-based quantum computer.

[0009] In some embodiments, (a)-(c) are repeated at least once. In some embodiments, (a)-(c) and the preparing the eigenstates of the initial Hamiltonian on the quantum computer are repeated at least once. In some embodiments, (a)-(c) and the performing the measurements in an eigenbasis of the target Hamiltonian are repeated at least once. In some embodiments, (a)-(c) and the providing a representation of a superposition of the one or more target eigenstates are repeated at least once. In some embodiments, (a) includes receiving the reflection path from a user. In some embodiments, (b) includes receiving the sequence of reflections from a user. In some embodiments, (b) includes using an optimization protocol to obtain the sequence of reflections, the optimization protocol including at least one member of the group consisting of a gradient-based optimization procedure and a derivative free optimization (DFO) procedure.

[0010] In some embodiments, (b) includes using an optimization protocol to obtain the sequence of reflections, the optimization protocol being based at least in part on at least one method selected from the group consisting of gradient descent, stochastic gradient descent, steepest descent, Bayesian optimization, random search, and local search. In some embodiments, (b) includes using a machine learning technique to obtain the sequence of reflections. In some embodiments, (a) or (b) or both include using prior information to obtain the sequence of reflections, the reflection path, or both. In some embodiments, (a) includes using an adiabatic path to obtain the reflection path. In some embodiments, the objective Hamiltonian represents at least one member of the group consisting of an optimization problem, a kSAT problem, a spin glass problem, and a quadratic unconstrained binary optimization problem.

[0011] In some embodiments, the target Hamiltonian represents at least one of a quantum many-body system, a fermionic system, and a bosonic system. In some embodiments, the target Hamiltonian represents an optimization problem having at least one constraint. In some embodiments, the eigenstates of the initial Hamiltonian are ground states of the initial Hamiltonian, the ground states defining a region that represents the at least one constraint of the optimization problem. In some embodiments, the preparing an eigenstate of the initial Hamiltonian on a non-classical computer includes constructing the eigenstate from a unitary decomposition. In some embodiments, (c) includes using a classical computing system operatively connected to the non-classical computer to issue one or more instructions to the non-classical computer, the one or more instructions configured to execute the sequence of reflections along the reflection path. In some embodiments, prior to (a), the method includes obtaining a representation of the target Hamiltonian and a representation of the one or more target eigenstates. In some embodiments, prior to (a), the method includes obtaining a representation of the initial Hamiltonian. In some embodiments, the representation of the initial Hamiltonian comprises a domain of an optimization problem.

[0012] In another aspect, the disclosure provides a system for preparing eigenstates of a target Hamiltonian on a quantum computer, the system including a communications interface for providing instructions to the quantum computer and for obtaining quantum measurements, and a digital computer including a non-transitory computer-readable medium operatively coupled to the interface and a processor, the non-transitory computer-readable medium including instructions, the processor configured to execute the instructions, the instructions being at least (a) to obtain a reflection path between an initial Hamiltonian and a target Hamiltonian, (b) to obtain a sequence of reflections along the reflection path using one or more eigenstates, and (c) to provide instructions to the quantum computer using the communications interface to perform the sequence of reflections along the reflection path.

[0013] In some embodiments, the non-classical computer is a quantum computer. In some embodiments, the processor is configured to execute the instructions to prepare an eigenstate of the initial Hamiltonian on the quantum computer, the eigenstate being non-orthogonal to the one or more target eigenstates. In some embodiments, the quantum computer is configured to perform a measurement in an eigenbasis of the target Hamiltonian, optionally performing the measurement in the eigenbasis of the target Hamiltonian to confirm that the one or more target eigenstates have been achieved. In some embodiments, the measurement is a quantum measurement. In some embodiments, the processor is configured to execute the instructions to obtain a representation of a superposition of the one or more target eigenstates. In some embodiments, the representation of the one or more target eigenstates includes at least one of an energy interval, an integer representing a plurality of eigenstates having a lowest energy, an integer representing a plurality of eigenstates having a highest energy, a label, and a two-variable function indicating a target eigenstate.

[0014] In some embodiments, the quantum computer is configured to perform the sequence of reflections using a plurality of gating operations. In some embodiments, the plurality of gating operations comprises a phase kickback. In some embodiments, the plurality of gating operations comprises an energy comparison. In some embodiments, (c) comprises instructions directing the quantum computer to perform a quantum phase estimation without performing an energy measurement. In some embodiments, (c) comprises instructions directing the quantum computer to perform at least one of qubitization, quantum signal processing, and a partial energy measurement. In some embodiments, the quantum measurement comprises performing at least one of qubitization, quantum signal processing, and a partial energy measurement.

[0015] In some embodiments, the quantum computer comprises at least one member of the group consisting of a circuit-based quantum computer, a superconducting quantum computer, a trapped ion quantum computer, a quantum dot computer, an optical quantum computer, a nuclear magnetic resonance (NMR) quantum computer, a solid-state NMR Kane quantum computer, an electron-on-helium quantum computer, a cavity quantum electrodynamics-based quantum computer, a molecular magnet-based quantum computer, a fullerene-based ESR quantum computer, a diamond-based quantum computer, a Bose-Einstein condensate-based quantum computer, a transistor-based quantum computer, a rare earth metal ion doped inorganic crystal-based quantum computer, and a metallic carbon nanosphere-based quantum computer.

[0016] In some embodiments, the processor is further configured to repeat the instructions to perform (a)-(c) at least once. In some embodiments, the processor is further configured to repeat the instructions to perform (a)-(c) and preparing the eigenstates of the initial Hamiltonian on the quantum computer at least once. In some embodiments, the processor is further configured to repeat the instructions to perform (a)-(c) and performing the measurements in an eigenbasis of the target Hamiltonian at least once. In some embodiments, the processor is further configured to repeat the instructions to perform (a)-(c) and obtaining a representation of a superposition of the one or more target eigenstates at least once. In some embodiments, the processor is further configured to receive the reflection path from a user. In some embodiments, the processor is further configured to receive the sequence of reflections from a user. In some embodiments, the processor is further configured to use an optimization protocol to obtain the sequence of reflections, the optimization protocol comprising at least one member of the group consisting of a gradient-based optimization procedure, a derivative free optimization (DFO) procedure.

