Techniques for converting an optimization functional of a binary optimization problem into a cost function for quantum computing

Abstract

This disclosure relates to a method for converting an optimization functional of a binary optimization problem into a cost function for quantum computing. The method includes representing a first optimization variable among a plurality of binary optimization variables of the optimization functional as a first product of binary transformation variables, wherein the first product includes a first plurality of factors, each of which corresponds to a subset of a plurality of optimization variables including the first optimization variable. The method further includes converting each of the binary optimization variables or binary transformation variables into continuous variables; converting the optimization functional into a cost function, wherein the cost function includes the transformed continuous variables; and selecting a plurality of subsets for quantum computing.

Description

Technical Field

[0001] This disclosure relates to techniques for performing quantum computing using quantum computing networks, and in particular to techniques for converting the optimization functional of a binary optimization problem into a cost function for quantum computing. Background Technology

[0002] Quantum computers provide a platform for controllable quantum mechanical systems whose states and interactions can be controlled to perform computations. Computation is achieved through the evolution of the controllable quantum mechanical system, and the state of the quantum mechanical system can be measured to determine the computation result.

[0003] Quantum computers typically encode information into so-called "qubits," representing a quantum mechanical two-level system, the quantum mechanical equivalent of a classical bit. A qubit is a physical system whose quantum mechanical state can be (coherently) controlled and (essentially) maintained between two fundamental states during computation, denoted below by right-hand vector notation as |0> and |1>. As an example, qubits can be implemented by encoding information into the spin state of an electron (e.g., an electron in an "up" or "down" state), or into the polarization state of a photon, the state of a (superconducting) oscillator, the energy levels of an atom, and so on.

[0004] The manipulation of these qubits is commonly referred to as "quantum gates." Quantum gates can be coherently applied to qubits to change the state of a single qubit (so-called single-qubit gates) and to multiple qubits (so-called multi-qubit gates), such as entangled states of multiple qubits, and any combination thereof. For example, a single-qubit gate can rotate the spin state of an electron by an optional value, such as π / 2. Multi-qubit gates can be coherently applied to two or more qubits, such as performing a coherent CNOT operation on the states of two qubits. Multiple quantum gates can be applied in parallel or sequentially to the qubits of a quantum computer to perform computations. Finally, after applying a series of quantum gates, the states of the qubits can be measured (usually repeatedly) in a measurement unit to determine the probability of each possible outcome of the computation.

[0005] Quantum computers can utilize the unique properties of quantum mechanical states, particularly the superposition and entanglement of different quantum states, to find solutions with relatively few computational steps. These properties enable quantum computers to compute solutions to problems considered unsolvable on classical computers.

[0006] Combinatorial optimization is ubiquitous in many areas of industry and technology. Many of these optimization problems can be formulated as quadratic unconstrained binary optimization (QUBO) problems, or more generally, as polynomial unconstrained binary optimization (PUBO) problems. These are NP-hard problems, and quantum computers have shown the potential to outperform classical computers in solving them. Classical-quantum hybrid computers, particularly variable quantum computing networks, have been used for this task; see US patent application US2022 / 0101167 A1 and the paper "Towards large-scale quantum optimization solvers with few qubits" by M. Sciorilli et al., published on arXiv:2401.09421 v2, March 25, 2024.

[0007] However, current quantum computers have a finite number of logical qubits, so a major challenge is the number of qubits required for quantum solvers to enable quantum computers to outperform or even compete with classical computers. See the paper by M. Sciorilli et al. and the references cited therein for an overview of the current state of technology.

[0008] The papers by B. Tan et al., “Qubit-efficient encoding schemes for binary optimization problems” (published in Quantum 5 (2021) p. 454) and MR Perelshtein et al., “NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization” (published in Quantum 7 (2023) p. 1186), propose a simplified encoding (the so-called “minimum encoding”) to limit the number of qubits. However, minimum encoding comes at the cost of solution quality, which may be limited by local minima of the optimizing functional.

[0009] Given the current technology, there is a need for an improved encoding for binary optimization problems that would allow quantum computers to process these problems efficiently, particularly by reducing the number of qubits while still achieving high accuracy in the optimization results. Summary of the Invention

[0010] This objective is achieved by the method described in independent claim 1. The dependent claims relate to preferred embodiments.

[0011] According to a first aspect, this disclosure relates to a method for converting an optimization functional of a binary optimization problem into a cost function for quantum computing. The method includes: representing a first optimization variable among a plurality of binary optimization variables of the optimization functional as a first product of binary transformation variables, wherein the first product includes a first plurality of factors, wherein each of the first plurality of factors corresponds to a subset of a plurality of optimization variables including the first optimization variable; converting each of the binary optimization variables or binary transformation variables into continuous variables; converting the optimization functional into a cost function, wherein the cost function includes the transformed continuous variables; and selecting a plurality of subsets for quantum computing.

[0012] The optimization functional is sometimes referred to as the objective functional or objective function. The cost function of quantum computing also corresponds to the optimization functional of quantum computing, sometimes called the energy function or energy functional, although it does not necessarily represent the actual physical energy of the physical system.

