Combinatorial optimization via quantum interferometric sampling

Abstract

A method for optimizing a system that includes a set of degrees of freedom (DOF) and a set of constraints on 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 Subsequent to preparing the set of qubits in the first quantum state, the QCS performs a set of unitary operations on the set of qubits. Subsequent to 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 of the set of qubits. A set of values is determined based on the first set of observables.

Description

COMBINATORIAL OPTIMIZATION VIA QUANTUM INTERFEROMETRIC SAMPLING PRIORITY

[0001] This application claims priority to U.S. Provisional Application No 63 / 583,068, entitled COMBINATORIAL OPTIMIZATION VIA QUANTUM INTERFEROMETRIC SAMPLING, filed on September 15, 2023, the contents of which of herein incorporated in their entirety. FIELD

[0002] The present disclosure relates generally to quantum computing systems, and more particularly to combinatorial optimization via quantum interferometric sampling. BACKGROUND

[0003] Quantum computing is a computing method that takes advantage of quantum effects, such as superposition of basis states and entanglement to perform certain computations more efficiently than a classical digital computer. In contrast to a digital computer, which stores and manipulates information in the form of bits, e.g., a “1” or “0,” quantum computing systems can manipulate information using quantum bits (“qubits”). A qubit can refer to a quantum device that enables the superposition of multiple states, e.g., data in both the “0” and “1” state, and / or to the superposition of data, itself, in the multiple states. In accordance with conventional terminology, the superposition of a “0” and “1” state in a quantum system may be represented, e.g., as a |0〉 + b |1〉 The “0” and “1” states of a digital computer are analogous to the |0〉 and |1〉 basis states, respectively of a qubit. SUMMARY

[0004] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, or can be learned from the description, or can be learned through practice of the embodiments.

[0005] One example aspect of the present disclosure is directed to a method for optimizing a system that includes a set of degrees of freedom (DOF) and a set of constraints on the set of DOF. Each constraint of the set of constraints is associated with a subset of the set of DOF. The method includes determining a first polynomial function that corresponds to the set of DOF and the set of constraints. The first polynomial function is polynomial with respect to each DOF of the set of DOF. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum statethat 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 that is characterized by a cardinality of the set of qubits. Subsequent to 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, wherein the transformed state is representable by a second vector embedded in the Hilbert space. Subsequent to 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 of the set of qubits. A set of values is determined based on the first set of observables. Each value of the set of values corresponds to a DOF of the set of DOF and each value of the set of values simultaneously satisfies at least a portion of the set of constraints that is associated with the corresponding DOF.

[0006] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0007] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description and appended claims. The accompanying 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 the related principles. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0009] FIG.1 depicts an example quantum computing system according to example embodiments of the present disclosure.

[0010] FIG.2 illustrates a first table, which provides results from various embodiments.

[0011] FIG.3 illustrates a second table, which provides additional results based on the results shown in the first table 200 of FIG.2, according to various embodiments.

[0012] FIG.4 shows a quantum circuit employed by various embodiments.

[0013] FIG.5 depicts a flow diagram of an example method for analyzing a system of equations, according to various embodiments.DETAILED DESCRIPTION

[0014] Example aspects of the present disclosure are directed to enhanced systems and methods for performing combinatorial optimizations using quantum interferometric sampling in a quantum computing system. One non-limiting example of a combinatorial optimization problem is a max-XORSAT problem. However, the embodiments are not so limited, and other combinatorial optimization problems may be solved via the various embodiments. As described below, the max- XORSAT problem is related to a Nearest Codeword Problem (NCP) in classical error detection and correction. In a classical code (ℒ), the generator matrix comes from the clauses of the max- XORSAT instance. The embodiments present a quantum reduction of approximate max-XORSAT to an NCP problem for the dual code ℒ⊥. If the NCP instance on ℒ⊥can be solved in polynomial time out to distance ℓ then the quantum reduction finds with high probability an assignmentsatisfying at least ^^^^� 2 +�ℓ^^^^� 2 clauses. In some circumstances, the NCP problem on ℒ⊥may be easier than the original NCP problem on ℒ. In particular, if a max-XORSAT instance is chosen uniformly at random amongst all instances in which each variable appears in D clauses and each clause contains k variables then, with high probability, the NCP problem on ℒ⊥can be solved in polynomial time out to ℓ = Θ(m) using classical algorithms for LDPC decoding. In this case, the quantum approach yields an approximate solution satisfying (1 / 2 + ^)m clauses for constant ^. For sufficiently large k the average-case results obtained via the embodiments constitute an exponential speedup over the (worst case) classical randomized approximation.

[0015] More particularly, given a list of constraints that cannot all be simultaneously satisfied, the embodiments approach the optimization problem of simultaneously satisfying as many as possible. One such problem contemplated by the embodiments is a combinatorial optimization problem, such as but not limited to a max-XORSAT problem, in polynomial time. The embodiments employ quantum computing, and more specifically, quantum interferometric sampling, to solve a combinatorial optimization problem in polynomial time.

[0016] Conventional methods of solving a combinatorial optimization problem are NP- complete problems, and thus involve non-polynomial time-complexities for solving. Even more particularly, for a max-XORSAT problem, one is given a list of linear equations modulo 2. If a solution exists which satisfies all the equations, then it can be found in polynomial-time. But if such a solution does not exist, then finding a solution which satisfies as many equations as possible is NP-complete. This implies two things. The first is that 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. The second isthat exact optima will sometimes require exponential time to compute. In such cases one may seek approximately optimal solutions.

[0017] One example aspect of the present disclosure is directed to a method for optimizing a system of equations that includes a set of degrees of freedom (DOF) and a set of constraints on the set of DOF. Each constraint of the set of constraints is associated with a subset of the set of DOF. The method includes determining a first polynomial function that corresponds to the set of DOF and the set of constraints. The first polynomial function is polynomial with respect to each DOF of the set of DOF. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum state that 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 that is characterized by a cardinality of the set of qubits. Subsequent to 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, wherein the transformed state is representable a second vector embedded in the Hilbert space. Subsequent to 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 of the set of qubits. A set of values is determined based on the first set of observables. Each value of the set of values corresponds to a DOF of the set of DOF and each value of the set of values simultaneously satisfies at least a portion of the set of constraints that is associated with the corresponding DOF.