[0017] In some embodiments, the processor is further configured to use an optimization protocol to obtain the sequence of reflections, the optimization protocol being based at least in part on at least one method selected from the group consisting of gradient descent, stochastic gradient descent, steepest descent, Bayesian optimization, random search, and local search. In some embodiments, the processor is further configured to use a machine learning technique to obtain the sequence of reflections. In some embodiments, at least one of the sequence of reflections and the reflection path is obtained using prior information. In some embodiments, the reflection path is obtained using an adiabatic path. In some embodiments, the target Hamiltonian represents at least one member of the group consisting of an optimization problem, a kSAT problem, a spin glass problem, and a quadratic unconstrained binary optimization problem. In some embodiments, the target Hamiltonian represents at least one of a quantum many-body system, a fermionic system, and a bosonic system.

[0018] In some embodiments, the target Hamiltonian represents an optimization problem having at least one constraint. In some embodiments, the eigenstates of the initial Hamiltonian are ground states of the initial Hamiltonian, the ground states defining a domain representing the at least one constraint of the optimization problem. In some embodiments, the processor is further configured to construct the eigenstates from a unitary decomposition. In some embodiments, prior to (a), the processor is further configured to obtain a representation of the target Hamiltonian and a representation of the one or more target eigenstates. In some embodiments, prior to (a), the processor is further configured to obtain a representation of the initial Hamiltonian. In some embodiments, the representation of the initial Hamiltonian comprises a domain of the optimization problem.

[0019] In another aspect, the disclosure provides a method for preparing eigenstates of a target Hamiltonian using a non-classical computer, which may include (a) obtaining a representation of the target Hamiltonian and a representation of one or more target eigenstates, (b) obtaining a representation of an initial Hamiltonian, (c) obtaining a reflection path between the initial Hamiltonian and the target Hamiltonian, (d) using the representation of the one or more target eigenstates to obtain a sequence of reflections along the reflection path, and (e) using the non-classical computer to execute the sequence of reflections along the reflection path.

[0020] In some embodiments, the non-classical computer is a quantum computer. In some embodiments, prior to (c), the method includes preparing an eigenstate of the initial Hamiltonian on a quantum computer, the eigenstate being non-orthogonal to one or more target eigenstates. In some embodiments, following (e), the method includes performing a measurement in an eigenbasis of the target Hamiltonian to confirm that one or more target eigenstates have been achieved. In some embodiments, the measurement is a quantum measurement. In some embodiments, the method further includes providing a representation of a superposition of one or more target eigenstates. In some embodiments, the representation of the one or more target eigenstates includes at least one of an energy interval, an integer representing a plurality of eigenstates having a lowest energy, an integer representing a plurality of eigenstates having a highest energy, a label, and a two-variable function indicative of the target eigenstate. In some embodiments, the sequence of reflections is performed using a gating operation. In some embodiments, the gating operation includes a phase kickback. In some embodiments, the gating operation includes an energy comparison. In some embodiments, performing the sequence of reflections includes a quantum phase estimation without energy measurements. In some embodiments, performing the sequence of reflections includes at least one of qubitization, quantum signal processing, and partial energy measurement. In some embodiments, performing the quantum measurement includes at least one of qubitization, quantum signal processing, and partial energy measurement.In some embodiments, the quantum computer comprises at least one member of the group consisting of a circuit-based quantum computer, a superconducting quantum computer, a trapped ion quantum computer, a quantum dot computer, an optical quantum computer, a nuclear magnetic resonance quantum computer, a solid-state NMR Kane quantum computer, an electron-on-helium quantum computer, a cavity quantum electrodynamics-based quantum computer, a molecular magnet-based quantum computer, a fullerene-based ESR quantum computer, a diamond-based quantum computer, a Bose-Einstein condensate-based quantum computer, a transistor-based quantum computer, a rare earth metal ion doped inorganic crystal-based quantum computer, and a metallic carbon nanosphere-based quantum computer.

[0021] In some embodiments, (c)-(e) are repeated multiple times. In some embodiments, (c)-(e) and preparing eigenstates of the initial Hamiltonian on the quantum computer are repeated multiple times. In some embodiments, (c)-(e) and performing measurements in an eigenbasis of the target Hamiltonian to confirm that one or more target eigenstates are achieved are repeated multiple times. In some embodiments, (c)-(e) and providing a representation of a superposition of one or more target eigenstates are repeated multiple times. In some embodiments, the reflection path is obtained from a user. In some embodiments, the sequence of reflections is obtained from a user. In some embodiments, the sequence of reflections is obtained using an optimization protocol that includes at least one member of the group consisting of a gradient-based optimization procedure and a derivative free optimization (DFO) procedure. In some embodiments, the sequence of reflections is obtained using an optimization protocol based on at least one method selected from the group consisting of gradient descent, stochastic gradient descent, steepest descent, Bayesian optimization, random search, and local search. In some embodiments, the sequence of reflections is obtained using a machine learning technique.

[0022] In some embodiments, at least one of the sequence of reflections and the reflection path is obtained using prior information. In some embodiments, the reflection path is obtained using an adiabatic path. In some embodiments, the target Hamiltonian represents at least one member of the group consisting of an optimization problem, a kSAT problem, a spin glass problem, and a quadratic unconstrained binary optimization problem. In some embodiments, the target Hamiltonian represents at least one of a quantum many-body system, a fermionic system, and a bosonic system. In some embodiments, the target Hamiltonian represents an optimization problem having at least one constraint. In some embodiments, an eigenstate of the initial Hamiltonian is a ground state of the initial Hamiltonian, further wherein the ground state defines a domain representing the at least one constraint of the optimization problem. In some embodiments, preparing the eigenstate of the initial Hamiltonian on the quantum computer comprises construction from a unitary decomposition. In some embodiments, (e) comprises using a classical computing system operatively connected to the non-classical computer to issue non-classical computer instructions that execute the sequence of reflections along the reflection path. In some embodiments, the representation of the initial Hamiltonian comprises a domain of the optimization problem.