[0013] The optimization of the transformed functional can be viewed as a simulation of local search, i.e., an algorithm that starts with candidate solutions and iteratively searches for better solutions within a predefined set (especially a neighborhood set). The optimized functional is particularly well-suited for implementation on quantum computers, capable of handling neighborhood sizes that are classically intractable. By appropriately selecting subsets, the cost function can be efficiently optimized on quantum computers using limited qubit resources without significantly sacrificing the accuracy of the optimization results, as will be explained in more detail below. In particular, the technique disclosed herein can construct a simulation of local search with a significantly larger number of neighbors than the minimum coding techniques in the prior art, thereby overcoming the limitations of minimum coding.

[0014] At first glance, the transformation of the optimization functional seems counterintuitive and disadvantageous, as it introduces additional complexity, which seems to make optimization more cumbersome. However, it turns out that the transformed optimization functional is more suitable for implementation on quantum computers, especially on variable quantum computing networks, which can compensate for the additional complexity. Therefore, the technique disclosed herein can allow for the fast and accurate solving of binary optimization problems on quantum computers with limited qubit resources.

[0015] According to the embodiments, the binary optimization is either quadratic unconstrained binary optimization (QUBO) or polynomial unconstrained binary optimization (PUBO).

[0016] Many combinatorial binary optimization problems can be formulated or reformulated as QUBO or PUBO problems. Furthermore, it is well known that QUBO is equivalent to solving the Ising model, a well-known physical model for studying spin-spin interactions.

[0017] Therefore, according to the embodiment, the optimized functional is the Ising energy function. The Ising energy function can represent the energy of multiple interacting spins, which may optionally be placed in an external force field, such as an external magnetic field.

[0018] Therefore, the technology of this invention allows for the efficient solution of the Ising model on a quantum computer with limited qubit resources.

[0019] According to the embodiment, each binary transformation variable may correspond to the flipping of multiple optimization variables in a corresponding subset. In other words, each binary transformation variable may correspond to the flipping of a specific subset of optimization variables, corresponding to a specific spin subset in the Ising model.

[0020] The variable transformation has already been described above with reference to the first optimization variable among multiple binary optimization variables. However, in the context of this disclosure, multiple or all optimization variables of the optimization functional may be transformed accordingly.

[0021] Specifically, the method may include representing a second optimization variable among a plurality of binary optimization variables of an optimized functional as a second product of binary transformation variables, wherein the second product includes a second plurality of factors, wherein each of the second plurality of factors corresponds to a subset of a plurality of optimization variables and includes the second optimization variable.

[0022] Typically, the method may include representing each of the multiple binary optimization variables of the optimized functional as a corresponding product of binary transformation variables, wherein the corresponding product includes a plurality of corresponding factors, wherein each factor corresponds to a subset of the plurality of optimization variables including the corresponding optimization variable.

[0023] The total number of subsets corresponds to the total number of binary transformation variables.

[0024] When selecting multiple subsets for quantum computing, one can consider making the quantum computing feasible on quantum computing networks, especially on variable quantum computing networks with a finite number of qubits.

[0025] Multiple subsets may include all subsets. However, in other embodiments, multiple subsets may include only a finite number of subsets, less than the total number of subsets.

[0026] According to an embodiment, selecting a plurality of subsets associated with a first optimization variable may include selecting a subset that includes optimization variables adjacent to the first optimization variable, particularly those adjacent to the first optimization variable in the graphical representation of the optimization functional.

[0027] Specifically, selecting multiple subsets related to the first optimization variable may include selecting only a subset of optimization variables that are adjacent to the first optimization variable, particularly in the graphical representation of the optimization functional.

[0028] Similarly, multiple subsets can be selected for all binary optimization variables.

[0029] Therefore, according to an embodiment, selecting multiple subsets associated with the corresponding optimization variable may include selecting a subset that includes optimization variables adjacent to the corresponding optimization variable, particularly those adjacent to the corresponding optimization variable in the graphical representation of the optimization functional.

[0030] Specifically, selecting multiple subsets related to the corresponding optimization variable includes selecting only a subset of optimization variables that are adjacent to the corresponding optimization variable, in particular selecting those adjacent to the corresponding optimization variable in the graphical representation of the optimization functional.

[0031] According to an embodiment, the method also includes representing each continuous variable with quantum probability, particularly through a transformation function.

[0032] In the context of this disclosure, quantum probability can correspond to the probability that the output state of a quantum computation will produce a specific measurement result.

[0033] By representing continuous variables using quantum probabilities, cost functions can be efficiently implemented on quantum computing networks. Quantum probabilities can establish a link between the measurement results of quantum computing and the cost function to be optimized.

[0034] According to the embodiment, the transformation function is a monotonic transformation function.

[0035] The monotonic transformation function makes the transformation from quantum probability to cost function particularly direct, and specifically guarantees that the transformation will not produce additional unnecessary local minima.

[0036] According to the embodiments, the transformation function is a differentiable function, especially a smooth function.

[0037] According to the embodiments, the transformation function is a step function, particularly a differentiable or smooth step function.

[0038] In the context of this disclosure, a step function can be a function that takes a first value or close to a first value (such as the value 1) in a continuous part of the parameter space. The step function can then descend to a second value or close to a second value (such as the value -1) in another part of the parameter space. This descent can be continuous.

[0039] Ladder-type transformation functions can effectively serve as filters for low quantum probabilities, thus making quantum computing more efficient, particularly by reducing the number of layers and measurements required for quantum computing.

[0040] According to embodiments, the method may also include optimizing the cost function, particularly through quantum computing.