[0018] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the embodiments enable a quantum computing system (QCS) to solve a combinatorial optimization problem, such as but not limited to a max-XORSAT problem, in polynomial time. Classical methods of solving such problems require non-polynomial time (e.g., exponential time) to solve. Accordingly, large scale combinatorial optimization problems may be solved in practical amounts of time. Solving such combinatorial optimization problems has practical applications in various scenarios, such as but not limited to optimal routing and optimal scheduling applications. Quantum Computing Systems

[0019] FIG.1 depicts an example quantum computing system 100. The system 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, willunderstand that other quantum computing devices or systems can be used without deviating from the scope of the present disclosure.

[0020] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 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. The quantum hardware 102 includes components for performing quantum computation. For example, the quantum hardware 102 includes a quantum system 110, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubits 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.

[0021] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 114 attached to one or more superconducting qubits, e.g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices or superconducting cavities (e.g., with which states may be prepared without requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits.

[0022] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 110 via multiple control lines that are coupled to one or more control devices 112. Example control devices 112 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 may be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 112 may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0023] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices may be provided to the classical processors 104 for processing and analyzing. In some implementations, the quantum hardware 102 may include a quantum circuit and the control device(s) 112 and readout devices(s) 114 may implement one or more quantum logic gates that operate on the quantum system 102through physical control parameters (e.g., microwave pulses) that are sent through wires included in the quantum hardware 102. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0024] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104. In addition, the quantum hardware 102 may be configured to receive data specifying physical control qubit parameter values 106 from the classical processors 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the action of the control device(s) 112 and readout devices(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing voltage strengths of one or more DACs included in the control devices 112 and may update the action of the DACs on the quantum system 110 accordingly. The classical processors 104 may be configured to initialize the quantum system 110 in an initial quantum state, e.g., by sending data to the quantum hardware 102 specifying an initial set of parameters 106.

[0025] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0〉 and |1〉 states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readout resonator can take on different values when a qubit is in the state |0〉 or the state |1〉, due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to impede microwave propagation at the qubit frequency.

[0026] In some embodiments, the quantum system 110 can include a plurality of qubits 120 arranged, for instance, in a two-dimensional grid 122. For clarity, the two-dimensional grid 122 depicted in FIG.1 includes 4x4 qubits, however in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 can interact with each other through multiple qubit couplers, e.g., qubit coupler 124. The qubit couplers can define nearest neighbor interactions between the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0027] In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit 128. A data qubit is a qubit that participates in a computation being performed by the system 100. A measurement qubit is a qubit that may be usedto determine an outcome of a computation performed by the data qubit. That is, during a computation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0028] In some implementations, each qubit in the multiple qubits 120 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 120 can be chosen before a computation is performed.

[0029] FIG.1 depicts one example quantum computing system that can be used to implement the methods and operations according to example aspects of the present disclosure. Other quantum computing systems can be used without deviating from the scope of the present disclosure. The Max-XORSTAT Problem Statement

[0030] In an n-variable instance of the max-XORSAT problem, one is given a list of m clauses, each specifying the parity of a subset of the n binary variables. The max-XORSAT problem is to find the assignment to the n variables that maximizes the number of satisfied clauses. This can be thought of as a problem of maximizing the number of linear constraints satisfied over ^^^^2(e.g., {0,1}).

[0031] Let be the matrix whose columns are b1,...,bm. One can thus define thecode (2)

[0032] and its dual

[0033] The codes ℒ and ℒ⊥play a central role in these notes.

[0034] One can view max-XORSAT as a nearest codeword problem on ℒ. Specifically, max-XORSAT is equivalent to finding the element of ℒ(^^^^^^^^) with minimum Hamming distance from v = (v1, ... ,vm). As described in the below “Quantum Interferometric Sampling” section, a quantum reduction is given from max-XORSAT to a nearest codeword problem on ℒ⊥.

[0035] It is sometimes convenient to rephrase the max-XORSAT problem as maximizing the following objective function.where Jj= (−1)vj. (5)

[0036] When each clause is restricted to contain 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 called Max EkLIN2, Max-kXOR, or max-k-Lin-2 in by various sources. The special case of max-XORSAT where every clause contains exactly two variables and all weights are negative is MAXCUT, which is already NP-hard. Consequently, exact solutions for some worst-case instances of max-XORSAT may not be found by polynomial-time classical or quantum algorithms. However, for the embodiments and explained below, classical and quantum algorithms have been devised to approximate max-k-XORSAT for various k. Quantum Interferometric Sampling

[0037] Let b be a bit string of length n. The quantum Hadamard transform applied to the basis state |b^ yields

[0038] Comparing Eq.6 to Eq.4, and using the linearity of H⊗n, one obtainswhere , as dictated by normalization.

[0039] Measuring the state from Eq.7 in the computational basis yields bit string x with probability

[0040] Relative to classical uniform sampling, in which every bit string has probability 1 / 2n, this (slightly) boosts the probability of sampling bit strings for which the objective function f has large magnitude. The initial step of preparing the initial statecan be achieved using a quantum circuit of O(nm) elementary quantum gates, using a method such as a quantum random oracle model (QROM). It may be significant, however, to achieve a larger boost by preparing a statefor larger ℓ. By construction,

[0041] Eq.11 holds because raising f to a power ℓ produces all ℓ-fold products of monomials in Eq.4, which compose according to the rule (−1)bi·x(−1)bj·x= (−1)(bi⊕bj)·x).

[0042] In the following discussion, it is assumed that there are no nontrivial ^^^^2-linear dependencies at weight ℓ.

[0043] That is, there are no solutions toother than whenboth sides are equal to 0 is a reordering of i1, . . . , iℓ. This is equivalent to assumingthat the code L⊥(B) has distance greater than ℓ.