[0023] In another aspect, the disclosure provides a system for preparing eigenstates of a target Hamiltonian on a quantum computer, which may include: (a) a communications interface for providing instructions to the quantum computer and for obtaining quantum measurements; and (b) a digital computer including a non-transitory computer-readable medium operatively coupled to the interface and a processor, the non-transitory computer-readable medium including instructions, the processor configured to execute the instructions, the instructions being at least instructions to obtain a representation of a target Hamiltonian and a representation of one or more target eigenstates, obtain a representation of an initial Hamiltonian, obtain a reflection path between the initial Hamiltonian and the target Hamiltonian, obtain a sequence of reflections along the reflection path, provide instructions to the quantum computer using the communications interface to perform the sequence of reflections, perform quantum measurements in an eigenbasis of the target Hamiltonian, and obtain a superposition of one or more target eigenvalues ​​from the quantum computer using the communications interface.

[0024] Another aspect of the present disclosure provides a system including one or more computer processors and a computer memory coupled thereto, the computer memory including machine-executable code that, upon execution by the one or more computer processors, performs any of the methods described above or elsewhere herein.

[0025] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in the art from the following detailed description. In the following detailed description, only exemplary embodiments of the present disclosure are shown and described. As will be recognized, the present disclosure is capable of other various embodiments, and its several details are capable of modification 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 restrictive.

[0026] Incorporation by Reference All publications, patents, and patent applications mentioned herein are incorporated by reference herein 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 a publication, patent, or patent application incorporated by reference conflicts 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]

[0027] 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 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 in which the accompanying drawings (also referred to herein as "Figure" and "FIG.") are shown: [Figure 1] FIG. 1 is a diagram of a system for preparing eigenstates of a target Hamiltonian on a quantum computer. [Diagram 2] FIG. 1 is a flow diagram of a method for preparing eigenstates of a target Hamiltonian on a quantum computer. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0028] 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 will occur to those skilled in the art without departing from the invention. It is understood that various alternatives to the embodiments of the invention described herein may be used.

[0029] Whenever the terms "at least," "greater than," or "greater than or equal to" are associated with the first number in a series of two or more numbers, the terms "at least," "greater than," or "greater than or equal to" apply to 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.

[0030] Whenever the terms "no more than," "less than," or "less than or equal to" are accompanied by the first number in a series of two or more numbers, the terms "no more than," "less than," or "less than" apply to each 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.

[0031] Certain inventive embodiments herein contemplate numerical ranges. When a range is present, the range includes the endpoints of the range. In addition, all subranges and values ​​within the range are included as if expressly set forth.

[0032] The term "about" or "approximately" may mean within an acceptable error range for a particular value, which depends in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, "about" may mean within one or more standard deviations, as is customary in the art. Alternatively, "about" may mean within a range of 20% or less, 10% or less, 5% or less, or 1% or less of a given value. When a particular value is described in this application and claims, unless otherwise indicated, it can be assumed that the term "about" means within an acceptable error range for the particular value.

[0033] As used herein, the term "classical" in the context of computing or calculations generally refers to calculations performed with binary values ​​using discrete bits without the use of quantum mechanical superposition and entanglement. A classical computer may be a digital computer, such as a computer that uses discrete bits (e.g., 0s and 1s) without the use of quantum mechanical superposition and entanglement.

[0034] As used herein, the term "non-classical" used in the context of computing or computation generally refers to any method or system for performing computational procedures outside the classical computing paradigm.

[0035] As used herein, the term "quantum device" generally refers to a device or system for performing computations using any of the quantum mechanical phenomena, such as quantum mechanical superposition and quantum mechanical entanglement.

[0036] As used herein, the terms “quantum computation,” “quantum procedure,” “quantum operation,” and “quantum computer” generally refer to any method or system for performing computations using quantum mechanical operations (such as unitary transformations on quantum channels or completely positive definite trace-preserving (CPTP) maps) on Hilbert spaces represented by quantum devices.

[0037] As used herein, the term "qubit" generally refers to a unit of quantum information processing whose quantum state is a two-dimensional complex unit vector. These two dimensions are typically referred to as "0" and "1." When quantum error correction is used, a logical qubit refers to a set of physical qubits that encode one fault-tolerant quantum bit.

[0038] As used herein, the term "data qubit" generally refers to one of the quantum bits used to encode quantum information for quantum computation. It may include part of the input or part of the output state. When quantum error correction is used, it refers to the logical quantum bit, otherwise it refers to the physical quantum bit.

[0039] As used herein, the term "register" generally refers to a set of quantum bits used to perform a quantum computation. Different registers may refer to different parts of the computation.

[0040] As used herein, a "quantum gate" generally refers to a manipulation of a qubit that can be represented by a unitary operation on the quantum state of the qubit.

[0041] As used herein, the term "quantum gate operation" generally refers to a quantum gate, a sequence of quantum gates, or a combination of a quantum gate and a quantum measurement that performs an isometry on the quantum state of a qubit.

[0042] As used herein, the term "ancilla qubit" generally refers to one of the additional quantum bits used to make a quantum gate operation more efficient or to perform an intermediate calculation. When quantum error correction is used, it refers to the logical qubit; otherwise, it refers to the physical qubit.

[0043] As used herein, the term "optimization problem" generally refers to any problem that involves minimizing or maximizing an objective function defined over a given domain.

[0044] As used herein, the term "optimization protocol" generally refers to a protocol, algorithm, or method for solving an optimization problem exactly or approximately.

