Combinatorial optimization via quantum interferometry sampling

Abstract

A method for optimizing a system that includes a set of degrees of freedom (DOF) and a set of constraints with respect to the set of DOF is disclosed. Each constraint is associated with a subset of the set of DOF. The method includes determining a first polynomial function. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum state. After preparing the set of qubits in the first quantum state, the QCS performs a set of unitary operations on the set of qubits. After performing the set of unitary operations on the set of qubits, the QCS generates a first set of observable quantities by performing a first measurement operation on each qubit in the set of qubits. A set of values is determined based on the first set of observable quantities.

Description

[0001] Priority

[0002] This application claims priority to U.S. Provisional Application No. 63 / 583,068, filed September 15, 2023, entitled “COMBINATORIAL OPTIMIZATION VIA QUANTUM INTERFEROMETRIC SAMPLING,” the contents of which are incorporated herein in their entirety. TECHNICAL FIELD

[0003] The present disclosure relates generally to quantum computing systems, and more particularly, to combinatorial optimization via quantum interferometric sampling. BACKGROUND

[0004] Quantum computing is a method of computing that leverages quantum effects, such as ground state superposition and entanglement, to perform certain computations more efficiently than classical digital computers. Unlike digital computers, which store and process information in the form of bits (e.g., “1” or “0”), quantum computing systems can use qubits to process information. A qubit can refer to a quantum device that can superpose multiple states, e.g., data in “0” and “1” states, and / or to the superposition of the data itself in multiple states. According to conventional terminology, the superposition of “0” and “1” states in a quantum system can be represented as, for example, + b The “0” and “1” states of a digital computer are analogous to the and ground states of a qubit, respectively. SUMMARY

[0005] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, or can be obvious from the description, or can be learned through practice of the embodiments.

[0006] One example aspect of the present disclosure relates to a method for optimizing a system comprising a set of degrees of freedom (DOFs) and a set of constraints on the set of DOFs. Each constraint in the set of constraints is associated with a subset of the set of DOFs. The method comprises determining a first polynomial function corresponding to the set of DOFs and the set of constraints. The first polynomial function is a polynomial with respect to each DOF in the set of DOFs. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum state, which corresponds to an initial state based on the first polynomial function. The initial state is represented by a first vector embedded in a Hilbert space characterized by a cardinality of the set of qubits. After preparing the set of qubits in the first quantum state, the QCS performs a set of unitary operations on the set of qubits such that the first quantum state of the set of qubits is transformed to a transformed state, where the transformed state is representable by a second vector embedded in the Hilbert space. After performing the set of unitary operations on the set of qubits, the QCS generates a first set of observables by performing a first measurement operation on each qubit in the set of qubits. A set of values is determined based on the first set of observables. Each value in the set of values corresponds to a DOF in the set of DOFs, and each value in the set of values simultaneously satisfies at least a portion of the set of constraints associated with the corresponding DOF.

[0007] Other aspects of the present disclosure relate to various systems, methods, apparatus, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0008] These and other features, aspects, and advantages of various embodiments of the present disclosure will be better understood when read with reference to the following description and when considered in connection with the accompanying drawings. The drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain related principles. BRIEF DESCRIPTION OF DRAWINGS

[0009] With reference to the drawings, a detailed discussion of embodiments oriented to one of ordinary skill in the art follows, in which:

[0010] Figure 1 An example quantum computing system is depicted in accordance with example embodiments of the present disclosure.

[0011] Figure 2 A first table is shown providing results from various embodiments.

[0012] Figure 3 A second table is shown providing additional results based on the results shown in the first table 200 in accordance with various embodiments. Figure 2 A second table is shown providing additional results based on the results shown in the first table 200 in accordance with various embodiments.

[0013] Figure 4 Quantum circuits employed by various embodiments are shown.

[0014] Figure 5 A flowchart depicting an example method for analyzing a system of equations is described in accordance with various embodiments. DETAILED DESCRIPTION

[0015] Example aspects of the present disclosure relate to enhanced systems and methods for performing combinatorial optimization using quantum interference measurement sampling in a quantum computing system. One non-limiting example of a combinatorial optimization problem is the max-XORSAT problem. However, embodiments are not limited to this and other combinatorial optimization problems can be solved via various embodiments. As described below, the max-XORSAT problem is related to the nearest codeword problem (NCP) in classical error detection and correction. In classical codes , the generator matrix comes from the clauses of a max-XORSAT instance. Embodiments present a quantum reduction of the approximate max-XORSAT to the NCP problem for a dual code . If the NCP instance on can be solved in polynomial time to distance , then the quantum reduction finds assignments that satisfy at least instances of the NCP problem on can be easier than the original NCP problem on . In particular, if a max-XORSAT instance is chosen uniformly at random among all instances in which each variable appears in instances of the NCP problem on In this case, for a constant , the quantum method produces an approximate solution that satisfies instances of the clauses. For sufficiently large k, the average-case results obtained via embodiments constitute an exponential speedup over (worst-case) classical randomized approximation.

[0016] More specifically, given a list of constraints that cannot be simultaneously satisfied, embodiments solve an optimization problem of simultaneously satisfying as many as possible. One such problem contemplated by embodiments is a combinatorial optimization problem in polynomial time, such as but not limited to the max-XORSAT problem. Embodiments employ quantum computing, and more specifically, quantum interference measurement sampling, to solve combinatorial optimization problems in polynomial time.

[0017] The conventional approach to solving a combinatorial optimization problem is an NP-complete problem and thus involves non-polynomial time complexity for solving. Even more specifically, for the max-XORSAT problem, a list of linear equations modulo 2 is given. If a solution exists that satisfies all the equations, then that solution can be found in polynomial time. However, if such a solution does not exist, then finding a solution that satisfies as many equations as possible is NP-complete. This means two things. First, other common combinatorial optimization problems such as optimal routing and scheduling can be re-expressed as max-XORSAT and thus solvers for max-XORSAT can be applied to other more directly practical problems. Second, exact optimal solutions will sometimes require exponential time to compute. In such cases, an approximately optimal solution can be sought.

[0018] One example aspect of the present disclosure relates to a method for optimizing a system of equations, the system of equations comprising a set of degrees of freedom (DOF) and a set of constraints on the set of DOF. Each constraint in the set of constraints is associated with a subset of the set of DOF. The method comprises determining a first polynomial function corresponding to the set of DOF and the set of constraints. The first polynomial function is polynomial with respect to each DOF in the set of DOF. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum state, the first quantum state corresponding to an initial state based on the first polynomial function. The initial state is represented by a first vector embedded in a Hilbert space, the Hilbert space characterized by a cardinality of the set of qubits. After the set of qubits is prepared in the first quantum state, the QCS performs a set of unitary operations on the set of qubits such that the first quantum state of the set of qubits is transformed to a transformed state, where the transformed state is representable by a second vector embedded in the Hilbert space. After performing the set of unitary operations on the set of qubits, the QCS generates a first set of observables by performing a first measurement operation on each qubit in the set of qubits. A set of values is determined based on the first set of observables. Each value in the set of values corresponds to a DOF in the set of DOF, and each value in the set of values simultaneously satisfies at least a portion of the set of constraints associated with the corresponding DOF.