[0041] Optimizing the cost function can include minimizing or maximizing the cost function.

[0042] According to an embodiment, optimizing the cost function may include determining the gradient of the cost function.

[0043] Gradient-based methods are ubiquitous and practical tools in the quantum implementation of binary optimization problems.

[0044] According to an embodiment, optimizing the cost function includes optimizing the cost function on a quantum computing network.

[0045] According to an embodiment, the cost function can be optimized on a quantum computing network including: a quantum register suitable for storing multiple qubits; multiple quantum gates suitable for acting on the multiple qubits, particularly quantum gates acting on the multiple qubits according to a selected subset; and a measurement unit.

[0046] Optimizing the cost function on a quantum computing network may include: initializing the qubits; applying multiple quantum gates to the qubits, particularly according to a selected subset, wherein the multiple quantum gates may include multi-qubit quantum gates acting on multiple qubits; determining the output state of the quantum computing network; and measuring the output state at a measurement unit.

[0047] Specifically, applying multiple quantum gates can include sequentially applying multiple layers of quantum gates to qubits, wherein each layer includes a multi-qubit quantum gate acting on multiple qubits, particularly depending on a selected subset.

[0048] Measurement results and output states can be used to determine the set of continuous variables after the transformation of the optimization cost function.

[0049] According to the embodiments, the quantum computing network is a quantum-classical hybrid network, particularly a variable quantum computing network.

[0050] The structure of the cost function (including the transformed continuous variables selected according to the chosen subset) is particularly well-suited for efficient computation in quantum computing with a finite number of qubits, especially in variable quantum computing networks.

[0051] According to an embodiment, the qubits and / or quantum gates of a quantum computing network can be selected based on a chosen subset.

[0052] For example, qubits and / or quantum gates can be selected to restrict the interaction between qubits to nearest-neighbor interactions, or to the interaction of qubits within a finite predefined neighborhood.

[0053] In an embodiment, the plurality of quantum gates include parameterized quantum gates, and the method includes optimizing a cost function based on parameters of the parameterized quantum gates, particularly by iteratively applying multiple layers of parameterized quantum gates to the qubits for different values ​​of the parameters.

[0054] According to the embodiments, parameters can be selected by means of optimization (such as optimization on neural networks).

[0055] According to the embodiments, the number of qubits is logarithmically proportional to the number of binary transformation variables, and / or logarithmically proportional to the number of selected subsets.

[0056] This logarithmic ratio makes it possible to run optimizations on quantum computers with limited qubit resources, while still allowing for a larger neighborhood compared to traditional minimum coding techniques, thereby improving computational accuracy.

[0057] According to an embodiment, the method includes determining an optimal set of continuous variables from measurements of the output state and converting the optimal set of continuous variables into an optimal set of binary optimization variables.

[0058] The optimal set of continuous variables can be the set of variables for optimizing (e.g., minimizing or maximizing) the cost function of a quantum computation. This solution can be transformed into the optimal set of binary optimization variables. The optimal set of binary optimization variables can be the set of variables for optimizing (e.g., minimizing or maximizing) the optimizing functional of a binary optimization problem.

[0059] The extrema (i.e. minimum and / or maximum) of the cost function in quantum computing can correspond to the extrema (i.e. minimum and / or maximum) of the optimization functional of a binary optimization problem, and vice versa.

[0060] According to an embodiment, converting the optimal set of continuous variables into the optimal set of binary optimization variables involves multiplying the continuous variables, particularly according to their respective selected subsets.

[0061] Therefore, by further transformation, the solution to the cost function (which can be found using quantum computing networks) can be transformed back into the solution to the original binary optimization problem.

[0062] In particular, the extreme points of the cost function (such as maximum and / or minimum) can correspond to the extreme points of the optimization functional (such as maximum and / or minimum), and vice versa.

[0063] According to the second aspect, this disclosure relates to a computer program and / or computer program product and / or storage component, the computer program and / or computer program product and / or storage component including computer-readable instructions, wherein, when executed on a computer system, the computer-readable instructions implement on the computer system a method having some or all of the features of the first aspect described above.

[0064] According to a third aspect, this disclosure relates to a quantum computing network, comprising: a quantum register suitable for storing a plurality of qubits; a plurality of quantum gates suitable for acting on the qubits; a measurement unit suitable for measuring the output state of the quantum computing network; and a control unit, wherein the control unit is coupled to the plurality of qubits, the plurality of quantum gates and the measurement unit, and is suitable for controlling the plurality of qubits and the plurality of quantum gates; wherein the control unit is suitable for implementing a method having some or all of the features of the first or second aspect described above.

[0065] In this embodiment, the quantum computing network is a quantum-classical hybrid network, specifically a variable quantum computing network.

[0066] According to an embodiment, multiple quantum gates can be grouped into multi-layer quantum gates, wherein each layer includes multi-qubit quantum gates suitable for acting on multiple qubits.