[0045] The preparation of eigenstates for both classical and quantum Hamiltonians can be important in a variety of fields. It may be used to solve problems in the statistical zero-knowledge complexity class (see, for example, Aharonov et al., "Adiabatic quantum state generation and statistical zero knowledge," STOC'03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pp. 20-29, 2003, which is incorporated herein by reference for all purposes). It may be used to perform approximate computations (see, e.g., Han et al., “Approximate computing: An emerging paradigm for energy-efficient design,” 2013 18th IEEE European Test Symposium (ETS), IEEE, 2013, which is incorporated by reference for all purposes). The eigenstate preparation may be used, for example, as a subroutine for solving quantum linear systems (see, e.g., An et al., “Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm,” arXiv:1909.05500, 2019, which is incorporated by reference for all purposes).Eigenstate preparation may be used as part of or in place of quantum searching (see, e.g., Grover, "A fast quantum mechanical algorithm for database search," Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pp. 212-219, 1996, and Brassard et al., "An exact quantum polynomial-time algorithm for Simon's problem," Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems, IEEE, 1997, each of which is incorporated herein by reference for all purposes). It may be used, for example, to create or reduce a database or as a preprocessing algorithm for finding a direct solution. It may be used for quantum metropolis sampling (see, for example, Temme et al., "Quantum Metropolis sampling," Nature 471, pp. 87-90, 2011, which is incorporated by reference for all purposes), where the algorithm allows direct sampling from the eigenstates of the Hamiltonian.It may also be used to solve NP-hard problems (see, e.g., Kaminsky et al., "Scalable architecture for adiabatic quantum computing of NP-hard problems," Quantum computing and quantum bits in mesoscopic systems, pp. 229-236, 2004, which is incorporated by reference for all purposes). Quantum simulation may use eigenstate preparation to initialize a quantum computer in a quantum many-body eigenstate (see, e.g., Whitfield et al., "Simulation of electronic structure Hamiltonians using quantum computers," Molecular Physics 109:5, pp. 735-750, 2011, which is incorporated by reference for all purposes).

[0046] There are various algorithms and heuristics for the preparation of eigenstates. For example, Grover's algorithm (for quantum search, see, e.g., "A fast quantum mechanical algorithm for database search" by Grover, Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pp. 212-219, 1996, which is incorporated by reference herein for all purposes), adiabatic state preparation, in which eigenstates of the instantaneous Hamiltonian of the time-dependent Hamiltonian are prepared by adiabatic evolution (for continuous quantum computing, see, e.g., "Quantum computation by adiabatic evolution" by Farhi et al., arXiv:quant-ph / 0001106, 2000, which is incorporated by reference herein for all purposes), discrete adiabatic state preparation, in which instead of evolution, projected measurements from a discretization of the adiabatic path may be used (for example, "Resource estimates" by Lemieux et al., 2002, which is incorporated by reference herein for all purposes), and Resource estimation for quantum many-body ground-state preparation on a quantum computer, Physical Review A103, no. 5:052408, 2021, which is incorporated by reference for all purposes.), eigenpath traversal by phase randomization instead of projective measurements (see, e.g., Boixo et al., "Quantum state preparation by phase randomization," arXiv:0903.1652, 2009, which is incorporated by reference for all purposes).), or eigenpath traversals by alternating between reflections or Grover iterations and projections (see, e.g., Boixo et al., "Fast quantum algorithms for traversing paths of eigenstates," arXiv:1005.3034, 2010, which is incorporated by reference for all purposes), quantum approximate optimization algorithms in which unitary operators defined by a set of angles and projective measurements are also used (see, e.g., Farhi et al., "A quantum approximate optimization algorithm," arXiv:1411.4028, 2014, which is incorporated by reference for all purposes), and quantum walks in which stable states are prepared from a sequence of reflections defined by a reversible Markov chain (see, e.g., Lemieux et al., "Efficient Quantum Walk Circuits for Metropolis-Hastings See, “An Efficient Quantum Walk Circuit for the Metropolis-Hastings Algorithm,” Quantum4, p. 287, 2020, which is incorporated by reference for all purposes.

[0047] However, the above-mentioned methods may have at least some drawbacks: the success probability of Grover's algorithm may be periodic, e.g., increasing the number of iterations may decrease the success probability; quantum error correction may not be useful for better scalability in continuous quantum computing; projective measurements may cause the wave function to collapse into undesired subspaces at intermediate steps in any of the above-mentioned methods.

[0048] Optimization of variational parameters may result in exponential decay of the Baren plateau, which requires exponential growth of resources for a desired level of accuracy. Eigenstate traversal paths may require a significant degree of overlap between states. Problems solved by the Quantum Metropolis-Hastings method may have to obey detailed balance conditions.

[0049] SUMMARY OF THE DISCLOSURE There is herein recognized a need for improved methods and systems that may overcome at least one of the above-identified shortcomings.

[0050] Quantum Devices Any type of quantum computer may be suitable for the techniques disclosed herein. A quantum processor or quantum computer may include one or more of adiabatic quantum computers, quantum gate arrays, one-way quantum computers, topological quantum computers, quantum Turing machines, superconducting-based quantum computers, trapped ion quantum computers, trapped atom quantum computers, optical lattices, quantum dot computers, spin-based quantum computers, space-based quantum computers, Loss-DiVincenzo quantum computers, nuclear magnetic resonance (NMR)-based quantum computers, solution-state NMR quantum computers, solid-state NMR quantum computers, solid-state NMR Kane quantum computers, electron-on-helium quantum computers, cavity quantum electrodynamics-based quantum computers, molecular magnet quantum computers, fullerene-based quantum computers, linear optical quantum computers, diamond-based quantum computers, nitrogen-vacancy (NV) diamond-based quantum computers, Bose-Einstein condensate-based quantum computers, transistor-based quantum computers, and rare earth metal ion-doped inorganic crystal-based quantum computers. The quantum processor or quantum computer may include one or more of a quantum annealer, an Ising solver, an optical parametric oscillator (OPO), and a gate model quantum computer.

[0051] A quantum processor or quantum computer may include one or more qubits, which may include superconducting qubits, trapped ion qubits, trapped atom qubits, photon qubits, quantum dot qubits, electron spin-based qubits, nuclear spin-based qubits, molecular magnet qubits, fullerene-based qubits, diamond-based qubits, nitrogen-vacancy (NV) diamond-based qubits, Bose-Einstein condensate-based qubits, transistor-based qubits, or rare earth metal ion-doped inorganic crystal-based qubits.