[0019] Aspects of the present disclosure provide a number of technical effects and benefits. For example, embodiments enable a quantum computing system (QCS) to solve combinatorial optimization problems, such as but not limited to max-XORSAT problems, in polynomial time. Classical approaches to solving such problems require non-polynomial time (e.g., exponential time) to solve. Thus, large-scale combinatorial optimization problems can be solved in a practical amount of time. Solving such combinatorial optimization problems has practical applications in a variety of scenarios, such as but not limited to optimal routing and optimal scheduling applications.

[0020] Quantum computing system

[0021] Figure 1 An example quantum computing system 100 is depicted. System 100 is an example of a system of one or more classical computers and / or quantum computing devices located in one or more locations in which the systems, components, and techniques described below can be implemented. Using the disclosure provided herein, those of ordinary skill in the art will appreciate that other quantum computing devices or systems can be used without departing from the scope of the present disclosure.

[0022] System 100 includes quantum hardware 102 in data communication with one or more classical processors 104. Classical processor 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations, such as any of the operations described herein. Quantum hardware 102 includes components for performing quantum computations. For example, quantum hardware 102 includes a quantum system 110, a control device 112, and a readout device 114 (e.g., a readout resonator). Quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubit 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, spin-based qubits, and the like.

[0023] The type of multi-level quantum subsystems used by system 100 can vary. For example, in some cases, it can be convenient to include one or more readout devices 114 attached to one or more superconducting qubits (e.g., transmon qubits, flux qubits, gmon qubits, xmon qubits, or other qubits). In other cases, ion traps, photonic devices, or superconducting cavities (e.g., with which a qubit can not be needed to prepare a state) can be used. Further examples of implementations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots, or phosphorus quantum bits.

[0024] A quantum circuit can be constructed and applied to a register of qubits included in quantum system 110 via a plurality of control lines coupled to one or more control devices 112. An example control device 112 operating on a register of qubits can be used to implement a quantum gate or a quantum circuit having a plurality of quantum gates, such as a Pauli gate, a Hadamard gate, a controlled not (CNOT) gate, a controlled phase gate, a T gate, a multi-qubit quantum gate, a coupler quantum gate, and the like. One or more control devices 112 can be configured to operate on quantum system 110 by one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems can be superconducting qubits, and control device 112 can be configured to provide control pulses to the control lines to produce a magnetic field to adjust the frequency of the qubits.

[0025] Quantum hardware 102 may further include a readout device 114 (e.g., a readout resonator). Measurement results 108 obtained via the measurement device can be provided to classical processor 104 for processing and analysis. In some implementations, quantum hardware 102 may include quantum circuitry, and control device 112 and readout device 114 may implement one or more quantum logic gates that operate the quantum system 102 via physical control parameters (e.g., microwave pulses) transmitted via wires included in the quantum hardware 102. Further examples of the control device include an arbitrary waveform generator, where a DAC (digital-to-analog converter) creates the signal.

[0026] The readout device 114 can be configured to perform a quantum measurement on the quantum system 110 and send the measurement result 108 to the classical processor 104. Additionally, the quantum hardware 102 can be configured to receive data from the classical processor 104 specifying physical control qubit parameter values ​​106. The quantum hardware 102 can use the received physical control qubit parameter values ​​106 to update the actions of the control device 112 and the readout device 114 on the quantum system 110. For example, the quantum hardware 102 can receive data specifying a new value representing the voltage intensity of one or more DACs included in the control device 112, and the quantum hardware can update the actions of the DACs on the quantum system 110 accordingly. The classical processor 104 can be configured, for example, to initialize the quantum system 110 in an initial quantum state by sending data specifying an initial parameter set 106 to the quantum hardware 102.

[0027] In some implementations, the readout device 114 may utilize elements of a quantum system (such as qubits). and The impedance difference between states is used to measure the state of an element (e.g., a qubit). For example, due to the nonlinearity of the qubit, when the qubit is in a state... or state The resonant frequency of the readout resonator can be different. Therefore, the microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depends on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device 114 to block microwave propagation at the qubit frequency.

[0028] In some embodiments, the quantum system 110 may include, for example, a plurality of qubits 120 arranged in a two-dimensional grid 122. For clarity, Figure 1The two-dimensional grid 122 depicted includes 4x4 qubits; however, in some implementations, system 110 may include fewer or more qubits. In some embodiments, multiple qubits 120 may interact with each other via multiple qubit couplers (e.g., qubit coupler 124). A qubit coupler can define the nearest-neighbor interaction between the multiple qubits 120. In some implementations, the strength of the multiple qubit couplers is an adjustable parameter. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0029] In some implementations, the plurality of qubits 120 may include data qubits (such as qubit 126) and measurement qubits (such as qubit 128). Data qubits are qubits that participate in computations performed by system 100. Measurement qubits are qubits that can be used to determine the result of a computation performed by the data qubits. That is, during computation, the unknown state of a data qubit is transferred to a measurement qubit using appropriate physical operations and measured via appropriate measurement operations performed on the measurement qubit.

[0030] In some implementations, each of the multiple qubits 120 can operate using a corresponding operating frequency, such as an idle frequency and / or an interaction frequency and / or a readout frequency and / or a reset frequency. The operating frequencies of different qubits can be different. For example, each qubit may be idle at a different operating frequency. The operating frequencies of the qubits 120 can be selected before computation is performed.

[0031] Figure 1 An example quantum computing system is described that can be used to implement the methods and operations according to the example aspects of this disclosure. Other quantum computing systems can be used without departing from the scope of this disclosure.

[0032] Max-XOR SAT problem statement

[0033] Given an n-variable instance of the max-XORSAT problem, a list of m clauses is provided, each clause specifying the parity of a subset of n binary variables. The max-XORSAT problem is to find assignments for n variables that maximize the number of satisfied clauses. This can be considered as maximizing the parity of the subset of n binary variables. (For example, The problem of maximizing the number of linear constraints satisfied on a given surface.

[0034] (1)

[0035] set up It is listed as The matrix. Therefore, a code can be defined.

[0036] (2)

[0037] And its dual

[0038] (3)

[0039] code and It plays a central role in these explanations.

[0040] max-XORSAT can be viewed as being about The recent coding problem. Specifically, max-XORSAT is equivalent to from Find the element with the minimum Hamming distance. As described in the "Quantum Interferometric Measurement Sampling" section below, the range from max-XORSAT to approximately... Quantum reduction of the recent codeword problem.

[0041] Sometimes it is convenient to reformulate the max-XORSAT problem as maximizing the following objective function.

[0042] (4)

[0043] in

[0044] (5)

[0045] When each clause is restricted to containing at most k variables, the resulting problem is called max-k-XORSAT. Unfortunately, many other naming conventions exist in the literature; max-k-XORSAT is referred to as... in various sources. , or A special case of max-XORSAT where each clause contains exactly two variables and all weights are negative is MAXCUT, which is already NP-hard. Therefore, it is impossible to find exact solutions for some worst-case instances of max-XORSAT using polynomial-time classical or quantum algorithms. However, for the embodiments explained below, classical and quantum algorithms have been designed to approximate solutions for various k. .