[0044] In the absence of dependencies, Eq.10 can be rewritten aswhere θ is as given in section 12 and is the superposition over nonzero bitstrings from .

[0045] Specifically,where is the appropriate normalization constant.

[0046] Given a quantum circuit of G gates to prepare , may be prepared using Oe(G + n) gates, as discussed in the below “Adjusting the All-Zeros Amplitude.” Thus, one mayprepare , which some embodiments approach as follows. First, prepare the state

[0047] Second, reversibly compute bi1⊕...⊕biℓinto the last register and kick back the appropriate phase Ji1...Jiℓ∈ {+1,−1}, yielding

[0048] Third, uncompute the classical bit string yielding

[0049] The initial registers in the state |0^ can be discarded, yielding the desired state .

[0050] The first step can be achieved in O(ℓ poly(logm, logℓ)) gates. The second step can be achieved in O(nm) quantum gates using methods such as QROM. (In a RAM model, this cost would be only poly-logarithmic in m but in a circuit model, whether classical or quantum, one is faced with a cost linear in m.) The dominant contribution to the complexity, and the most nontrivial aspect, is the third step in which the indices are uncomputed. This problem can be abstracted as follows. Problem 1. We are given vectors ^^^^ , .... ^^^^1 ,^^^^^^^^ ∈ ^^^^^^^^, a positive integer ℓ < m, and aWe are promised that s is uniquely obtained as a linear combination of the formwhere y = (y1, ... , ym) has Hamming weight ℓ. Find y.

[0052] Problem 1 can be reduced to the nearest codeword problem on the code L⊥definedin Eq. 3. Given ^^^^ ∈ ^^^^^^^^^^^^, problem 1 asks one to find x satisfying |x| = ℓ and xB = s, under thepromise that this is unique. Under the additional promise that ∄y with |y| < ℓ satisfying xB = s, problem 1 is equivalent to finding the lowest Hamming weight solution to xB = s. It is not hard to find a solution to xB = s if we ignore the constraint that the solution should have Hamming weight ℓ. This can be done in O(m3) time using Gaussian elimination. Given any such solution x0, one can pose the following problem: find the element y ∈ L⊥(B) with smallest Hamming distance from x0. This is an instance of the nearest codeword problem. Given y one can recover the solution x to problem 1 via x = x0⊕ y.

[0053] To summarize, the embodiments may employ the following theorem.

[0054] Theorem 1. Let C be a class of codes for which there is a (classical or quantum) decoding algorithm that solves the Nearest Codeword Problem out to distance ℓ using Cℓelementary gates. (ℓ must be less than half the distance of the code for the decoding problem to have a guaranteed unique solution.) Suppose the matrix B defined by the clauses of the max- XORSAT instance defines a parity check matrix for a code from this class. Then, the reductiongiven here provides a way to produce the state using ^^�^^(^^^^ 3ℓ + ^^^^ ) elementary gates.

[0055] The nearest codeword problem is NP-hard in general. However, under specific circumstances, such as for LDPC codes, efficient classical algorithms are known for solving the nearest codeword problem out to ℓ = Θ(m). In such cases, as discussed in the next two sections, thisreduction yields a polynomial time quantum algorithm that produces a solution to max-XORSAT satisfying (1 / 2 + ^)m clauses for some constant ^. Classical Decoding Algorithms

[0056] The best classical and quantum algorithms for worst case instances of the nearest codeword problem all have exponential complexity, albeit with mild exponential scaling. (These can be applied directly to XORSAT because XORSAT itself can be cast as a nearest codeword problem, as discussed in section 1). Various embodiments employ polynomial-time classical approximation algorithms that are configured to find a codeword whose distance is guaranteed to be within a factor of O(n / log n) of the nearest codeword.

[0057] An approach of the embodiments that can make the nearest codeword problem easier is sparsity of the parity check matrix. Linear codes defined by sparse parity check matrices are referred to as Low Density Parity Check (LDPC) codes. The nearest codeword problem for LDPC codes arises in the context of maximum likelihood decoding. An effective algorithm for LDPC decoding in most situations of practical interest is a message passing algorithm referred to as Belief Propagation (BP). It may be shown that when the Tanner graph is a tree (i.e. has no cycles), BP converges to the maximum likelihood value for each bit after a constant number of rounds. Analyzing the performance of BP for graphs with cycles is less straightforward but can be done in some cases. In particular, for (k,D)-regular Tanner graphs chosen uniformly at random, the average case performance of BP can be calculated by numerically finding the fixed point of a set of recursive distributional equations.

[0058] FIG.2 illustrates a first table 200, which provides results from various embodiments. More particularly, the first table 200 indicates some results from the above analysis. In the first table 200, ^BPis the fraction of flipped bits that Belief Propagation can decode from. ^optis the fraction of flipped bits that the optimal maximum-likelihood decoder could correct. This is a property only of the LDPC code itself. No efficient decoder achieving this is known. ^Shannonis the maximum number of flipped bits that can be corrected by any code with the same rate as a (k,D)- regular LDPC code, namely 1 − k / D.

[0059] Other decoding algorithms for LDPC codes have also been proposed such as ones based on linear programming and simple bit-flipping procedures. These are not necessarily better than BP in practice, but they can be somewhat easier to prove bounds on, particularly for large k and D. In particular, it has been proven that a simple bit flipping algorithm can solve average case LDPC decoding up to 1 / 32 times the maximum distance ^Shannonset by the Shannon bound. From the first table 200 of FIG.2, one sees that this is not a competitive bound for small k and D.However, the interesting feature of this bound is that it depends only on the ratio k / D rather than on k and D individually. In cases where k and D are both large but of fixed ratio, this may yield better bounds on approximation than prior quantum or classical algorithms, as discussed in the below “Other Classical Algorithms for LDPC Decoding” section.

[0060] Substituting ℓ = ^BPm into Eq.31 directly yields corresponding approximation ratios achieved by quantum interferometric sampling.