[0052] In accordance with the description herein, suitable quantum computers include, by way of non-limiting example with associated references each of which is incorporated by reference in its entirety, superconducting quantum computers (qubits implemented as small superconducting circuits - Josephson junctions) (Clarke et al., "Superconducting quantum bits," Nature 453, no. 7198, pp. 1031-1042, 2008), trapped ion quantum computers (qubits implemented as ion-trapped states) (Kielpinski et al., "Architecture for a large-scale ion-trap quantum computer," Nature 417, no. 6890, pp. 709-711, 2002), optical lattice quantum computers (qubits implemented as neutral atom states trapped in an optical lattice) (Deutsch et al., "Quantum computing with neutral atoms in an optical lattice," Fortschritte der Physik: Progress of Physics 48, no. 9-11, pp. 925-943, 2000), spin-based quantum dot computers (qubits implemented as spin states of trapped electrons) (Imamoglu et al., "Quantum information processing using quantum dot spins and cavity QED," Physical Review Letters 83, no. 20, p.4204, 1999), spatially based quantum dot computers (qubits implemented as electron positions in a double quantum dot) (Fedichkin et al., "Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot," arXiv:quant-ph / 0006097, 2000), coupled quantum wires (qubits implemented as pairs of quantum wires coupled by quantum point contacts) (Bertoni et al., "Quantum logic gates based on coherent electron transport in quantum wires," Physical Review Letters 84, no. 25, p. 5912, 2000), and nuclear magnetic resonance quantum computers (qubits implemented as nuclear spins and probed by radio waves) (Cory et al., "Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum "Nuclear magnetic resonance spectroscopy: an experimentally accessible paradigm for quantum computing," arXiv:quant-ph / 9709001, 1997), solid-state NMR Kane-type quantum computer (qubits implemented as nuclear spin states of phosphorus donors in silicon) (Kane, "A silicon-based nuclear spin quantum computer," Nature 393, no.6681, pp133-137, 1998), electron-on-helium quantum computers (qubits implemented as electron spins) (Lyon, "Spin-based quantum computing using electrons on liquid helium", arXiv:cond-mat / 0301581, 2006), cavity quantum electrodynamics-based quantum computers (qubits implemented as trapped atomic states coupled to high-finesse resonators) (Burell, "An Introduction to Quantum Computing using Cavity QED concepts", arXiv:1210.6512, 2012), molecular magnet-based quantum computers (qubits implemented as spin states) (Leuenberger et al., "Quantum Computing in Molecular "Quantum Computing in Molecular Magnets," arXiv:cond-mat / 0011415, 2001), and fullerene-based electron spin resonance (ESR) quantum computers (qubits implemented as the electron spins of atoms or molecules housed in fullerenes) (Harneit, "Quantum Computing with Endohedral Fullerenes," arXiv:1708.09298, 2017), linear optical quantum computers (qubits implemented as processing states of light in different modes via linear optical elements such as mirrors, beam splitters, and phase shifters) (Knill et al., "Efficient linear optics quantum computation," arXiv:quant-ph / 0006088, 2000), diamond-based quantum computers (qubits implemented as electron or nuclear spins in nitrogen-vacancy (NV) centers in diamond) (Nizovtsev et al., "A quantum computer based on NV centers in diamond: optically detected nutations of single electron and nuclear spins," Optics and spectroscopy 99, no. 2, pp. 233-244, 2005), Bose-Einstein condensate-based quantum computers (qubits implemented as two-component Bose-Einstein condensates) (Byrnes et al., "Macroscopic quantum computation using Bose-Einstein condensates," arXiv:quantum-ph / 1103.5512, 2011), and transistor-based quantum computers (qubits implemented as semiconductors coupled to nanophotonic cavities) (Sun et al., "A single-photon switch and transistor enabled by a solid-state quantum memory," arXiv:quant-ph / 1805.01964, 2018), rare-earth metal ion-doped inorganic crystal-based quantum computers (qubits implemented as ground-state hyperfine levels of atoms in rare-earth ion-doped inorganic crystals) (Ohlsson et al., "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals," Optics Communications 201, no. 1-3, pp. 71-77, 2002), and metallic-like carbon nanosphere-based quantum computers (qubits implemented as electron spins in conducting carbon nanospheres) (Nafradi et al., "Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres," arXiv:cond-mat / 1611.07690, 2016).

[0053] Classical Computers In some cases, the systems, media, networks, and methods described herein include a classical computer (e.g., a digital 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 units (CPUs) that perform the functions of the classical computer. In some cases, the classical computer further includes an operating system (OS) configured to execute executable instructions. 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 a cloud computing infrastructure. In some cases, the classical computer is connected to an intranet. In some cases, the classical computer is connected to a data storage device.

[0054] 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 in communication 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, whereby devices coupled to the computer system may act as clients or servers.

[0055] In accordance with the description herein, suitable classical computers may include, by way of non-limiting example, server computers, desktop computers, laptop computers, notebook computers, subnotebook computers, netbook computers, netpad computers, set-top computers, media streaming devices, handheld computers, Internet appliances, portable smart phones, tablet computers, personal digital assistants, gaming consoles, and vehicles. Smart phones may be suitable for use with the methods and systems described herein. In some cases, selected televisions, video players, and digital music players with computer network connectivity may be suitable for use in the systems and methods described herein. Suitable tablet computers may include tablet computers having booklet, slate, and convertible configurations.

[0056] In some cases, the classical computer includes an operating system configured to execute executable instructions. An operating system may be, for example, software 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 may 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 non-limiting example, Apple TV®, Roku®, Boxee®, Google TV®, Google Chromecast®, Amazon Fire®, and Samsung® HomeSync®. Suitable game console operating systems may include, by non-limiting example, Sony® PS3®, Sony® PS4®, Microsoft® Xbox 360®, Microsoft® Xbox One®, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®.