[0046] Quantum interferometric sampling

[0047] set up It is a bit string of length n. Applied to the ground state. The quantum Hadamard transformation generates

[0048] (6)

[0049] Compare Equation 6 with Equation 4, and use The linearity was obtained.

[0050] (7)

[0051] in , as specified by normalization.

[0052] Based on computation, the bit string is generated from the measured state in Equation 7 with the following probability.

[0053] (8)

[0054] Each bit string has a probability The classic uniform sampling method (slightly improves the sampling efficiency for its objective function). The probability of a bit string with a large number of values. Prepare the initial steps for the following initial state.

[0055] (9)

[0056] It can be used A quantum circuit with a basic quantum gate is implemented using methods such as the quantum random oracle model (QROM). However, for larger... It can be important to achieve greater improvement by preparing for the following conditions.

[0057] (10)

[0058] Through construction,

[0059] (11)

[0060] Equation 11 holds true because it will Rise to power Generate all the monomials in Equation 4 Products of multiples, these products according to the rules composition.

[0061] In the following discussion, it is assumed that in the weights There is nothing extraordinary about it. Linear correlation.

[0062] In other words, except when both sides are equal or yes During reordering, there is no The solution. This is equivalent to the hypothesis code. Having greater than The distance.

[0063] In the absence of correlation, Equation 10 can be rewritten as

[0064] (12)

[0065] in As given in Chapter 12, and It comes from The superposition of non-zero bits in the string.

[0066] Specifically,

[0067] (13)

[0068] in It is an appropriate normalization constant.

[0069] In the given preparation In the case of a quantum circuit with G gates, It can be used Each door can be prepared, as discussed below in "Adjusting All-Zero Amplitude". Therefore, preparation can be made. Some embodiments address this as follows. First, prepare the following state.

[0070] (14)

[0071] Second, reversibly The calculation is entered into the final register and the appropriate phase is reversed. Thus producing

[0072] (15)

[0073] Third, solve classic bit strings. Thus producing

[0074] (16)

[0075] state The initial register can be discarded to produce the desired state. .

[0076] The first step can be This is implemented within a single gate. The second step can be achieved using methods such as QROM. This is implemented in a quantum gate. (In the RAM model, this cost has only a multilogarithmic relationship with m, but in the circuit model, whether classical or quantum, we face a cost that has a linear relationship with m). The main contribution to complexity, and also the most nontrivial aspect, is the third step in which the index is solved. This problem can be summarized as follows.

[0077] Question 1. We are given a vector , positive integer and vectors We guarantee that s can be uniquely obtained as a linear combination of the following forms.

[0078] (17)

[0079] in Has Hamming weight Find y.

[0080] Problem 1 can be reduced to the definition of the code in Equation 3. The recent code word problem. Given... In this case, problem 1 requires finding a solution that satisfies... and x, provided that it is unique. Within it... of satisfy Under the additional premise, problem 1 is equivalent to finding The lowest Hamming weight solution. If we ignore the fact that the solution should have Hamming weights... It is not difficult to find constraints. The solution can be found using Gaussian elimination. Complete within time. Given any such solution. In this case, the following question can be asked: Find the one that has the same characteristics as... The elements of the minimum Hamming distance This is an example of a recent coding problem. Given y, it is possible to... Restore the solution x to problem 1.

[0081] In summary, the following theorems can be applied to the implementation examples.

[0082] Theorem 1. Suppose C is a type of code, for which there exists a... Solving the problem of finding the nearest code for a basic goalkeeper to the distance (Classical or quantum) decoding algorithms. It must be less than half the code distance so that the decoding problem has a guaranteed unique solution. Suppose that matrix B, defined by clauses of a max-XORSAT instance, defines the parity check matrix for codes from this class. Then, the reduction given here provides the use of... The basic gate generates a state. The method.

[0083] The closest codeword problem is generally NP-hard. However, in specific cases, such as for LDPC codes, a solution can be found that reduces the closest codeword problem to... Efficient classical algorithms for this are known. In this case, as discussed in the following two chapters, such reduction produces a polynomial-time quantum algorithm that generates a quantum algorithm for certain constants. satisfy The solution to the max-XORSAT of each clause.

[0084] Classical decoding algorithm

[0085] The best classical and quantum algorithms for worst-case instances of the nearest codeword problem all have exponential complexity, albeit with modest exponential scaling. (These can be directly applied to XORSAT, since XORSAT itself can be cast to the nearest codeword problem, as discussed in Chapter 1). Various embodiments employ polynomial-time classical approximations, which are configured to find a distance guaranteed to be within a certain range of the nearest codeword. The codeword within the factor.

[0086] One approach that can make the nearest codeword problem easier is the sparsity of the parity check matrix. Linear codes defined by sparse parity check matrices are called low-density parity-check (LDPC) codes. The nearest codeword problem for LDPC codes arises in the scenario of maximum likelihood decoding. In most cases of practical interest, the efficient algorithm for LDPC decoding is a message-passing algorithm called belief propagation (BP). It can be shown that when the Tanner graph is a tree (i.e., without cycles), BP converges to the maximum likelihood for each bit after a constant number of rounds. Analyzing the performance of BP for graphs with cycles is not straightforward, but it can be done in certain cases. In particular, for uniformly randomly selected... - With regular Tanner plots, the average-case performance of BP can be calculated by numerically finding the fixed points of a set of recursive distribution equations.

[0087] Figure 2 A first table 200 is shown, providing results from various embodiments. More specifically, the first table 200 indicates some results from the analysis described above. In the first table 200, It is the fraction of the flipped bits that the confidence propagation can decode from its decoding. It is the fraction of flipped bits that the best maximum likelihood decoder can correct. This is simply a property of LDPC codes themselves. No efficient decoder in this respect is known. It is possible to be with -The same rate as regular LDPC codes (i.e., The maximum number of flipped bits corrected by any code.

[0088] Other decoding algorithms for LDPC codes have also been proposed, such as those based on linear programming and simple bit-flipping processes. These are not necessarily better than BP in practice, but they are easier to prove their bounds for, especially for large k and D. In particular, it has been shown that simple bit-flipping algorithms can solve for the maximum distance defined by Shannon's bounds. 1 / 32 times the average case of LDPC decoding. From Figure 2 As seen in the first table 200, this is not a competing bound for small k and D. However, an interesting feature of this bound is that it depends only on the ratio k / D, rather than solely on k and D. In cases where both k and D are large but have a fixed ratio, this can produce a better bound about the approximation than existing quantum or classical algorithms, as discussed in the "Other Classical Algorithms for LDPC Decoding" section below.

[0089] Will Substituting into Equation 31 directly generates the corresponding approximate ratio obtained through quantum interference measurement sampling.

[0090] One might ask whether the power of this approach comes from quantum reduction or from belief propagation. By constructing a factor graph, belief propagation can be applied to the problem of decoding LDGM codes directly generated from the max-k-XORSAT problem. However, the degree distribution of this factor graph is typically very different from the degree distribution of the Tanner graph in the LDPC decoding problem addressed here. Therefore, directly applying belief propagation to max-k-XORSAT will not be expected to reproduce the performance of the quantum algorithm described here.

[0091] Statistical properties of the objective function

[0092] set up

[0093] (18)

[0094] Then the objective function of equation 4 can be written as

[0095] (19)

[0096] in This is a set of variables present in the j-th clause. Here, we can analyze the probability distribution of equation 20.