[0061] One may question whether the power of this approach comes from quantum reduction versus from Belief Propagation. Belief Propagation can be applied to the problem of decoding LDGM codes that directly arises from the max-k-XORSAT problem, by constructing the factor graph. However, the degree distribution of this factor graph is in general quite different from the degree distribution of the Tanner graph of the LDPC decoding problem treated here. So, the direct application of Belief Propagation to max-k-XORSAT would not be expected to reproduce the performance of the quantum algorithms described here. Statistical Properties of the Objective Function

[0062] Let

[0063] The objective function of Eq.4 can then be written aswhere Cj⊂ {1, ... , n} is the set of variables present in the jthclause. Here the probability distribution of Eq.20 can be analyzedwhere the variables z1,...,zn∈ {+1,−1} are distributed uniformly at random. This also tells us the number of assignments achieving objective value t, namely 2npt.

[0064] The expectation value of any nontrivial product of variables in this ensemble is zero due to equal likelihood of values +1 and −1.

[0065] Consequently, in this ensemble, the expected value of each term in f is zero, and its standard deviation is one. Interestingly, in the absence of dependencies of the form bi⊕ bj= 0n, all terms in f are pairwise uncorrelated. In fact, the expectation value of any product of k terms from f is zero provided k is smaller than the minimum number of bit strings from the set {b1,...,bm} thatone needs to XOR together to obtain the all zeros string. We will call this quantity dmin, which can be interpreted as the minimum distance of the code L⊥.

[0066] From the above, when z is chosen uniformly at random, the values of the clauses behave much like a set of m independent identically distributed variables equally likely to take the values +1 and −1. Thus, the distribution over objective function values should look approximately binomial. That is, the probability for τ clauses to be satisfied will be well approximated by . We call this approximation the independent and identically distributed (iid) random variable model (or simply the iid model). In the convention used in these notes, namely (4), the value of the objective function is given by the number of satisfied clauses minus the number of unsatisfied clauses, i.e. t = 2τ − m. Thus,

[0067] A more precise statement than the ≃ symbol in Eq.22 can be made as discussed below: Let be the expectation value of fkaccording to the approximation in Eq.22 and let ^fk^ be the true expectation value of fk, when z1, ... , zmare sampled uniformly at random. By the above arguments ^fk^ = ^fk^iidk = 1,2, ... , dmin− 1. (23)

[0068] Some further consequences of this are discussed in the below “Rigidity of the Objective Distribution” section. Maximum Likelihood Objective Value

[0069] Suppose we can solve NCP in polynomial time on ℒ⊥out to distance ℓ. This allowsus to efficiently prepare the state . Suppose we then measure the computational basis andevaluate f on the resulting bit string. Under the iid assumption, (22) and (11), this would yield the following probability distribution over objective values,

[0070] In this section, the implications of this are clarified by estimating the maximum likelihood value of t from the distribution .

[0071] If t were continuous, the peaks of the probability distribution over objective values would occur where the derivative of with respect to t is zero. Since t may be an even integer, this corresponds to . By Eq.24 this yieldswhere, for notational simplicity, τ = 2ℓ is introduced. Eq.25 rearranges towhich yields

[0072] By t ≫ τ, the lefthand side of Eq.27 may be approximated with

[0073] By m ≫ t ≫ 1, the righthand side of Eq.27 may be approximated with

[0074] Substituting Eq.28 and Eq.29 into Eq.27 yieldswhich can be solved for t to obtain: ^^^^ ≃ ± √^^^^^^^^ = ± √2^^^^ℓ. (31)

[0075] This is an approximate analysis based on the iid model. However, it will be made more precise in the next section. More Precise Estimates of Objective Distribution

[0076] In the preceding section we saw that, for 1≪ ℓ ≪ m the probability distributionhas two peaks at ^^^^ ≃ ± √2^^^^ℓ according to the iid model. Here, we estimate more precisely theshape of these peaks and make a rigorous statement about what the results within the iid model imply about the true distribution.

[0077] Let ^^�^^(ℓ)^^^^denote the approximation to obtained using Stirling’s approximation ln x! ≃ x ln x − x and no other simplifications. Then, by (24):where l0is the logarithm of the normalization constant that is implicit in the ∝ symbol in Eq.24.

[0078] Taking a derivative yieldsr t ≪ m we expand the logarithms as ln�1 = ± ^^^^ ^^^^2

[0079] Fo^^^^+ ^^^^^^^^. Applying this approximation and solving for the value of t that renders this derivative zero yields t2= 2ℓm in agreement with the calculation in the above “Maximum Likelihood Objective Value” section.

[0080] Taking another derivative of Eq.33 yields the following (exact) expression for the second derivative of ln^^�^^(ℓ)^^^^. ^^^^2ln ^^�^^(ℓ) = −2ℓ − 1+2^^^^− 1^^^^^^^^2^^^^22 2−2^^^^. (34)

[0081] Recalling 1 ≪ ℓ ≪ m,

[0082] A Taylor expansion of ln^^�^^(ℓ)^^^^to second order in the distance (t − 2ℓm) may be obtained from the peak. Exponentiating yields an approximation to ^^�^^(ℓ) (ℓ)^^^^and hence to ^^^^^^^^as a sum of two Gaussians, namely �^^ �2 ��2�exp�− ^^−√2ℓ^^^^2^^^^� + exp�− ^^^^+√2ℓ^^^^2^^^^ , (37) where the factor√is dictated by normalization. Of course, one could systematically additional corrections to ^^�^^(ℓ)^^^^by keeping higher orders in Taylor expansion.

[0083] From Eq.37 approximations to the moments of t in the distribution may be obtained. Specifically,with all odd moments being zero.