[0057] In some cases, the classical computer includes a storage and / or memory device. In some cases, the storage and / or memory device is one or more physical devices used to store data or programs, either temporarily or permanently. In some cases, the storage and / or memory device may have one or more additional data storage devices located outside the classical computer, e.g., on a remote server in communication with the classical computer via 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 some cases, the device is a storage device, including, by 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 and / or memory device is a combination of devices such as those disclosed herein.

[0058] In some cases, a classical computer includes 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) or an active matrix OLED (AMOLED) display. In some cases, the display is a plasma display. In some cases, the display is a video projector. In some cases, the display is a combination of devices such as those disclosed herein.

[0059] In some cases, the classical computer includes an input device 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 non-limiting example, 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 audio 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, etc. In some cases, the input device is a combination of devices, such as those disclosed herein.

[0060] 1, a diagram of a system for preparing eigenstates of a target Hamiltonian on a quantum computer is shown. The system includes a digital computer 100 and a non-classical computer (e.g., a quantum computer, quantum computing device, etc.) 104. The digital computer 100 includes at least one processing device 106, a display device 108, an input device 110, a communication port 114, and a memory 112 that includes a computer program executable by the processing device 106. The digital computer 100 may be of various types, such as any of the digital computers disclosed herein.

[0061] 1, quantum computer 104 includes a quantum processor 120 having a quantum memory 124. In some cases, quantum computer 104 includes a readout control system 122 for quantum measurement readout. Quantum computer 104 is operably connected to digital computer 100 by a connection between readout control system 122 and communication port 114. Quantum computer 104 may include any quantum computer, such as any quantum device disclosed elsewhere herein.

[0062] In some cases, the digital computer 100 is used to provide instructions to the quantum computer 104 using the communications port 114 and the readout control system 122 .

[0063] Referring now to FIG. 2, a flow diagram of a method for preparing eigenstates of a target Hamiltonian on a quantum computer is shown.

[0064] According to process operation 202, a representation of the target Hamiltonian and a representation of one or more target eigenstates are obtained. The representation of the target Hamiltonian may be of various kinds. In some cases, the representation of the target Hamiltonian is a mathematical operator that represents an energy observable.

[0065] The one or more target eigenstates may be of various types. In some cases, the representation of the one or more target eigenstates is represented via an energy interval. In some cases, the representation of the one or more target eigenstates is an integer representing one or more eigenstates having a lowest energy. In some cases, the representation of the one or more target eigenstates is an integer representing one or more eigenstates having a highest energy. In some cases, the representation of the one or more target eigenstates is represented using a label.

[0066] In one example, the target Hamiltonian is the k-body Ising Hamiltonian

[0067]

number

[0068]

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[0069] In some cases, the objective Hamiltonian may represent an optimization problem with constraints. In some cases, the objective Hamiltonian may represent a satisfiability problem. For example, the satisfiability problem may be satisfiability in conjunctive normal form (CNF). A type of CNF SAT problem may be a kSAT problem. The kSAT problem may have a number k of literals. The SAT problem may be constructed such that a number of literals between 1 and k must be true. For example, the kSAT problem may be a 3SAT problem. In some cases, the number k may be about 2, about 3, about 4, about 5, about 6, about 7, about 8, about 9, about 10, about 30, about 50, or more. In one example, the objective Hamiltonian may represent a MAX-SAT problem. The SAT problem may be an unrestricted SAT problem, a one-in-three 3SAT problem, a linear SAT problem, HORNSAT, XOR-SAT, or the like. The MAX-SAT problem may be a generalization of the kSAT problem. A MAX-SAT problem may involve maximizing the number of constraints that must be satisfied by a set of variables.

[0070] In some cases, the target Hamiltonian may represent an optimization problem. Examples of optimization problems include kSAT problems, spin glass problems, and quadratically unconstrained binary optimization problems. In some cases, the target Hamiltonian may represent a quantum many-body system. In some cases, the target Hamiltonian may represent a fermionic system. In some cases, the target Hamiltonian may represent a bosonic system.

[0071] The representation of the target Hamiltonian and the representation of the one or more target eigenstates may be obtained in a variety of ways. In some cases, the representation of the target Hamiltonian and the representation of the one or more target eigenstates may be obtained using a digital computer, such as any of the digital computers 100 disclosed herein with respect to FIG. 1. In some cases, the representation of the target Hamiltonian and the representation of the one or more target eigenstates may be stored in memory 112 of the digital computer 100. In some cases, the representation of the target Hamiltonian and the representation of the one or more target eigenstates may be obtained from a remote processing device operably coupled to the digital computer 100.

[0072] In one example, a kSAT problem can be solved by finding the ground state energy of the corresponding k-body Ising Hamiltonian. For example, in a 3SAT problem,

[0073]

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[0074]

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[0075]

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[0076] 2, according to process operation 204, a representation of the initial Hamiltonian is obtained. The representation of the initial Hamiltonian may be obtained in a variety of ways. In some cases, the representation of the initial Hamiltonian may be obtained using a digital computer, such as any of the digital computers 100 disclosed herein with respect to FIG. 1. In some cases, the representation of the initial Hamiltonian may be stored in memory 112 of the digital computer 100. In some cases, the representation of the initial Hamiltonian may be obtained from a remote processing device operably coupled to the digital computer 100.

[0077] The representation of the initial Hamiltonian may be of various kinds. In some cases, the representation of the initial Hamiltonian is a self-adjoint operator that represents the energy observables.

[0078] In some cases, for a classical objective Hamiltonian that is diagonal in the computational basis, the transverse magnetic field Hamiltonian

[0079]

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[0080]

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[0081]

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[0082] 2, according to process operation 206, eigenstates of the initial Hamiltonian are prepared on a quantum computer. In some cases, the prepared eigenstates of the initial Hamiltonian are not orthogonal to one or more target eigenstates. The eigenstates of the initial Hamiltonian may be such that they are easy to prepare on a quantum computer. The quantum computer may be of various types, such as any of the quantum computers 104 disclosed herein with respect to FIG. 1.