[0097] (20)

[0098] Among the variables Uniform random distribution. This also tells us the number of assignments needed to achieve the target value t, i.e. .

[0099] Due to the equal likelihood of the values ​​+1 and -1, the expected value of any nontrivial product of the variables in this set is zero.

[0100] (twenty one)

[0101] Therefore, in this set, The expected value of each term in the expression is zero, and its standard deviation is one. Interestingly, in the absence of a formal expression... In the case of correlation, All items in the set are uncorrelated in pairs. In fact, if k is less than the value from the set... The elements need to be XORed together to obtain the minimum number of bits in the all-zero string, which comes from... The expected value of any product of k terms is zero. We will call this quantity... This quantity can be interpreted as a code. The minimum distance.

[0102] Based on the above, when z is uniformly and randomly selected, the value of the clause behaves much like a set of m independent and identically distributed variables, each with an equal probability of taking the values ​​+1 and -1. Therefore, the distribution of the objective function value should appear approximately binomial. That is, The probability of each clause being satisfied will be well passed through Approximation. We call this approximation an independent and identically distributed (iid) random variable model (or simply an iid model). In the conventions used in these descriptions (i.e., (4)), the value of the objective function is the number of satisfied clauses minus the number of unsatisfied clauses (i.e., Therefore,

[0103] (twenty two)

[0104] Compared to Equation 22 A more precise statement of the symbols can be discussed as follows: when When uniformly sampled randomly, according to the approximation in Equation 22, let... for The expected value, and let yes The true expected value. Based on the above arguments...

[0105] .(twenty three)

[0106] Some further results on this topic are discussed in the following section, “Rigidity of Target Distribution”.

[0107] Maximum likelihood objective value

[0108] Suppose we can achieve this in polynomial time. General NCP solution to distance This allows us to prepare the state efficiently. Suppose we then measure the computational basis and evaluate it on the resulting bit string. Under the iid assumptions (22) and (11), this will produce the following probability distribution over the target value,

[0109] (twenty four)

[0110] In this chapter, we estimate the distribution... The maximum likelihood value of t is used to illustrate the meaning of this point.

[0111] If t is continuous, the peak of the probability distribution over the target value will appear at... The derivative with respect to t is zero in the following places. Since t can be even, this is related to... Correspondingly, this is achieved through Equation 24.

[0112] (25)

[0113] For the sake of simplicity, the notation was introduced. Equation 25 is rearranged as follows:

[0114] (26)

[0115] This produced

[0116] (27)

[0117] pass The left-hand side of equation 27 can be approximated by the following formula.

[0118] (28)

[0119] pass The right-hand side of equation 27 can be approximated by the following formula.

[0120] (29)

[0121] Substituting equations 28 and 29 into equation 27 produces...

[0122] (30)

[0123] This can be obtained by solving for t:

[0124] (31)

[0125] This is an approximate analysis based on the iid model. However, a more accurate analysis will be presented in the next chapter.

[0126] More accurate estimation of the objective distribution

[0127] In previous chapters, we saw that, according to the iid model, for probability distribution exist There are two peaks at this point. Here, we estimate the shape of these peaks more precisely and make a rigorous statement about what the results in the iid model mean for the true distribution.

[0128] set up Indicates the use of the Stirling approximation And there are no other approximations obtained through simplification. Then, through (24):

[0129] (32)

[0130] in It is in equation 24 The logarithm of the normalization constant implied by the sign.

[0131] Taking the derivative produces

[0132] (33)

[0133] for We will expand the logarithm as Applying this approximation and solving for the value of t that makes the derivative zero yields the same result as calculated in the "Maximum Likelihood Objective Value" section above. .

[0134] Taking another derivative of equation 33 produces the result used for The following (accurate) expression is given for the second derivative of .

[0135] (34)

[0136] review ,

[0137] (35)

[0138] (36)

[0139] In distance middle The second-order Taylor expansion can be obtained from the peak value. Powering the product yields pairs. An approximation of , and thus an approximation of The approximation is the sum of two Gaussian functions, i.e.

[0140] (37)

[0141] Among the factors This is specified through normalization. Of course, it can be systematically computed by preserving higher orders in the Taylor expansion. Additional corrections.

[0142] According to Equation 37, we can obtain the distribution. An approximation of the moment of t. Specifically,

[0143] (38)

[0144] (39)

[0145] All odd moments are zero.

[0146] In precise cases, a strict lower bound on the probability of obtaining a given cost function magnitude can be used as a function of the moments of the iid model. For this purpose, an embodiment may employ the Payley-Zigmund inequality, which states that for any nonnegative random variable Z and any ,

[0147] (40)

[0148] Now, apply this to , where t is the number of satisfied clauses minus the number of unsatisfied clauses. If the matrix B generated by the clauses is defined to have a value greater than 1, then t is a matrix that is satisfied by the number of satisfied clauses minus the number of unsatisfied clauses. If the distance to the code is given, then we guarantee that the state is... The generated and This is exactly equal to their values ​​in the iid model. In this case, combining equation 40 with equations 38 and 39 produces...

[0149] (41)

[0150] therefore,

[0151] (41)

[0152] Quantification performance

[0153] Various implementations can be achieved by using belief propagation. Among them Figure 2 The first table, 200, lists the values ​​for small k and D. Some values. According to Equation 31 (as further demonstrated in the "More Precise Estimation of the Target Distribution" section above), it can be shown that it is shown with high probability. Recall that t is the number of satisfied clauses minus the number of unsatisfied clauses; we see that the fraction of satisfied clauses is 1 / 2. ,in . Figure 3 A second table 300 according to various embodiments is shown, the table being based on Figure 2 The results shown in the first table 200 provide additional results. More specifically, in Figure 3 In the second form 300, use in Figure 2 The first table of 200 collected The known values ​​list includes some values.

[0154] Comparison with classical random algorithms

[0155] The following results have been previously proven.

[0156] Theorem 2. There exists a constant and in time The randomized algorithm running in the context of max-k-XORSAT instances When the m degree constraints are at most D, the algorithm finds the assignment with high probability. Make

[0157] .

[0158] Here Let x represent the fraction of the constraint satisfied by x. Specifically, for odd numbers k, by trying assignments and their negations, the algorithm can output x that satisfies the following...

[0159] .

[0160] In contrast, at any point during the solution process, quantum interference measurement sampling has a runtime that is polynomial to k.

[0161] A simple bit-flipping algorithm is implemented when When a complex set of nonlinear conditions dependent on k and D is satisfied, a random LDPC code satisfying k < 4 < D can be linearly decoded to distance in time with high probability. The lower limit below represents this aspect.

[0162] (43)

[0163] It is the following minimum correct solution

[0164] (44)

[0165] Here, H is defined as in Equation 83. The entropy function with base .

[0166] This method was applied to the solution step of the QIS algorithm, and it was implemented. Through equation 44, we can see that... It depends only on the ratio k / D. Therefore, for a fixed k / D, for a sufficiently large k, QIS will be satisfied in polynomial time. The solution to each clause, while the classical result of the theorem only satisfies The program has several clauses and a runtime that is exponential with respect to k. However, the results are not entirely comparable because we are considering the average case here.