[0084] A rigorous lower bound on the probability of obtaining a given cost function magnitude may be made in the exact case as a function of the moments of the iid model. For this purpose, the embodiments may employ the Payley-Zigmund inequality, which states that for any nonnegative random variable Z and any 0 ≤ θ ≤ 1, 2〈^^^^〉2 Pr[^^^^ > ^^^^〈^^^^〉] ≥ (1 − ^^^^)〈^^^^2〉. (40)

[0085] Now, apply this to Z = t2, where t is the number of satisfied clauses minus the number of unsatisfied clauses. If the matrix B arising from the clauses defines a code with distance more than 2ℓ + 4, then we are guaranteed that ^t2^ and ^t4^ arising from the stateare exactly equal to their values in the iid model. In this case, combining Eq.40 with Eq.38 and Eq.39 yields

[0086] Consequently,Quantitative Performance

[0087] By using Belief Propagation, various embodiments may achieve ℓ = ^BPm, with some values of ^BPfor small k and D tabulated in the first table 200 of Fig.2. From Eq.31 (as further justified in the above “More Precise Estimates of Objective Distribution” section), it can beshown that with high likelihood that ^^^^ ≥ √2^^^^BP^^^^. Recalling that t is the number of satisfiedclauses minus the number of unsatisfied clauses, we see that the fraction of satisfied clauses is1 / 2+^QIS with ^^^^QISFIG. 3 illustrates a second table 300, which provides additionalresults based on the results shown in the first table 200 of FIG.2, according to various embodiments. More particularly, in the second table 300 of FIG.3, some values are tabulated using known values of ^BPcollected in the first table 200 of FIG.2. Comparison Against Classical Randomized Algorithm

[0088] The following result has been previously proven.

[0089] Theorem 2. There is a constant c = exp(−O(k)) and a randomized algorithm running in time poly(m,n,exp(k)) that, given an instance I of max-k-XORSAT with m constraints of

[0090] Here valI denotes the fraction of constraints satisfied by x. In particular, for odd k, by trying the assignment and its negation, the algorithm can output an x satisfying val^^^^(^^^^) ≥1^^^^2+√^^^^.

[0091] In contrast, Quantum Interferometric Sampling has runtime polynomial in k whenever the uncomputation step does.

[0092] A simple bit-flipping algorithm achieves linear-time decoding of random LDPC codes satisfying k < 4 < D, with high probability, out to distance α∗m when α∗satisfies a certain set of complicated nonlinear conditions depending on k and D. The following lower bound characterizes this. α∗≥ α / 2 (43) α is the smallest positive solution to . (44)

[0093] Here, H is the base-e entropy function as defined in Eq.83.

[0094] Using this method for the uncomputation step in QIS algorithm, one achieves ^^^^ ≃√^^^^. By Eq.44, one sees that α depends only on the ratio k / D. Thus, for k / D fixed, for sufficiently large k, QIS will in polynomial time obtain solutions satisfying √m(1 / 2+Ω(1)) clauses, whereas the classical result of theorem satisfies only m(1 / 2 + exp(−k) / D) clauses and has runtime exponential in k. However, the results are not completely comparable, as we here consider an average case.

[0095] In practice, Belief Propagation is regarded as the best performing algorithm for decoding LDPC codes in most regimes of interest. The main advantage of the bit flipping method is the availability of simple bounds on its performance. Weighted max-XORSAT

[0096] In this section we consider how to solve weighted max-XORSAT problems in which the coefficientstake values outside of {+1,−1}. In this case, the quantum algorithm is identical, except that at the first step, the statedefined in Eq.13 can no longer be simply prepared using phase kickback.

[0097] To see how to prepare in the weighted case, the expression may be rewritten as

[0098] Analogous to the unweighted case, we can proceed by first preparingthen computing BTx into a second register of qubits through straightforward mod-2 matrix multiplication, and lastly uncomputing x using LDPC decoding algorithms. Thus, the only new task in the weighted case is the preparation of the state shown in Eq.46.

[0099] First, consider the case that the coefficientstake on only two magnitudes M1we canwhere(51)

[0100] The state shown in Eq.47 can be prepared by first preparing the statethen conditionally preparing the states |ϕ1^|ϕ2^ into additional registers, and lastly uncomputing |λ1^|λ2^ from the initial registers. The preparation of the state shown in Eq.52 can be performed efficiently because it is a state on only O(log(m)) qubits, and the individual amplitudes are efficiently computable. The uncomputation of |λ1^|λ2^ is straightforward, since these are simply the Hamming weights of the strings in the last two registers. The phasesQj:xrj=1 sign(Ji) can be efficiently applied to the states |ϕ1^ and |ϕ2^ using phase kickback. The efficient preparation of the uniform superposition over all bit strings of a given Hamming weight is also a solved problem, having drawn attention due to their significance in physics, where they are called Dicke states.

[0101] This procedure generalizes straightforwardly to the case where the coefficients take on any constant number of magnitudes. However, as the number of distinct magnitudes grows, the number of terms in the sum (51) grows exponentially, and the number of qubits in the state (52) grows linearly. We leave this general case as an open problem.

[0102] In weighted max-XORSAT (31) no longer applies. For example, in the case of two magnitudes, one has the following, under the iid model. Out of the s1clauses that have magnitudeM1let f1be the number that are satisfied, and similarly for f2. Then, the final measurement in ℓthorder QIS will yield the following probability distribution defined via Eq.53 and Eq.54 below.where ^^^^^^^^ = ∑^^^^1^^^1^ =0 ∑^^^^2^^^2^ =0^^^^(^^^1^ ,^^^2^ ). (54)

[0103] From f, the objective value t is given t = (2f1− s1)M1+ (2f2− s2)M2. (55)

[0104] Hence, from these formulas, the expected value of t can be calculated by direct summation. If the number of magnitudes of coefficients grows beyond two the number of terms in the sums becomes exponentially large and should instead use more indirect methods of evaluation. In particular, if the weights are all integers of polynomial magnitude then the expected value of t can be efficiently computed using dynamic programming. Beyond max-XORSAT

[0105] For quantum interferometric sampling to work efficiently, the embodiments may employ an optimization problem with the following properties: The objective function has a Fourier spectrum that is sparse or can be well-approximated as sparse. The coefficients in the sparse Fourier spectrum can be computed efficiently.