[0067] Quantum computing networks can be implemented using various techniques. For example, qubits can be realized as the spin state of an electron in a magnetic field, the energy level of an atom, the energy state of a superconducting oscillator, or the polarization state of a photon. This variability and versatility are particular advantages of the techniques described in this disclosure. Attached Figure Description

[0068] The features and advantages of the technology described herein will become most apparent from the description of exemplary embodiments with reference to the accompanying drawings, in which: Figure 1 This is a schematic diagram of a quantum computing network that can be used in the context of this disclosure; Figure 2 This is a schematic diagram of a variable quantum computing network that can be used in the context of this disclosure; Figure 3 This is a schematic diagram of a family of transformation functions that can be used in the context of this disclosure; Figure 4 This is a flowchart illustrating a method for converting the optimization functional of a binary optimization problem into a cost function for quantum computing, according to an embodiment. Figure 5 This is a schematic diagram illustrating the technique of finding a solution to an optimization problem related to the Ising ground state using a variable quantum circuit, according to an embodiment. Figure 6 Simulation results for the MaxCut optimization problem using the techniques disclosed herein are presented; and Figure 7 Simulation results obtained using the techniques disclosed herein for the graph coloring optimization problem are presented. Detailed Implementation

[0069] The techniques disclosed herein can be used to solve binary optimization problems on quantum computers. In the following description of exemplary embodiments, we focus on solutions to quadratic unconstrained binary optimization (QUBO) problems. However, we will also show that these techniques can be generalized to higher-order problems, particularly to solutions to polynomial unconstrained binary optimization (PUBO) problems in the same manner.

[0070] For QUBO, assume a given real upper triangular matrix For a certain positive integer n The challenge lies in finding a binary vector that minimizes the optimized functional. : (1) Through transformation It can be seen that QUBO is equivalent to the Ising model, which describes the energy of multiple interacting spins (possibly in an external force field): (2) in It is a spin variable.

[0071] The following equation relates the coefficients of equations (1) and (2): (3) Local search is a fundamental algorithm that has been successfully applied to many difficult combinatorial optimization problems. Local search algorithms start with candidate solutions and iteratively search for better solutions within a predefined set (usually a neighborhood), stopping when all neighbors are worse than the current solution. Many metaheuristic algorithms (such as simulated annealing, tabu search, and meme algorithms) are based on local search. Expanding the neighborhood may yield better solutions, but this increases computational complexity and runtime.

[0072] In the QUBO local search heuristic, neighbors are typically selected from a set of solutions that differ only in a few bits. When the Hamming distance is at most 1... (i.e., the difference does not exceed) r When all solutions (bits) are used as neighbors, the algorithm is called... - Flip or r - Local search. Parameters r A smaller value should generally be chosen because the number of neighbors and the required computing resources will increase accordingly. r Rapid growth. For example, in areas with nIn a problem with multiple variables, the number of neighbors equals the binomial coefficient. .

[0073] In the context of this disclosure, we propose a method for controlling quantum processing units by solving a combinatorial optimization problem using a variational algorithm, which achieves... One neighbor, only A quantum version of a local search for qubits, where It represents the "floor" of x, that is, the smallest integer not less than x.

[0074] (a) Conversion In the context of this embodiment, we focus on the Ising representation of the QUBO problem according to equation (2). First, we choose the initial state of the spins, where all spins are in the initial state. State. Then, we introduce a new one. A set of binary transformation variables Each variable corresponds to a specific spin subset. The flip. For each i = 1, …, n The state of the original Ising variable is decoded back through the following product: (4) This definition guarantees that the flip... Cause subset All spins A flip occurs.

[0075] Substituting equation (4) into equation (2), we get: (5) because Equation (5) can be rewritten as: (6) We use discrete variables Replace with continuous variables Continuing the derivation, we obtain the following cost function, which we can call... Auxiliary functions : (7) in This can be understood as representing The probability of.

[0076] (b) Properties of the transformed cost function The transformed cost function of equation (6) EIt is a multilinear function. C. Laneve et al., “The interval analysis of multilinear expressions,” published in Electronic Notes in Theoretical Computer Science, vol. 267, no. 2 (Oct. 2010), pp. 43-53, have shown that all local minima of a multilinear function occur at its vertices. Proposition 1 set up It is a multilinear function. Therefore, in the hypercube... Above, all local minima are At its apex—that is The point in the middle.

[0077] Proposition 1 describes the local minimum characteristics of multilinear functions on a hypercube. Specifically, Proposition 1 implies the use of continuous variables in equation (7). Replace the discrete variables in equation (6) It does not affect the local minimum, so in terms of optimization, equations (6) and (7) can be considered equivalent.

[0078] Proposition 2 set up (For example, for) ， (Is logical NOT). yes The local minimum is found if and only if for all ,have ， prove Without loss of generality, consider the vertices. . It is a local minimum if and only if for all partial derivatives Because a multilinear function is linear with respect to every variable when other variables are fixed: (8) Given We obtained the conclusions we needed.

[0079] Applying Proposition 2 to the cost function (7), we can conclude that the local minimum of this function is better than all the functions derived from the set. The neighbor solution is obtained by spin flipping the encoding in the middle. Therefore, we construct a similar method for local search, which has more neighbors than the minimum encoding.

[0080] Note that each product in equation (7) includes at most [number of products]. The expression includes 1 multiplier, and the entire expression includes 1 multiplier. The summation term, i.e., the complexity of estimating the cost function (7) does not exceed 1000. .

[0081] (c) Extending to higher-order problems The techniques described in Sections (a) and (b) above can be applied to polynomial unconstrained binary optimization (PUBO) problems, and are equivalently applied to generalized Ising models with higher-order spin interactions.

[0082] set up Indicates the number of variables. Therefore, the Ising energy can be written as: (9) in, For model parameters. Equation (9) extends Equation (2) to higher-order spin interactions.