[0167] In fact, in most areas of interest, belief propagation is considered the best performing algorithm for decoding LDPC codes. A major advantage of the bit-flipping method is the availability of simple limits regarding its performance.

[0168] Weighted max-XORSAT

[0169] In this chapter, we consider how to solve the weighted max-XORSAT problem, where the coefficients... Pick Other than the values. In this case, the quantum algorithm is the same, except that in the first step, the state defined in Equation 13. We can no longer simply use phase backlash for preparation.

[0170] To understand how to prepare in a weighted situation The expression can be rewritten as

[0171] (45)

[0172] Similar to the unweighted case, we can continue as follows: First, prepare...

[0173] (46)

[0174] Then, through simple mod-2 matrix multiplication, The calculations are stored in the second register of the qubits, and finally, the LDPC decoding algorithm is used to solve for x. Therefore, the only new task in the weighted case is to prepare the state shown in Equation 46.

[0175] First, consider the coefficients. Only two values ​​are taken and In this situation, let's assume... It is a quantity value The number of coefficients. Without loss of generality, we can imagine these clauses being numbered such that as well as Then, state (46) can be expressed as

[0176] (47)

[0177] in

[0178] (48)

[0179] (49)

[0180] (50)

[0181] (51)

[0182] The state shown in Equation 47 can be prepared by first preparing the following state:

[0183] (52)

[0184] Then conditionally set the state Prepare for loading into the additional registers, and finally solve from the initial register. The preparation of the state shown in Equation 52 can be performed efficiently because it is only... The states are located on qubits, and each magnitude is efficiently computable. The solution is simple because these are merely Hamming weights of the strings in the last two registers. Using phase reflex, phase... It can be applied efficiently and Efficient preparation of uniform superpositions over all bit strings with given Hamming weights is also a solved problem that has attracted attention due to their importance in physics, where they are known as Dicke states.

[0185] This process can be directly extended to cases where the coefficients take any constant value. However, as the number of different values ​​increases, the number of terms in (51) increases exponentially, and the number of qubits in state (52) increases linearly. We treat this general case as an open problem.

[0186] This is no longer applicable in weighted max-XORSAT (31). For example, in the case of two magnitudes, under the iid model, the following applies. In the case of magnitudes... of In each clause, let It is the quantity that is satisfied, and for Similar. Then, the first The final measurement in the QIS will produce the following probability distribution as defined by Equations 53 and 54.

[0187] (53)

[0188] in

[0189] (54)

[0190] according to The target value t is given by the following formula.

[0191] (55)

[0192] Therefore, according to these formulas, the expected value of t can be calculated by direct summation. If the number of coefficient values ​​increases to more than two, the number of terms in the sum will increase exponentially, and a more indirect evaluation method should be used instead. In particular, if all weights are integers of polynomial values, dynamic programming can be used to efficiently calculate the expected value of t.

[0193] Hypermax-XORSAT

[0194] To enable efficient sampling in quantum interference measurements, an implementation can employ an optimization problem with the following properties: the objective function has a sparse or well-approximated sparse Fourier spectrum. The coefficients in the sparse Fourier spectrum can be computed efficiently.

[0195] Here, The Fourier transform on the Boolean function (also known as the Hadamard transform) is called the maximum-XORSAT. The objective function used for max-XORSAT clearly possesses these properties. However, many other functions of interest also possess these properties. It is of interest to study the application of quantum interference measurement sampling in this more general setting. The observations given in these descriptions regarding max-XORSAT hint at a deeper and broader connection between the sparsity of the Fourier spectrum of Boolean functions and quantum complexity.

[0196] Hyperaverage case

[0197] Here, we primarily consider the average-case performance of QIS on uniformly random instances of degree D with max-k-XORSAT, since such instances yield a well-studied set of LDPC codes. In this section, we make two observations that go beyond the general case. The first observation is that, given the ring length of the Tanner graph, variants of QIS can achieve a nontrivial lower bound on the performance of the worst-case instances used for adversarial selection. The second observation is the method of constructing instances that superficially appear random but will cause QIS to fail.

[0198] The performance of belief propagation on average-case instances of LDPC decoding is well understood, but its performance on adversarially constructed instances is much worse. For adversarially constructed instances of max-k-XORSAT, we can use a variant of QIS, where the solution steps are completed using a linear programming approach for LDPC decoding.

[0199] The most general statement about the performance of a linear programming decoder is that the algorithm can run in polynomial time, and the output can be the nearest codeword if the distance between the input and the nearest codeword is less than the fractional distance of the code. This is complex to define, but can be evaluated in polynomial time using a linear program. A more convenient, but more relaxed, characterization of the performance of a linear programming decoder is given based on the Tanner graph of the code. Let... This is the length of the cycle in the Tanner graph of the code. Assuming the cycle length is at least 3, find the minimum degree of each vertex. The minimum degree of the vertices is at least 3, and the minimum degree of the odd and even vertices. The value is at least 2. Then, the linear programming decoder will successfully reach the distance... ,in The lower limit is .

[0200] Therefore, the following result can be shown through Equation 31. Let I be an arbitrary instance of max-XORSAT, where the minimum number of variables in any clause is k. min The minimum number of clauses in which any variable appears is D. min .set up It is used for corresponding LDPC codes The length of the ring in the Tanner graph (where the roles of variables and clauses are reversed). Consider QIS, where a linear programming decoder is reversibly implemented to perform the solution steps. If as well as Then this variant of QIS will be implemented with a high probability. (Here, "with high probability" refers to a high probability relative to repeated sampling. It can be a set of probabilities without instances.)

[0201] This result is characterized by not explicitly depending on k (i.e., the maximum number of variables per clause) or D (i.e., the maximum number of clauses in which variables can appear), which are more conventional measures of the difficulty of max-XORSAT instances. However, these quantities play an indirect role, since a Tanner graph with a higher height will necessarily have a smaller loop length. To be precise,

[0202] (56)

[0203] Where N is the number of vertices. Let c be the average degree, and the closest possible value of c is unknown. The known closest possible value of c satisfies c ≤ 2 and c ≥ 4 / 3.

[0204] Consider how to construct instances of bounded max-k-XORSAT that appear random but will cause QIS failure. By determining the implanted codewords with weights of O(1) and following the standard procedure for generating random LDPC codes, but discarding any potential parity checks incompatible with the implanted codewords, the LDPC codes can be ensured. The distance is O(1). In this case, for values ​​greater than a constant... Because it has less than The Hamming weights, however, produce multiple linear combinations of codewords with the same bit string, making the solution steps inexplicable. If the number of such linear combinations is polynomial, then a list of all these combinations can be classically computed, and then their uniform superposition can be solved. However, this is beyond the scope of these explanations.

[0205] Adjusting all-zero amplitudes

[0206] Figure 4 A quantum circuit 400 employed in various embodiments is shown. More specifically, in the "Quantum Interference Measurement Sampling" section above, the efficient circuit is functionally equivalent to the non-limiting quantum circuit 400. The quantum circuit 400 implements... Take as The unitary U operator. The unitary operator is described below. Given a quantum circuit 400 implementing such a unitary U operator, the desired state can be prepared with minimal additional cost using the construction shown below. .