[0083] In some cases, the Hadamard gate

[0084]

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[0085]

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[0086] In some cases, the eigenstates of the initial Hamiltonian are prepared using unitary decomposition. The initial Hamiltonian may be constructed from unitary decomposition. An example of a unitary decomposition procedure may be found in Krol, A. M. et al., "Efficient decomposition of unitary matrices in quantum circuit compilers," arXiv:2101.02993 (2021), which is incorporated by reference herein for all purposes. State

[0087]

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[0088]

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[0089]

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[0090]

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[0091]

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[0092]

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[0093] In some cases, the eigenstates of the initial Hamiltonian are ground states of the initial Hamiltonian. In some cases, the ground states may define a domain that represents a constraint of the optimization problem. For example, in a MAX2SAT problem, such a constraint may be that two particular variables v1 and v2 cannot both be true. Instead of starting with an equal superposition of all states (such as the ground state of the transverse field Hamiltonian), one may eliminate states of the initial superposition where v1=v2=1. One may then construct an initial Hamiltonian whose ground states are defined by the constraints.

[0094] According to processing operation 208, a reflection path between the initial Hamiltonian and the target Hamiltonian is obtained. The reflection path may be obtained in a variety of ways. In some cases, the reflection path may be obtained using a digital computer, such as any of the digital computers 100 disclosed herein with respect to FIG. 1. In some cases, the reflection path may be stored in memory 112 of the digital computer 100. In some cases, the reflection path may be obtained from a remote processing device operably coupled to the digital computer 100.

[0095] In some cases, the reflected path is obtained from a user. In some cases, the reflected path is obtained using an adiabatic path. In some cases, the reflected path is obtained using a priori information.

[0096] A reflection may be a quantum gate operation that changes the phase of a subset of states in a given orthonormal basis. In some cases, each reflection is performed using a gate operation such as a phase kickback. A reflection may be performed by replacing a projected measurement with a (multi-)controlled-NOT (CNOT) gate, where the control qubit is one or more data qubits and the destination qubit is a negative-state ancillary qubit.

[0097]

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[0098]

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[0099]

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[0100]

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[0101]

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[0102] A reflection path may be a continuous function defined from a bounded interval of real numbers to a Hilbert space that contains both the initial Hamiltonian and the target Hamiltonian. For example, define a as the lower bound of the interval and b as the upper bound of the interval. Then, a reflection path is a continuous function f(x) such that f(a) is equal to the initial Hamiltonian and f(b) is equal to the target Hamiltonian. A reflection path may be used to define a sequence of reflections for an algorithm.

[0103] For example, a reflection route for solving the kSAT problem is linear interpolation between the transverse field Hamiltonian and the corresponding k-body Ising Hamiltonian, e.g., for w between 0 and 1:

[0104]

number

[0105] Still referring to FIG. 2, a sequence of reflections along a reflection path may be obtained according to processing operation 210. The sequence of reflections may be obtained in various ways. In some cases, the sequence of reflections may be obtained using a digital computer, such as any of the digital computers 100 disclosed herein with respect to FIG. 1. In some cases, the sequence of reflections may be stored in memory 112 of the digital computer 100. In some cases, the sequence of reflections may be obtained from a remote processing device operatively coupled to the digital computer 100. In some cases, the sequence of reflections may be obtained from a user. In some cases, the sequence of reflections may be obtained using an optimization protocol, such as a gradient-based optimization procedure or a derivative free optimization (DFO) procedure. The optimization protocol may be used on either classical simulations of quantum algorithms or results obtained from quantum computations. The protocol may optimize a cost function calculated using samples of the final energy of the system. The reflection path, a discretization of the reflection path, the (eigen)states defining the reflections, or the energy threshold for each reflection may then be updated. In some cases, the sequence of reflections is obtained using an optimization protocol. For example, the optimization protocol may be based at least in part on a method selected from the group consisting of gradient descent, stochastic gradient descent, steepest descent, Bayesian optimization, random search, and local search. In some cases, the sequence of reflections is obtained using a machine learning method. The machine learning method can be trained on a particular class of problems to find, for example, a reflection path, a discretization of the reflection path, (eigen)states that define the reflection, or an energy threshold for each reflection. In some cases, the sequence of reflections may be obtained using prior information.

[0106] A sequence of reflections may be defined as reflections around some eigenstates of the selected respective Hamiltonian on the reflection path. For example, a constant step discretization of the reflection path may be selected to solve the kSAT problem. The reflections may be performed with respect to the ground states of the respective instantaneous Hamiltonian. For example, the reflection path

[0107]

number

[0108] 2, a sequence of reflections along a reflection path may be performed using a quantum computer, according to process operation 212. The quantum computer may be of various types, such as any of the quantum computers 104 disclosed herein with respect to FIG.

[0109] Quantum measurements in an eigenbasis of the target Hamiltonian may be performed according to process operation 214. A quantum measurement may be an operation of a physical system (e.g., of a qubit) that results in a numerical result representing the state of the qubit. A quantum measurement may be performed to verify that a target eigenstate has been achieved. An indication of one or more target eigenstates obtained according to process operation 202 may be used to verify that a target eigenstate has been achieved. For example, quantum phase estimation may be performed when the eigenstates correspond to energy observables. This may involve calculating the energy (in a computational basis) of each eigenstate of the target Hamiltonian using a register of an ancillary qubit. Prior to the measurement, the two registers may be entangled, where the second register contains the energy corresponding to the eigenstate in the first register. Measuring the energy register (e.g., the second register) may collapse the state into the corresponding eigenstate. The energy may be obtained in the computational basis and may be used to verify that a target eigenstate has been achieved. In some cases, quantum measurements (e.g., quantum phase estimation) may be performed to perform an energy measurement or a partial energy measurement.

[0110]

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[0111]

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[0112] In some cases, energy measurements (such as measurements for performing reflection or processing operation 214) may be replaced with fractional energy measurements.