[0083] Substituting the same variables from equation (4) into equation (9), we obtain: (10) Using the identity again The energy function (10) is simplified to a multilinear form, and the discrete variables are replaced with continuous variables in accordance with the same scheme described in Sections (a) and (b) above for quadratic problems.

[0084] (d) Neighborhood selection The set of coefficients in equation (7) Export the graph, whose nodes are: , side is , corresponding to the non-zero coefficient in equation (2) The inherent graph structure of the QUBO problem allows for the use of " Adjacent This term is used to describe index pairs. .

[0085] For optimization problems with fully interconnected interactions, given... Local search, without a general rule for selecting a neighbor set. Therefore, in this case, the Hamming distance is at most 1. All solutions can be included in the neighborhood set. However, we will show below that for sparse problems, it makes sense to select only a problem-specific subset of neighbors.

[0086] Let's assume we use minimal encoding and want to encode some pairs to increase the neighborhood set, thereby improving the accuracy of the solution. Consider reversing the... The increment of the energy function (2) after each spin: (11) From equation (11), we can see that depending on And those who are satisfied spin Therefore, reversing two non-adjacent spins... The increase in energy is For the local minimum of the minimum code, we have and ,therefore The result is that for non-adjacent pairs Encoding typically does not change the local minima of the minimum code. Therefore, to eliminate local minima, it makes sense to encode only adjacent pairs.

[0087] The same approach can be generalized to larger subsets: for any , If a set of spins Disjoint groups , and The union of the sets, then the reversal The energy increment after all spins is: .if and Already encoded, for Encoding does not contribute to eliminating local minima.

[0088] (e) Quantum circuits As shown in sections (a) through (d) above, we can achieve a local search of the original Ising problem by minimizing the auxiliary function derived from equation (7). The form of equation (7) makes it very suitable as a cost function for quantum computing. However, the variable (neighbor) The number can still be very large: from the smallest encoded In extreme cases In extreme cases It is a subset of all spins (all solutions are neighbors of each other). Therefore, in practically relevant cases, Limited by computational power. This can be addressed by introducing parameterized quantum circuits and controlling the variables. Encoding this into a circuit yields a quantum algorithm capable of handling neighbor numbers that are difficult to manage with classical methods. Our algorithm uses... A quantum circuit with 100 qubits.

[0089] Generally, quantum algorithms become heuristic algorithms because they optimize the parameters of quantum circuits rather than directly modifying them. With optimization, the above analysis of local minima is no longer directly applicable. Nevertheless, our numerical experiments on small problems show that, if the circuit depth is sufficient, the obtained solution is similar to that obtained by classical local search.

[0090] Figure 1 An example of a quantum computing network 10 that can be used to optimize equation (7) is schematically shown. The quantum computing network 10 includes a qubit register 12 suitable for storing multiple qubits. Multiple quantum gates 14 can act on the qubits in the qubit register 12 to perform quantum computation. The result of the quantum computation can be measured by a measurement unit 16, which can project the state of the qubits onto the computational fundamental state of the quantum computing network 10. The operation of the qubit register 12, the quantum gates 14, and the measurement unit 16 can be controlled by a control unit 18, which may include a (classical) processor and memory.

[0091] Control unit 18 can be configured to repeatedly execute a computation sequence on the qubits. This computation sequence may include initializing the qubits in qubit register 12 before each computation, such as initializing each qubit to a ground state, for example forming an initial state of |00…0>. Control unit 18 can then apply multiple quantum gates 14 to the qubits in qubit register 12 to drive the coherent evolution of the qubits. Initially, the controller can generate a superposition state of all qubits, for example, by applying a Hadamard gate to each qubit, and subsequently apply more quantum gates 14 as desired for the computation. After the coherent evolution, the state of the qubits in qubit register 12 can be measured by means of measurement unit 16. The measurement result is provided to control unit 18. Based on the measurement result, control unit 18 can classically compute the “energy” / “cost” of the solution using the cost function of the QUBO problem to be solved. Control unit 18 can then repeat the computation sequence based on the result, such as progressively improving the solution to the QUBO-type problem associated with the measurement result.

[0092] Figure 2 The implementation of the variable quantum computing network 20 is illustrated in the diagram below. The variable quantum computing network 20 is a combination of the above... Figure 1A specific implementation of the quantum computing network 10 described herein, with the corresponding components represented by the same reference numerals.

[0093] In the variable quantum computing network 20, quantum gates 14 include variable quantum gates with variable actions, whose physical parameters are adjusted according to the input of control unit 18. These physical parameters can be optimized by control unit 18 during classical optimization processes (e.g., using neural network techniques). Control unit 18 can analyze the measurement results of measurement unit 16 and then repeat the calculation sequence with adjusted variable actions based on those results, thereby progressively improving the solution to the QUBO-type problem associated with the measurement results. Specifically, control unit 18 can repeat the calculation sequence for the adjusted operating parameters of the variable quantum gates to determine the gradients of multiple quantum gates 14 from the measurement results, and can update the variable actions based on the estimated gradients, thereby progressively adjusting the quantum computing network 20 to a better solution.