[0207] If all n bottom bits are zero, then the final unitary operator indicates the flip of the first bit. From this circuit, we obtain... Then the state can be discarded. The initial qubits below.

[0208] Rigidity of the objective distribution

[0209] We can reiterate one of the conclusions from the "Statistical Properties of the Objective Function" section above.

[0210] Theorem 3. Consider a max-XORSAT instance with n variables and m clauses. Let... It is the minimum number of monomials from the cost function required to achieve the trivial product (i.e., all variables raised to even powers). Let... Let be a random variable representing the number of clauses satisfied when n input variables are uniformly and randomly assigned. Then, the moments... binomial distribution The former All the rectangles are the same.

[0211] This chapter establishes a general upper limit on how the distribution of cost function values ​​can deviate from the binomial distribution. Consider the following relevant facts.

[0212] Theorem 4. Let and They have the same front Given any two cumulative distribution functions of moments x, then for all values ​​of x,

[0213] (57)

[0214] in

[0215] (58)

[0216] as well as

[0217] (59)

[0218] Theorem 5. Let x be a number distributed according to the binomial distribution. A random variable with a distribution. Then for any power... = 1,2,3...,

[0219] (60)

[0220] as well as

[0221] (61)

[0222] Theorem 6. Let A be a theorem with entries Let be a complex n×n matrix.

[0223] (62)

[0224] set up Therefore Center or radius A is a closed disk. Each eigenvalue of A lies within the disk. Inside at least one of the disks.

[0225] Next, we will prove the following:

[0226] Lemma 1. The matrix defined in (59) Maximum eigenvalue satisfy

[0227] (63)

[0228] as well as

[0229] (64)

[0230] Proof. The upper limit (63) can be proven as follows.

[0231] (65)

[0232] (66)

[0233] Through (59)(67)

[0234] Passed (60). (68)

[0235] The upper limit (64) can be proven as follows. According to Theorem 6,

[0236] (69)

[0237] By examining (59)(70)

[0238] (71)

[0239] Through (61)(72)

[0240] (73)

[0241] (74)

[0242] Using this lemma, we can prove the main result of this chapter.

[0243] Theorem 7. Consider a max-XORSAT instance with n variables and m clauses. Let... It is the minimum number of monomials derived from the cost function required to achieve the trivial product. Let F be the cumulative distribution function for the number of clauses satisfied when the input variables are uniformly and randomly assigned. Let G be the function for the binomial distribution. The cumulative distribution function. Then,

[0244] (75)

[0245] Proof. Let... According to Theorem 3 and Theorem 4

[0246] (76)

[0247] (77)

[0248] (78)

[0249] (79)

[0250] Through (64)(80)

[0251] (81)

[0252] Other classical algorithms for LDPC decoding

[0253] The above text discusses three classic algorithms for decoding LDPC codes: confidence propagation, bit-flipping algorithms, and linear programming methods. In fact, confidence propagation can be considered the optimal algorithm for LDPC decoding. However, other classic algorithms are also noteworthy because they offer slightly different provable performance guarantees.

[0254] A variant of the bit-flipping algorithm can solve the nearest codeword problem for LDPC codes whose Tanner graph is an extender. Specifically, consider max-k-XORSAT, where each variable appears in exactly D clauses. The corresponding Tanner graph will be a (k, D)-regular bipartite graph, where each of the m variable vertices has degree k, and each of the n constraint vertices has degree D. The number of constraints will then be... (Note that the roles of variables and constraints are reversed in the LDPC code compared to the original max-k-XORSAT instance.)

[0255] set up Let E represent the vertex set of the Tanner graph, and let E represent the edge set of the Tanner graph. If for all... ,in We have Make Then we say that the corresponding LDPC code is Expander code. (Note that, as a result of reduction from the max-XORSAT problem, we inherit n as the number of constraints. In most compilation scenarios, n representing the number of variables would be more conventional.) It has been proven that:

[0256] Theorem 8. If the LDPC code is - The expander code, at a given distance, is at most Given the given conditions, the nearest codeword problem can be solved using a classic multinomial-time algorithm.

[0257] Theorem 9. From a set of m k-regular vertices and... From the set of all (k,D)-regular bipartite graphs with D-regular vertices, B is uniformly and randomly selected. Then, for all... Using this as the corresponding LDPC code for its Tanner graph will have a high probability of being - Expander, where

[0258] (82)

[0259] Where H is... Entropy function with base as

[0260] (83)

[0261] Perhaps there is concern that if What happens when the distance is greater than or equal to half the codeword distance? In this case, the nearest codeword problem is not guaranteed to have a unique solution. However, this is impossible because the condition in Theorem 8 is sufficient to guarantee that the codeword distance is greater than half the codeword distance. .

[0262] Numerical studies revealed that only when k and D are very large... Very close to 1, When the value is very small, the guarantee from Theorem 9 is sufficient to satisfy the conditions of Theorem 8.

[0263] Non-limiting method of embodiments

[0264] Figure 5A flowchart of an example method 500 for analyzing a system of equations according to various embodiments is depicted. At block 502, a first polynomial function corresponding to the set of degrees of freedom and the set of constraints is determined. The first polynomial function is a polynomial with respect to each DOF in the set. At block 504, a set of qubits for a quantum computing system (QCS) is prepared in quantum states. The quantum states of this set of qubits correspond to an initial state based on the first polynomial function. The initial state is represented by a first vector embedded in a Hilbert space, characterized by the cardinality of the set of qubits. At block 506, and after preparing the set of qubits in quantum states, the QCS performs a set of unitary operations on the set of qubits, transforming the quantum states of the set of qubits to the transformed state. The transformed state can be represented by a second vector embedded in the Hilbert space. At block 508, and after performing the set of unitary operations on the set of qubits, the QCS generates / updates a set of observables by performing measurement operations on each qubit in the set. In decision box 510, it is determined whether the threshold conditions for this group of observables have been met. If the threshold conditions for this group of observables have been met, method 500 proceeds to box 512. Otherwise, method 500 returns to box 504. At box 512, a set of values ​​is determined based on this group of observables. Each value in this set of values ​​corresponds to a DOF in this group of DOFs. Each value in this set of values ​​simultaneously satisfies at least a portion of the constraints in this set associated with the corresponding DOF.

[0265] In some embodiments, the first polynomial function is based on an objective function for the system. The objective function is based on the set of DOFs and the set of constraints. The objective function is another polynomial. The first polynomial can be equivalent to another polynomial function raised to the power of a positive integer. The positive integer can be greater than 1.

[0266] In various embodiments, each DOF in the set of DOFs is a binary DOF. Each constraint in the set of constraints may include an XOR logical operation on two or more DOFs in the set. The cardinality of the set of DOFs may be less than the cardinality of the set of constraints, resulting in an over-constrained system, and each constraint in the set of constraints cannot be satisfied simultaneously via the set of values. The set of values ​​maximizes the number of constraints that can be satisfied simultaneously. The first vector corresponds to a linear superposition of the ground states in the Z-basis of the set of qubits. Each term in the linear superposition corresponds to a constraint in the set of constraints. The magnitude of each term in the linear superposition corresponds to the parity associated with the corresponding constraint.