[0113] 2, in accordance with processing operation 216, a representation of a superposition of one or more target eigenstates is obtained. The representation of the superposition of one or more target eigenstates may be obtained in a variety of ways. In some cases, the representation of the superposition of one or more target eigenstates may be obtained using a quantum computer, such as any quantum computer 104 disclosed herein with respect to FIG. 1. In some cases, the superposition of one or more target eigenstates may be stored in a quantum memory, such as quantum memory 124 disclosed herein with respect to FIG. 1. In some cases, the superposition of one or more target eigenstates may be obtained by a remote processing device operably coupled to quantum computer 104.

[0114] In some cases, the representation of the superposition of one or more target eigenstates includes partial information or an approximation of the superposition of one or more target eigenstates is obtained. The partial information may be obtained in a variety of ways, including but not limited to any output of a process such as energy, label, or any information obtained by sampling the superposition. In some cases, the partial information may be obtained using a digital computer, such as any digital computer 100 disclosed herein with respect to FIG. 1. In some cases, the partial information may be stored in memory 112 of the digital computer 100. In some cases, the partial information may be obtained by a remote processing device operably coupled to the digital computer 100.

[0115] 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. It is not intended that the present invention be limited by the specific examples described herein. Although the present invention has been described with reference to the foregoing specification, the description and illustration of the embodiments herein are not to be construed in a limiting sense. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the present invention. Furthermore, it is to 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 is to be understood that various alternatives to the embodiments of the present invention described herein may be used in carrying out the present invention. It is therefore contemplated that the present invention shall embrace any such alternatives, modifications, variations, or equivalents. The following claims define the scope of the present invention, and methods and configurations within the scope of these claims and equivalents are intended to be encompassed within the scope of the present invention.

Claims

A method for preparing an eigenstate of a target Hamiltonian using a quantum computer, comprising: (a) obtaining a reflection path between an initial Hamiltonian and the target Hamiltonian; (b) obtaining a sequence of reflections along the reflection path based at least in part on one or more target eigenstates; and (c) executing the sequence of reflections along the reflection path on the quantum computer. A method comprising the steps above. The method according to claim 1, further comprising, after (c), performing a measurement in the eigenbasis of the target Hamiltonian on the quantum computer to confirm that the one or more target eigenstates have been achieved. The method according to claim 1, further comprising obtaining a representation of a superposition of the one or more target eigenstates. The method according to claim 1, wherein the representation of the one or more target eigenstates comprises at least one of an energy interval, an integer representing a plurality of eigenstates having the lowest energy, an integer representing a plurality of eigenstates having the highest energy, a label, or a two-variable function indicating the one or more target eigenstates. The method according to claim 1, wherein (c) comprises executing the sequence of reflections on the quantum computer using a plurality of gate operations. The method according to claim 5, wherein the plurality of gate operations comprises a phase kickback or an energy comparison. The method according to claim 1, wherein (c) comprises performing quantum phase estimation on the quantum computer without performing an energy measurement. The method according to claim 7, wherein (c) comprises performing at least one of quantum bitization, quantum signal processing, or partial energy measurement. The method according to claim 2, wherein the measurement comprises a quantum measurement comprising at least one of quantum bitization, quantum signal processing, or partial energy measurement. **Claim 10**: The method according to claim 1, wherein the quantum computer includes at least one member selected from the group consisting of a circuit-based quantum computer, a superconducting quantum computer, an ion trap quantum computer, a quantum dot computer, an optical quantum computer, a nuclear magnetic resonance (NMR) quantum computer, a solid NMR Kane-type quantum computer, a quantum computer based on electrons on helium, a cavity quantum electrodynamics-based quantum computer, a molecular magnet-based quantum computer, a fullerene-based ESR quantum computer, a diamond-based quantum computer, a Bose-Einstein condensate-based quantum computer, a transistor-based quantum computer, a rare earth metal ion-doped inorganic crystal-based quantum computer, and a metal-like carbon nanosphere-based quantum computer. **Claim 11**: The method according to claim 1, wherein (a) to (c) are repeated at least once. **Claim 12**: The method according to claim 2, wherein (a) to (c) and the performing of the measurement in the eigenbasis of the target Hamiltonian are repeated at least once. **Claim 13**: The method according to claim 3, wherein (a) to (c) and the obtaining of the superposition representation of the one or more target eigenstates are repeated at least once. **Claim 14**: The method according to claim 1, wherein (b) includes obtaining the sequence of reflections based at least in part on an optimization protocol, and the optimization protocol is based at least in part on one or more methods selected from the group consisting of a gradient-based optimization procedure, a DFO (derivative free optimization) procedure, a gradient descent method, a stochastic gradient descent method, a steepest descent method, a Bayesian optimization method, a random search method, and a local search method. **Claim 15**: The method according to claim 1, wherein (b) includes using a machine learning technique to obtain the sequence of reflections. **Claim 16**: The method according to claim 1, wherein (a) or (b) or both include using prior information to obtain the sequence of reflections, the reflection path, or both. **Claim 17**: The method according to claim 1, wherein (a) includes using an adiabatic path to obtain the reflection path.

18. The method according to claim 1, wherein the target Hamiltonian represents at least one member of the group consisting of an optimization problem, a kSAT problem, a spin glass problem, a binary optimization problem without quadratic constraints, an optimization problem having at least one constraint, a quantum many-body system, a fermionic system, and a bosonic system.

19. The method according to claim 18, wherein the eigenstate of the initial Hamiltonian is the ground state of the initial Hamiltonian, and the ground state defines a region representing the at least one constraint of the optimization problem.

20. The method according to claim 1, wherein the representation of the initial Hamiltonian includes the domain of the optimization problem.

21. A system for preparing an eigenstate of a target Hamiltonian on a quantum computer, comprising: a communication interface configured to give instructions to the quantum computer; a digital computer including an interface and a non-transitory computer-readable medium communicatively coupled to a processor, the non-transitory computer-readable medium including instructions, the processor being configured to execute the instructions, the instructions including at least (a) obtaining a reflection path between an initial Hamiltonian and a target Hamiltonian; (b) obtaining a sequence of reflections along the reflection path, at least partially based on one or more target eigenstates; (c) instructions for giving, using the communication interface, instructions to the quantum computer to execute a sequence of reflections along the reflection path; a digital computer.

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