[0106] Here, the Fourier transform over , also called the Hadamard transform is referredto. The objective function for max-XORSAT manifestly has these properties. However, many other functions of interest have these properties too. Application of quantum interferometric sampling in this more general setting would be interesting to investigate. The observations given in these notes about max-XORSAT are a hint at a deeper and more general connection between sparsity of Fourier spectra of Boolean functions and quantum complexity. Beyond Average Case

[0107] Here we have largely considered the average-case performance of QIS on uniformly random instances of max-k-XORSAT with degree D, as such instances give rise to well-studied ensembles of LDPC codes. In this section we make two observations beyond average case. The first of these is that a variant of QIS can achieve a nontrivial lower bound on performance for adversarially chosen worst-case instances given a promise on the girth of the Tanner graph. Thesecond observation is a method to construct instances that superficially look like random instances but which will cause QIS to fail.

[0108] The performance of Belief Propagation on average-case instances of LDPC decoding is well understood, but its performance on adversarially constructed instances is much less so. For adversarially constructed instances of max-k-XORSAT we can use a variant of QIS in which the uncomputation step is done using linear programming approaches to LDPC decoding.

[0109] The most general statement of the performance of the linear programming decoder is that the algorithm may run in polynomial time and the output may be the nearest codeword if the distance of the input from the nearest codeword is below the fractional distance of the code, which is complicated to define but which can be evaluated in polynomial time using a linear program. A more convenient, though looser, characterization of the performance of the linear programming decoder is given in terms of the Tanner graph of the code. Let g be the girth of the code’s Tanner graph. Suppose the girth is at least three, the minimum degree δlof the bit vertices is at least three, and the minimum degree δrof the parity vertices is at least two. Then, the linear programming decoder will succeed out to distance[^^^^maxfrac / 2] − 1, where ^^^^maxfracis lower bounded by (δl− 1)[g / 4]-1.

[0110] Hence, by Eq.31, the following result may be shown. Let I be any instance of max- XORSAT in which the minimum number of variables in any clause is and the minimumnumber of clauses in which any variable appears is D . Let g be the of the Tanner graph the corresponding LDPC code L⊥(in which the role of variables and clauses are reversed). Consider QIS in which the linear programming decoder is implemented reversibly to perform the uncomputation step. If kmin≥ 3 and Dmin≥ 2 then this variant of QIS will with high probabilityachieve(Here, “with high probability” refers to high probability respect to repeated sampling. There may be no probabilistic ensemble of instances.)

[0111] A feature of this result is that it does not explicitly depend on k, the maximum number of variables per clause, or D the maximum number of clauses in which a variable can appear, which are more conventional metrics of the difficulty of a max-XORSAT instance. However, these quantities play an indirect role because Tanner graphs with higher degree will necessarily have smaller girth. To be precise, g ≤ clogd¯−1N, (56) where N is the number of vertices, ^̅^^^ is the average degree, and tightest possible value of c is not known. It is known that the tightest possible c satisfies c ≤ 2 and c ≥ 4 / 3.

[0112] It is informative to consider how to construct instances of max-k-XORSAT with bounded degree that superficially appear random but which will cause QIS to fail. One can ensurethat the distance of the LDPC code L⊥is O(1) by deciding upon a planted codeword of weight O(1) and following a standard procedure for generating a random LDPC code but discarding any potential parity checks that are incompatible with the planted codeword. In this case, for ℓ larger than constant, the uncomputation step will be ambiguous due to multiple linear combinations of codewords that have Hamming weight less than ℓ but yield the same bit string. If the number of such linear combinations is polynomial, one may classically compute the list of all of them and then uncompute their uniform superposition. However, this is beyond the scope of these notes. Adjusting the All-Zeros Amplitude

[0113] FIG.4 shows a quantum circuit 400 employed by various embodiments. More particularly, in the above “Quantum Interferometric Sampling” section, an efficient circuit that is functionally equivalent to the non-limiting quantum circuit 400. Quantum circuit 400 implements a unitary U operator that takesThe unitary operator is described below. Given the quantum circuit 400, which implements such a unitary U, the desired state may be prepared with minimal extra cost using the construction shown below.

[0114] The final unitary indicates flipping first bit if all n bottom bits are zero. From this circuit, one obtainsThe initial qubit in state |0^ can then be discarded. Rigidity of the Objective Distribution

[0115] One of the conclusions from the above “Statistical Properties of the Objective Function” section may be restated.

[0116] Theorem 3. Consider a max-XORSAT instance of n variables and m clauses. Let dminbe the minimum number of monomials from the cost function needed to achieve a trivial product (i.e. all variables raised to even powers). Let f be the random variable representing theB(m,1 / 2).

[0117] In this section, a universal upper bounds on how much the distribution of cost function values can deviate from the binomial distribution is set. Considering the below relevant facts.

[0118] Theorem 4. Let F and G be any two cumulative distribution functions, which have the the same first 2p moments. Them, for all values of x,whereand

[0119] Theorem 5. Let x be a random variable distributed according to the binomial distribution B(m,1 / 2). Thenand

[0120] Theorem 6. Let A be a complex n x n matrix with entries aij. Let

[0121] Let D(aii,Ri) ⊂ C be the closed disc centered at aiior radius Ri. Every eigenvalue of A lies within at least one of the discs D(aii,Ri).

[0122] We next prove the following,

[0123] Lemmaobeysand

[0124] Proof. The upper bound (63) can be proven as follows. u (65)

[0125] The upper bound (64) can be proven as follows. By theorem 6,

[0126] Using this lemma, the main result of this section can be proved.

[0127] Theorem 7. Consider a max-XORSAT instance of n variables and m clauses. Let dminbe the minimum number of monomials from the cost function needed to achieve a trivial product. Let F be the cumulative distribution function for the number of clauses satisfied when the input variables are assigned uniformly at random. Let G be the cumulative distribution function for

[0128] Proof. Let p = dmin− 1. By Theorems 3 and 4Other Classical Algorithms for LDPC Decoding

[0129] In the above text, the following three classical algorithms for decoding LDPC codes are discussed: Belief Propagation, a bit flipping algorithm, and linear programming approaches. In practice Belief Propagation may be the best algorithm for LDPC decoding. However, otherclassical algorithms are also worth noting, as they lend themselves to slightly different provable performance guarantees.