[0267] Additional embodiments

[0268] One example aspect of this disclosure relates to a method for optimizing a system of equations comprising a set of degrees of freedom (DOFs) and a set of constraints relating to the set of DOFs. Each constraint in the set of constraints is associated with a subset of the set of DOFs. The method includes determining a first polynomial function corresponding to the set of DOFs and the set of constraints. The first polynomial function is a polynomial with respect to each DOF in the set of DOFs. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum state corresponding to an initial state based on the first polynomial function. The initial state is represented by a first vector embedded in a Hilbert space, which is characterized by the cardinality of the set of qubits. After preparing the set of qubits in the first quantum state, the QCS performs a set of unitary operations on the set of qubits such that the first quantum state of the set of qubits is transformed into a transformed state, wherein the transformed state can be represented as a second vector embedded in the Hilbert space. After performing the set of unitary operations on the set of qubits, the QCS generates a first set of observables by performing a first measurement operation on each of the qubits in the set of qubits. A set of values ​​is determined based on the first set of observables. Each value in this set corresponds to a DOF in this set of DOFs, and each value in this set simultaneously satisfies at least a portion of the constraints in this set associated with the corresponding DOF.

[0269] The method may further include, after generating the first set of observables, preparing the set of qubits in a second quantum state corresponding to the initial state. The method may further include, after preparing the set of qubits in the second quantum state, performing the set of unitary operations on the set of qubits via a QCS, such that the second quantum state of the set of qubits is transformed to the transformed state. The method may further include, after performing the set of unitary operations on the set of qubits, generating a second set of observables by performing a second measurement operation on each of the qubits in the set via a QCS. The values ​​of this second set of observables are then determined based on the first set of observables and the second set of observables.

[0270] In various embodiments, the unitary operations include a Hadamard operation for each qubit in the set of qubits. The system of equations can logically be equivalent to the nearest codeword problem (NCP). The system of equations can be a max-XORSAT problem. The system of equations can be a combinatorial optimization problem. The combinatorial optimization problem can be an optimal routing problem. In other embodiments, the combinatorial optimization problem is an optimal scheduling problem.

[0271] The first polynomial function can be based on an objective function for the system of equations, where the objective function is based on the set of DOFs and the set of constraints.

[0272] The objective function can be another polynomial, and the first polynomial is equivalent to another polynomial function raised to the power of a positive integer.

[0273] Positive integers can be greater than 1.

[0274] Each DOF in this group can be a binary DOF.

[0275] Each constraint in this group of constraints includes an XOR logical operation on two or more DOFs in the group.

[0276] The cardinality of this set of DOFs is less than the cardinality of this set of constraints, causing the system of equations to be over-constrained, and each constraint in this set of constraints cannot be satisfied simultaneously via this set of values.

[0277] This set of values ​​maximizes the number of constraints that can be satisfied simultaneously.

[0278] The first vector corresponds to the linear superposition of the ground states in the Z basis of this set of qubits.

[0279] Each term in the linear superposition corresponds to a constraint in that set of constraints.

[0280] In linear superposition, the magnitude of each term corresponds to the parity associated with the constraint.

[0281] The unitary operations of this group include the Hadamard operation for each qubit in the group.

[0282] The system of equations is logically equivalent to the nearest codeword problem (NCP).

[0283] The system of equations is a characteristic of the max-XORSAT problem.

[0284] The system of equations is a combinatorial optimization problem.

[0285] Combinatorial optimization is the optimal routing problem.

[0286] Combinatorial optimization problems are optimal scheduling problems.

[0287] The implementations of the digital, classical, and / or quantum themes, as well as digital function operations and quantum operations described in this specification, may be implemented in digital electronic circuit systems, suitable quantum circuit systems, or more generally in quantum computing systems, in tangibly implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more of these. The term "quantum computing system" may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0288] The implementation of the digital and / or quantum themes described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or for controlling the operation of a data processing device. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubit / qubit structures, or a combination thereof. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagation signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, said artificially generated propagation signal being generated to encode the digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing device.

[0289] The terms quantum information and quantum data refer to information or data carried, stored, or preserved by quantum systems, where the smallest nontrivial system is a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, for example, systems with two or more levels. Examples of such systems include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational ground state is identified using the ground state and a first excited state; however, it should be understood that other settings where the computational state is identified using higher-level excited states (e.g., qubits) are also possible.

[0290] The term "data processing device" refers to digital and / or quantum data processing hardware, and includes all types of devices, apparatuses, and machines for processing digital and / or quantum data, such as programmable digital processors, programmable quantum processors, digital computers, quantum computers, or multiple digital and quantum processors or computers, and combinations thereof. The device may also be or include dedicated logic circuit systems, such as FPGAs (Field-Programmable Gate Arrays) or ASICs (Application-Specific Integrated Circuits), or quantum simulators, i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a dedicated quantum computer without the ability to perform general-purpose quantum computing. In addition to hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, protocol stacks, database management systems, operating systems, or combinations thereof.

[0291] Digital or classical computer programs, which can also be referred to or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages), and can be deployed in any form, including as standalone programs or as modules, components, subroutines, or other units suitable for use in a digital computing environment. Quantum computer programs, which can also be referred to or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages) and translated into a suitable quantum programming language, or can be written in quantum programming languages ​​such as QCL, Quipper, Cirq, etc.

[0292] Digital and / or quantum computer programs may, but do not necessarily, correspond to files in a file system. Programs may be stored as a portion of a file containing other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinating files (e.g., files storing one or more modules, subroutines, or code sections). Digital and / or quantum computer programs may be deployed to execute on a single digital or quantum computer, or on multiple digital and / or quantum computers located at a single site or distributed across multiple sites and interconnected via digital and / or quantum data communication networks. A quantum data communication network is understood as a network that can transmit quantum data using quantum systems (e.g., qubits). Generally, digital data communication networks cannot transmit quantum data; however, quantum data communication networks can transmit both quantum data and digital data.

[0293] The processes and logic flows described in this specification can be executed by one or more programmable digital and / or quantum computers (which may operate using one or more digital and / or quantum processors, as appropriate) executing one or more digital and / or quantum computer programs to perform functions by manipulating input digital and quantum data and generating outputs. The processes and logic flows can also be executed by a dedicated logic circuit system (e.g., an FPGA or ASIC or a quantum simulator) or by a combination of a dedicated logic circuit system or a quantum simulator and one or more programmable digital and / or quantum computers, and the device can also be implemented as said dedicated logic circuit system or said combination.

[0294] For a system of one or more digital and / or quantum computers or processors that is “configured” or “operable to” perform a specific operation or action, it means that the system has software, firmware, hardware, or a combination thereof installed thereon that causes the system to perform the operation or action in operation. For one or more digital and / or quantum computer programs to be configured to perform a specific operation or action, it means that the one or more programs include instructions that cause the digital and / or quantum data processing device to perform the operation or action when executed by the device. A quantum computer can receive instructions from a digital computer that cause the quantum computing device to perform the operation or action when executed by the quantum computing device.

[0295] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs may be based on general-purpose or special-purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, or random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0296] Some example elements of a digital and / or quantum computer are a central processing unit (CPU) that makes or executes instructions and one or more memory devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented by or incorporated into a dedicated logic circuit system or quantum simulator. Generally, a digital and / or quantum computer will also include one or more mass storage devices for storing digital and / or quantum data, such as magnetic disks, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information, or operatively coupled to receive digital and / or quantum data from or to said one or more mass storage devices, or both. However, a digital and / or quantum computer need not have such devices.