[0094] Variable quantum computing network 20 includes a qubit register 12, which includes integers N q Each of the qubits is represented by q0, q1, ..., q5. Figure 2 Showing N q = 6 computational qubits, but this is just an example; other quantum computing networks may include more or fewer computational qubits. Furthermore, quantum computing network 20 may include so-called auxiliary qubits ( Figure 2 (Not shown in the image), they can assist in state preparation to realize quantum gate 14 or perform quantum measurement in measurement unit 16. Figure 2 In the diagram, the evolution of the state of each qubit is shown as a horizontal line extending from the qubit register 12 to the measurement unit 16.

[0095] Quantum bits can be initialized to their ground state, for example, |0>. Figure 2 As shown, the multiple quantum gates 14 may include multiple Hadamard gates (H), which act on each quantum bit in the quantum bit register 12 after the quantum bit is initialized, so that each quantum bit is prepared into a superposition state.

[0096] like Figure 2 As further shown, multiple quantum gates 14 can be arranged into layers 22a, 22b with similar or identical structures, and then the control unit 18 can apply layers 22a, 22b with their respective variational parameters. Each layer 22a, 22b may include multi-qubit gates for entanglement of multiple or all qubit states, as well as variational quantum gates that affect all qubit states. Figure 2An implementation of two layers of quantum gates 22a and 22b with identical structures is shown, where layers 22a and 22b represent multiple quantum operations performed on the qubits in the qubit register 12, which are then applied. However, in general, depending on the specific application, any number L of layers can be used in the variable quantum computing network 20.

[0097] In each layer 22a and 22b, variational single-qubit gates drive each qubit around... Z Axis single-qubit rotation R Z (θ) and around Y Axis single-qubit rotation R Y (θ), the two rotations each have their own variable angles. θ i The variational angles θ1-θ on all layers 22a and 22b N The variational parameters that constitute the quantum computing network 20 The set of parameters can be adjusted and optimized by the control unit 18. Therefore, the resulting quantum circuit has trainable parameters .

[0098] exist Figure 2 In this configuration, each layer 24a and 24b additionally includes multiple two-qubit gates that operate on pairs of all adjacent qubits in the qubit register 12. Figure 2 In the examples, the two-qubit gate includes the entangled ECR ("echo cross resonance") gate. The ECR gate is a native gate on many IBM quantum devices and can be represented by the following matrix: (12) After applying layers 22a and 22b of quantum gates to the qubit, the state of the qubit can be measured by measurement unit 16. Measurement unit 16 may include multiple single-qubit state detectors for measuring the state of each qubit after evolution according to multiple quantum gates 14. Repeated state evolution and measurement processes can determine the probability of each measurement result.

[0099] Control unit 18 can additionally use adjusted operating parameters of variable quantum gates. The sequence of calculations is repeated to determine the gradients of multiple quantum gates 14 based on the measurement results, and the variable operations can be updated based on the estimated gradients to gradually adjust the quantum computing network 20 to a better solution of equation (7).

[0100] Those skilled in the art will understand that Figure 2The types and arrangements of the gates are for illustrative purposes only, and the architecture of the quantum computing network 20 may differ from the depicted representation. For example, quantum CNOT (controlled NOT) gates can be used instead of ECR ​​gates as entanglement gates.

[0101] (f) Encode variables as quantum amplitudes In order to Figure 1 and Figure 2 The quantum circuit used for the optimization of the QUBO problem requires linking the measurement results obtained by the measurement unit 16 and processed by the control unit 18 with the optimization parameters of the cost function (7). For this purpose, we can use a transformation function that represents each variable in the cost function (7) with quantum probabilities. .

[0102] set up This is the state vector. Here we will encode... Introducing quantum probability , where index k Corresponding to the set of measurement results, such as those corresponding to the computational basis and having inherent normalization. The measurement result set. As an example, we choose the following nonlinear transformation function: (13) here, and These are additional hyperparameters.

[0103] Examples of transformational functionalities are in Figure 3 The middle is shown as MP k The function, for (Solid line) and (Dashed line) Two different values. Hyperparameters The smoothness of the step-type transformation function (13) is affected. We found that adjusting... This improves trainability. The transition function of equation (13) is monotonic, so it does not produce additional local minima.

[0104] In practical quantum devices, quantum probability is typically not directly obtainable. Typically, this can only be achieved through computational basis: Below The estimation is performed using the first measurement (shot), where the first... The result appeared This is because we want the algorithm to be able to handle tasks that are difficult to process using classical methods. Therefore, the number of measurements This should be sufficient to obtain a good estimate of the cost function. The cost function (7) is unaffected by... The impact. From Figure 3 It can be seen that when hour, Therefore, only It contributes to the cost function (7). One measurement should be sufficient to estimate it. Thus, the step-type transformation function (13) can serve as a filter with low quantum probability.

[0105] The limitations may involve the following challenges: due to And only It contributes to the cost function, therefore the number of variables with values ​​less than 0 will not exceed [a certain limit]. In the context of local search, this means we are effectively limiting the number of steps to 1. ,because This will lead to the corresponding group in the most likely solution. The reversal. However, this problem can be overcome through iteration, that is, starting a new round of optimization from the solution found in the previous round.

[0106] go through After the measurement, non-zero The quantity does not exceed One. Due to It does not contribute to the cost function in formula (7), therefore the function can be limited to no more than Efficient computation within a time step (in contrast, typically requires) (Time step).