[0130] A variant of the bit flipping algorithm can solve the nearest codeword problem for LDPC codes whose Tanner graphs are expanders. Specifically, consider max-k-XORSAT in which each variable appears in exactly D clauses. The corresponding Tanner graph will be a (k,D)-regular bipartite graph in which 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 n = (k / D)m. (Note that the role of variables and constraints is reversed in the LDPC code relative to in the original max-k-XORSAT instance.)

[0131] Let V = Vvar∪ Vcondenote the Tanner graph’s set of vertices and let E denote its set of edges. If, for all S ⊂ Vvarwith |S| ≤ αm we have |{y : ∃x ∈ S such that (x,y) ∈ E}| > δ|S| then we say that the corresponding LDPC code is a (k,D,α,δ)-expander code. (Note that we inherit n as the number of constraints as a result of the reduction from a max-XORSAT problem. In most coding contexts it would be more conventional for n to denote the number of variables.) It has been proven that:

[0132] Theorem 8. If an LDPC code is a (k,D,α,3k / 4)-expander code then the nearest codeword problem can be solved by a polynomial-time classical algorithm given the promise that the distance is at most (α / 2)m. and

[0133] Theorem 9. Choose B uniformly at random from the set of all (k,D)-regular bipartite graphs with m k-regular vertices and n = (k / D)m D-regular vertices. Then, with high probability, for all 0 < α < 1, the corresponding LDPC code with this as its Tanner graph will be a (k, D, α, δ)-expander withwhere H is the base-e entropy function H(α) = −α ln α − (1 − α) ln(1 − α). (83)

[0134] One might worry about what happens if αm is greater than or equal to than half the distance of the code. In such a case the Nearest Codeword Problem is not guaranteed to have a unique solution. However, this cannot happen because the conditions in Theorem 8 also suffice to guarantee that the distance of the code is larger than 2αm.

[0135] Through numerical investigation, one finds that the guarantees from Theorem 9 suffice to meet the conditions of theorem 8 only when k and D very large, k / D is very close to one, and α is very small.Non-Limiting Methods of the Embodiments

[0136] FIG.5 depicts a flow diagram of an example method 500 for analyzing a system of equations, according to various embodiments. At block 502, a first polynomial function that corresponds to the set of DOF and the set of constraints is determined. The first polynomial function is polynomial with respect to each DOF of the set of DOF. At block 504, a set of qubits of a quantum computing system (QCS) is prepared in a quantum state. The quantum state of the set of qubits 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 that is characterized by a cardinality of the set of qubits. At block 506 and subsequent to preparing the set of qubits in the quantum state, the QCS performs a set of unitary operations on the set of qubits such that the quantum state of the set of qubits is transformed to a transformed state. The transformed state is representable by a second vector embedded in the Hilbert space. At block 508 and subsequent to performing the set of unitary operations on the set of qubits, the QCS generated / updated a set of observables by performing a measurement operation on each qubit of the set of qubits. At decision block 510, it is determined whether a threshold condition for the set of observables has been met. If the threshold condition for the set of observables has been met, then method 500 proceeds to block 512. Otherwise, method 500 returns to block 504. At block 512, a set of values is determined based on the set of observables. Each value of the set of values corresponds to a DOF of the set of DOF. Each value of the set of values simultaneously satisfies at least a portion of the set of constraints that is associated with the corresponding DOF.

[0137] 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 DOF and the set of constraints. The objective function is another polynomial. The first polynomial may be equivalent to the other polynomial function raised to a power that is equivalent to a positive integer. The positive integer may be greater than 1.

[0138] In various embodiments, each DOF of the set of DOF is a binary DOF. Each constraint of the set of constraints may include an exclusive OR (XOR) logic operation on two or more DOF of the set of DOF. A cardinality of the set of DOF may be less than a cardinality of the set of constraints such that the system is overly constrained and each constraint of the set of constraints is not simultaneously satisfiable via the set of values. The set of values maximizes a number of the set of constraints that is simultaneously satisfiable. The first vector corresponds to a linear superposition of basis states in a Z-basis of the set of qubits. Each term in the linear superposition corresponds to a constraint of the set of constraints. An amplitude of each term in the linear superposition corresponds to a parity associated with the corresponding constraint.Additional Embodiments

[0139] One example aspect of the present disclosure is directed to a method for optimizing a system of equations that includes a set of degrees of freedom (DOF) and a set of constraints on the set of DOF. Each constraint of the set of constraints is associated with a subset of the set of DOF. The method includes determining a first polynomial function that corresponds to the set of DOF and the set of constraints. The first polynomial function is polynomial with respect to each DOF of the set of DOF. A set of qubits of a quantum computing system (QCS) is prepared in a first quantum state that 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 that is characterized by a cardinality of the set of qubits. Subsequent to 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, wherein the transformed state is representable as a second vector embedded in the Hilbert space. Subsequent to 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 of the set of qubits. A set of values is determined based on the first set of observables. Each value of the set of values corresponds to a DOF of the set of DOF and each value of the set of values simultaneously satisfies at least a portion of the set of constraints that is associated with the corresponding DOF.

[0140] The method may further include subsequent to generating the first set of observables, preparing the set of qubits in a second quantum state that corresponds to the initial state. The method may further include subsequent to preparing the set of qubits in the second quantum state, performing, via the QCS, the set of unitary operations on the set of qubits such that the second quantum state of the set of qubits is transformed to the transformed state. The method may further include subsequent to performing the set of unitary operations on the set of qubits, generating a second set of observables by performing, via the QCS, a second measurement operation on each qubit of the set of qubits. The set of values is determined based on the first set of observables and the second set of observables.

[0141] In various embodiments, the set of unitary operations includes a Hadamard operation for each qubit of the set of qubits. The system of equations may be logically equivalent to a Nearest Codeword Problem (NCP). The system of equations may be a max-XORSAT problem. The system of equations of equations may be a combinatorial optimization problem. The combinatorial optimization problem may be an optimal routing problem. In other embodiments, the combinatorial optimization problem is an optimal scheduling problem.