[0297] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memories, media, and memory devices, including, for example, semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable hard disks; magneto-optical disks; and CD-ROM and DVD-ROM disks; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data for a long period with high fidelity and high efficiency, for example, using light for transmission and using matter for storage and preservation of quantum characteristics (such as superposition or quantum coherence) of the quantum data at an optical-material interface.

[0298] Control of the various systems or portions thereof described in this specification may be implemented using digital and / or quantum computer program products, which include instructions stored on one or more tangible, non-transitory, machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification may each be implemented as an apparatus, method, or electronic system, which may include one or more digital and / or quantum processing devices and memory for storing executable instructions to perform the operations described in this specification.

[0299] While this specification contains numerous details of specific implementations, these details should not be construed as limiting the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features described in this specification within the context of individual implementations may also be implemented in combination within a single implementation. Conversely, individual features described within the context of a single implementation may also be implemented individually or in any suitable sub-combination in multiple implementations. Furthermore, although features are described above as functioning in certain combinations, and even initially claimed to be so, one or more features from a claimed combination may, in some cases, be removed from said combination, and the claimed combination may be for a sub-combination or a variation thereof.

[0300] Similarly, although operations are depicted in a specific order in the accompanying drawings, this should not be construed as requiring such operations to be performed in the specific order shown or in sequential order, or requiring all shown operations to achieve the desired result. In some cases, multitasking and parallel processing can be advantageous. Furthermore, the separation of the various system modules and components in the implementation described above should not be construed as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or encapsulated in multiple software products.

[0301] Specific implementations of this subject matter have been described. Other implementations are within the scope of the appended claims. For example, the actions described in the claims can be performed in different orders and still achieve the desired result. As an example, the processes depicted in the figures do not necessarily require a specific order or sequence shown to achieve the desired result. In some cases, multitasking and parallel processing can be advantageous.

Claims

1. A method for analyzing a system of equations comprising a set of degrees of freedom (DOF) and a set of constraints relating to the set of DOF, wherein each constraint in the set of constraints is associated with a subset of the set of DOF, the method comprising: Determine a first polynomial function corresponding to the group of DOFs and the group constraints, wherein the first polynomial function is a polynomial with respect to each DOF in the group of DOFs; A set of qubits for a quantum computing system (QCS) is prepared in a first quantum state, which corresponds to an initial state based on a first polynomial function, wherein the initial state is represented by a first vector embedded in a Hilbert space, the Hilbert space being characterized by the cardinality of the set of qubits. After preparing the group of qubits in the first quantum state, a set of unitary operations is performed on the group of qubits via the QCS, such that the first quantum state of the group of qubits is transformed into a transformed state, wherein the transformed state can be represented by a second vector embedded in the Hilbert space; After performing the group unitary operation on the group of qubits, a first set of observables is generated by performing a first measurement operation on each of the group of qubits via the QCS. as well as A set of values ​​is determined based on the first set of observables, wherein each value in the set corresponds to a DOF in the set of DOFs, and each value in the set simultaneously satisfies at least a portion of the set constraints associated with the corresponding DOF.

2. The method of claim 1, further comprising: After generating the first set of observables, the set of qubits is prepared in a second quantum state corresponding to the initial state; After preparing the group of qubits in the second quantum state, the group unitary operation is performed on the group of qubits via the QCS, so that the second quantum state of the group of qubits is transformed into the transformed state; After performing the group unitary operation on the group of qubits, a second set of observables is generated by performing a second measurement operation on each qubit in the group of qubits via the QCS; and The group value is determined based on the first group of observables and the second group of observables.

3. The method of claim 1, wherein the first polynomial function is based on an objective function for the system of equations, wherein the objective function is based on the group of DOFs and the group constraints.

4. The method of claim 3, wherein the objective function is another polynomial, and the first polynomial is equivalent to another polynomial function raised to the power of a positive integer.

5. The method of claim 4, wherein the positive integer is greater than 1.

6. The method of claim 1, wherein each DOF in the group of DOFs is a binary DOF.

7. The method of claim 1, wherein each constraint in the group constraints comprises an XOR logical operation on two or more DOFs in the group.

8. The method of claim 1, wherein the cardinality of the group DOF is less than the cardinality of the group constraints, such that the system of equations is over-constrained, and each constraint in the group constraints cannot be satisfied simultaneously via the group value.

9. The method of claim 1, wherein the group value maximizes the number of group constraints that can be satisfied simultaneously.

10. The method of claim 1, wherein the first vector corresponds to a linear superposition of the ground states in the Z basis of the group of qubits.

11. The method of claim 10, wherein each term in the linear superposition corresponds to a constraint in the group constraint.

12. The method of claim 11, wherein the magnitude of each term in the linear superposition corresponds to the parity associated with the corresponding constraint.

13. The method of claim 1, wherein the group unitary operation includes a Hadamard operation for each qubit in the group of qubits.

14. The method of claim 1, wherein the system of equations is logically equivalent to the nearest codeword problem (NCP).

15. The method of claim 1, wherein the system of equations is a characteristic of the max-XORSAT problem.

16. The method of claim 1, wherein the system of equations is a combinatorial optimization problem.

17. The method of claim 16, wherein the combinatorial optimization problem is an optimal routing problem.

18. The method of claim 16, wherein the combinatorial optimization problem is an optimal scheduling problem.

19. A quantum computing system (QCS), comprising: A set of qubits; Multi-qubit logic gates; One or more processors; One or more memory devices storing computer-readable instructions that, when executed by the one or more processors, cause the one or more processors to perform operations characterizing the multi-qubit logic gate, the operations including: Determine a first polynomial function corresponding to a set of DOFs and a set of constraints for the system of equations, wherein the first polynomial function is a polynomial with respect to each DOF in the set of DOFs; The group of qubits is prepared in a first quantum state, which corresponds to an initial state based on the first polynomial function, wherein the initial state is represented by a first vector embedded in a Hilbert space, which is characterized by the cardinality of the group of qubits. After preparing the group of qubits in the first quantum state, a set of unitary operations is performed on the group of qubits via the QCS, such that the first quantum state of the group of qubits is transformed into a transformed state, wherein the transformed state can be represented by a second vector embedded in the Hilbert space; After performing the group unitary operation on the group of qubits, a first set of observables is generated by performing a first measurement operation on each of the group of qubits via the QCS. as well as A set of values ​​is determined based on the first set of observables, wherein each value in the set corresponds to a DOF in the set of DOFs, and each value in the set simultaneously satisfies at least a portion of the set constraints associated with the corresponding DOF.

20. The QCS of claim 19, wherein the operation further comprises: After generating the first set of observables, the set of qubits is prepared in a second quantum state corresponding to the initial state; After preparing the group of qubits in the second quantum state, the group unitary operation is performed on the group of qubits via the QCS, so that the second quantum state of the group of qubits is transformed into the transformed state; After performing the group unitary operation on the group of qubits, a second set of observables is generated by performing a second measurement operation on each qubit in the group of qubits via the QCS; and The group value is determined based on the first group of observables and the second group of observables.

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