[0107] (g) Optimization process Substitute the transformation function (13) into the cost function (7) and introduce as follows Figure 2 After parameterizing the quantum circuit as shown, we obtain the composite auxiliary function. To estimate the cost function, a formula can be prepared. Figure 2 Quantum circuits in computing bases and sampling Next, we obtain the estimation empirical probability distribution ,in Is providing the first The number of times each result is measured.

[0108] To optimize the cost function (7), we can use a gradient-based method and optimize the variational parameters. Therefore, we also need to calculate the cost function (7) relative to the variational parameters. The gradient. We pay particular attention to... (14) partial derivatives and The third partial derivative can be calculated by directly differentiating equations (7) and (13). The so-called Parameter offset rules To calculate: (15) See the paper “Evaluating analytic gradients on quantum hardware” by M. Schuld et al., published in Phys. Rev. A 99 (2019) 032331.

[0109] The optimization process can start from the parameters The process begins with random initialization. Then, the parameters are updated until convergence. Afterward, the circuit is measured again to obtain... The estimated value, i.e. The probabilities. These probabilities can be used to determine the most likely... Sampling is performed, and the corresponding solution is obtained using the transformation in equation (4). From this, the optimal solution can be selected. That is, according to equation (2), it has the minimum energy. The solution.

[0110] As described in section (f) above, one round of optimization can provide from The most at the beginning Continue the local search. To continue the local search, if... We can perform transformations. This transformation will map to The Ising coefficient is transformed as follows: .as well as These transformations introduce new optimization problems, for which... This becomes the initial solution. By repeating the above process on the transformed problem, the local search can continue iteratively.

[0111] (h) flow chart Figure 4 This is a flowchart illustrating a method for converting an optimization functional of a binary optimization problem into a cost function for quantum computing, according to an embodiment, such as the conversion described in sections (a) through (i) above.

[0112] In the first step S10, the first optimization variable among the multiple binary optimization variables is represented as a first product of binary transformation variables, wherein the first product includes a first plurality of factors, and each of the first plurality of factors corresponds to a subset of the plurality of optimization variables including the first optimization variable.

[0113] In the second step S12, each of the binary optimization variables or binary transformation variables is converted into a continuous variable.

[0114] In the third step S14, the optimization functional is transformed into a cost function for quantum computing, wherein the cost function includes the transformed continuous variables.

[0115] In the fourth step S16, multiple subsets are selected for quantum computing.

[0116] Figure 4 The flowchart necessarily shows the method steps in a certain order. However, those skilled in the art should understand that this disclosure is not limited to a specific temporal order, and the method steps according to this disclosure can be implemented in any feasible order. For example, the step of selecting multiple subsets for quantum computing can be performed at any time during the method, particularly before or after variable transformation. Furthermore, those skilled in the art should understand that two or more steps in steps S10 to S16 may not be implemented in chronological order, but can be combined into one step.

[0117] Figure 5 This is a conceptual diagram illustrating the conversion technique of this disclosure according to an embodiment, wherein the initial discrete Ising ground state problem is transformed into a continuous parameter optimization problem of a variable quantum computing network. The cost function of the quantum computation obtained by the Ising energy functional transformation serves as an auxiliary function for the original Ising ground state problem. Quantum computation, such as by means of a variable quantum computing network 20, can find approximate local minima of the auxiliary function with limited qubit resources. These local minima are precisely the Ising Hamiltonian in terms of... l The local minimum value on a custom neighborhood of a neighbor.

[0118] Since the variational quantum computing network 20 can be used N q = Achieved using qubits, and the number of optimized parameters for the variable quantum computing network 20. Usually less than l (at least for larger ones) l Therefore, the search space dimension for parameter optimization is smaller than that of other parameters. Groups and number of layers Number of measurements and value and It is a hyperparameter for the implementation method, which can be selected heuristically.

[0119] (i) Example implementation: Maximum cut problem One of the classic and well-studied combinatorial optimization problems is the so-called maximum cut problem, which searches for a way to partition the nodes of a graph into two complementary sets such that the total weight of the edges between these two sets is maximized. The maximum cut problem is equivalent to... and The Ising model at that time, in which It is the first one in the diagram. The and the first The weight of the edges between nodes.

[0120] In our experiment, we chose a 3-regular graph with 250 nodes. The weights were calculated from intervals. The samples were randomly sampled from a uniform distribution. We tested our algorithm by encoding different numbers of neighbors. Since the problem is sparse, we selected all connected groups. This makes it possible for four distinct values All The algorithm starts from 100 random initial solutions. Startup begins. L-BFGS-B is used as the optimizer; see RH Byrd et al.'s paper "A limited memory algorithm for bound constrained optimization," published in SIAM J. Sci. Comput. 16(1995) pp. 1190-1208. After optimization, the found solution is used as a new initial point, and the algorithm is restarted. Several rounds of this process have been performed. For comparison, we start with the same initial solution and run a classic local search using the same neighbors.

[0121] The table below shows four different values. The experimental parameters are as follows. The table shows the number of qubits. N q As a function of the group number l, it exhibits a logarithmic proportional relationship.

[0122]

[0123] The empirical cumulative distribution function (ECDF) of the objective function is calculated using the solution set of the final solution obtained from 100 independent runs. The results are as follows: Figure 6 As shown, the graph displays four different values. Pairs of quantum and classical solutions (from right to left, solid lines represent quantum solutions and dashed lines represent classical solutions).