[0142] The first polynomial function may be based on an objective function for the system of equations, wherein the objective function is based on the set of DOF and the set of constraints.

[0143] The objective function may be another polynomial and the first polynomial is equivalent to the other polynomial function raised to a power that is equivalent to a positive integer.

[0144] The positive integer may be greater than 1.

[0145] Each DOF of the set of DOFs may be a binary DOF.

[0146] Each constraint of the set of constraints includes an exclusive OR (XOR) logic operation on two or more DOF of the set of DOF.

[0147] A cardinality of the set of DOF is less than a cardinality of the set of constraints such that the system of equations is overly constrained and each constraint of the set of constraints is not simultaneously satisfiable via the set of values.

[0148] The set of values maximizes a number of the set of constraints that is simultaneously satisfiable.

[0149] The first vector corresponds to a linear superposition of basis states in a Z-basis of the set of qubits.

[0150] Each term in the linear superposition corresponds to a constraint of the set of constraints.

[0151] An amplitude of each term in the linear superposition corresponds to a parity associated with the corresponding constraint.

[0152] The set of unitary operations includes a Hadamard operation for each qubit of the set of qubits.

[0153] The system is logically equivalent to a Nearest Codeword Problem (NCP).

[0154] The system of equations is characteristic of a max-XORSAT problem.

[0155] The system of equations is a combinatorial optimization problem.

[0156] The combinatorial optimization problem is an optimal routing problem.

[0157] The combinatorial optimization problem is an optimal scheduling problem.

[0158] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational 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 combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computingsystems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0159] Implementations of the digital and / or quantum subject matter 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, or to control the operation of, data processing apparatus. 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 qubits / qubit structures, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0160] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states (e.g., qudits) are possible.

[0161] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantumcomputer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0162] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, 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 a quantum programming language, e.g., QCL, Quipper, Cirq, etc..

[0163] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds 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 coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0164] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0165] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause thesystem to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0166] Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0167] Some example elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0168] 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 memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD- ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0169] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. Thesystems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0170] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0171] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood 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 packaged into multiple software products.

[0172] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

WHAT IS CLAIMED IS:

1. A method for analyzing a system of equations that includes a set of degrees of freedom (DOF) and a set of constraints on the set of DOF, wherein each constraint of the set of constraints is associated with a subset of the set of DOF, the method comprising: determining a first polynomial function that corresponds to the set of DOF and the set of constraints, wherein the first polynomial function is polynomial with respect to each DOF of the set of DOF; preparing a set of qubits of a quantum computing system (QCS) in a first quantum state that 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 that is characterized by a cardinality of the set of qubits; subsequent to preparing the set of qubits in the first quantum state, performing, via the QCS, 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, wherein the transformed state is representable by a second vector embedded in the Hilbert space; subsequent to performing the set of unitary operations on the set of qubits, generating a first set of observables by performing, via the QCS, a first measurement operation on each qubit of the set of qubits; and determining a set of values based on the first set of observables, wherein each value of the set of values corresponds to a DOF of the set of DOF and each value of the set of values simultaneously satisfies at least a portion of the set of constraints that is associated with the corresponding DOF.

2. The method of claim 1, further comprising: subsequent to generating the first set of observables, preparing the set of qubits in a second quantum state that corresponds to the initial state; subsequent to preparing the set of qubits in the second quantum state, performing, via the QCS, the set of unitary operations on the set of qubits such that the second quantum state of the set of qubits is transformed to the transformed state; subsequent to performing the set of unitary operations on the set of qubits, generating a second set of observables by performing, via the QCS, a second measurement operation on each qubit of the set of qubits; anddetermining the set of values based on the first set of observables and the second set 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 set of DOF and the set of constraints.

4. The method of claim 3, wherein the objective function is another polynomial and the first polynomial is equivalent to the other polynomial function raised to a power that is equivalent to 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 of the set of DOFa is a binary DOF.

7. The method of claim 1, wherein each constraint of the set of constraints includes an exclusive OR (XOR) logic operation on two or more DOF of the set of DOF.

8. The method of claim 1, wherein a cardinality of the set of DOF is less than a cardinality of the set of constraints such that the system of equations is overly constrained and each constraint of the set of constraints is not simultaneously satisfiable via the set of values.

9. The method of claim 1, wherein the set of values maximizes a number of the set of constraints that is simultaneously satisfiable.

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

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

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

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

14. The method of claim 1, wherein the system of equations is logically equivalent to a Nearest Codeword Problem (NCP).

15. The method of claim 1, wherein the system of equations is characteristic of a 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; a multi-qubit logic gate; one or more processors; one or more memory devices, the 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 for characterizing the multi-qubit logic gate, the operations comprising: determining a first polynomial function that corresponds to a set of DOF and a set of constraints for a system of equations, wherein the first polynomial function is polynomial with respect to each DOF of the set of DOF; preparing the set of qubits in a first quantum state that 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 that is characterized by a cardinality of the set of qubits; subsequent to preparing the set of qubits in the first quantum state, performing, via the QCS, a set of unitary operations on the set of qubits such that the first quantumstate of the set of qubits is transformed to a transformed state, wherein the transformed state is representable a second vector embedded in the Hilbert space; subsequent to performing the set of unitary operations on the set of qubits, generating a first set of observables by performing, via the QCS, a first measurement operation on each qubit of the set of qubits; and determining a set of values based on the first set of observables, wherein each value of the set of values corresponds to a DOF of the set of DOF and each value of the set of values simultaneously satisfies at least a portion of the set of constraints that is associated with the corresponding DOF.

20. The QCS of claim 19, wherein the operations further comprise: subsequent to generating the first set of observables, preparing the set of qubits in a second quantum state that corresponds to the initial state; subsequent to preparing the set of qubits in the second quantum state, performing, via the QCS, the set of unitary operations on the set of qubits such that the second quantum state of the set of qubits is transformed to the transformed state; subsequent to performing the set of unitary operations on the set of qubits, generating a second set of observables by performing, via the QCS, a second measurement operation on each qubit of the set of qubits; and determining the set of values based on the first set of observables and the second set of observables.

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