[0124] from Figure 6 It can be seen that the more neighbors the encoding has, the better the solution obtained. Using... Encoding can find the optimal solution. Experiments confirm that the quantum algorithm can find solutions similar to classical local search in the corresponding neighborhood. Furthermore, we emphasize that our quantum algorithm has an advantage over classical optimization of the cost function in equation (7) in terms of the number of variational parameters. As can be seen from the table comparing the number of groups and circuit parameters above, with With the increase in [the number of users], this advantage becomes even more pronounced.

[0125] (j) Example implementation: Graph coloring problem To illustrate a more complex neighbor selection problem, we consider a constrained problem: graph coloring. Graph coloring is a method of coloring the vertices of a graph such that no two adjacent vertices are assigned the same color. The QUBO formula for the graph coloring problem can be written in the following form: (16) See the paper “Quantum optimization for the graph coloring problem with space-efficient embedding” by Z. Tabi et al., published on arXiv:2009.07314 v1, September 15, 2020.

[0126] In equation (16), This represents the graph adjacency matrix. The bits in this formula... Dual index ,in Mark vertices, Coloring is used. Therefore, a feasible solution (each vertex colored with a different color) is restricted to a solution for each index. Exactly one bit In the case of equation (16), there is a pre-factor. The first item aims to punish infeasible solutions. Then, in The desired coloring can be obtained at this location.

[0127] Existing minimum coding methods are ineffective for this problem because if a bit is reversed, any feasible solution becomes infeasible. Therefore, due to the penalty term in equation (16), any feasible solution is a local minimum.

[0128] To ensure that the neighborhood of one feasible solution includes another feasible solution, we choose variables that, for all and Each variable will flip bit pairs. This set includes all groups that recolor any vertex to a different color. Note that this selection is more efficient than selecting all connected pairs, because the latter includes pairs that do not preserve feasibility. .

[0129] We selected excerpts from the paper by Z. Tabi et al. myciel7 Example, and the following parameters were used in the experiment: number of vertices Number of sides: 2360, Number of colors: The experiment employed the same scheme as the maximum cut problem described in section (i) above. For quantum computing, 100 random feasible solutions were selected as the initial solutions, and the experimental parameters were as follows: 5348 groups, 13 qubits, 20 layers. , .

[0130] Figure 7 and Figure 6 Similarly, the empirical cumulative distribution function (ECDF) of the objective function for solutions after two, three, and four iterations (from right to left) is shown. After four rounds of optimization (using the final solution of the previous round as the initial solution for the next round), our algorithm finds the correct coloring with a 20% probability. Note that one-hot encoded algorithms (including QAOA) require 1528 qubits for this problem, far exceeding the capabilities of current quantum computing devices.

[0131] The description and accompanying drawings of the specific embodiments are for illustrative purposes only and are not intended to imply any limitations. The scope of the invention should be determined by the appended claims.

[0132] Figure Labels 10 Quantum Computing Networks 12-qubit register 14 Quantum Gates 16 Measurement Units 18 Control Unit 20 Variable Quantum Computing Networks 22a, 22b: The first and second layers of quantum gates.

Claims

1. A method for converting the optimization functional of a binary optimization problem into a cost function for quantum computing, the method comprising: The first optimization variable among the plurality of binary optimization variables of the optimization functional is represented (S10) as a first product of binary transformation variables, wherein the first product includes a first plurality of factors, wherein each of the first plurality of factors corresponds to a subset of the plurality of optimization variables including the first optimization variable; Transform each of the binary optimization variables or the binary transformation variables into a continuous variable (S12); The optimized functional is transformed (S14) into the cost function, wherein the cost function includes the transformed continuous variables; and Select (S16) a plurality of said subsets for said quantum computing.

2. The method according to claim 1, wherein, Each binary transformation variable corresponds to the flipping of the multiple optimization variables in the corresponding subset.

3. The method according to claim 1 or 2, wherein, Selecting (S16) the plurality of subsets associated with the first optimization variable includes selecting a subset of optimization variables that are adjacent to the first optimization variable, particularly in the graphical representation of the optimization functional.

4. The method according to any one of the preceding claims further includes: Each of the continuous variables is represented by quantum probability, particularly through a transformation function.

5. The method according to any one of the preceding claims further comprises: The cost function is optimized, particularly through quantum computing.

6. The method according to claim 5, wherein, Optimizing the cost function includes determining the gradient of the cost function.

7. The method according to claim 5 or 6, wherein, Optimizing the cost function includes optimizing the cost function on a quantum computing network (10, 20), particularly on a quantum computing network (10, 20) comprising a quantum register (12) suitable for storing multiple qubits, multiple quantum gates (14) suitable for acting on the multiple qubits, and a measurement unit (16).

8. The method according to claim 7, wherein, Optimizing the cost function on the quantum computing network (10, 20) includes: Initialize the qubits in the quantum register (12); The quantum gate (14) is applied to the qubit, wherein the quantum gate (14) includes a multi-qubit quantum gate that acts on a plurality of the qubits, particularly according to a selected subset; Determine the output state of the quantum computing network (10, 20); and The output state is measured at the measurement unit (16).

9. The method according to claim 7 or 8, wherein, The quantum computing networks (10, 20) are quantum-classical hybrid networks, particularly the variable quantum computing network (20).

10. The method according to any one of claims 7 to 9, wherein, The plurality of quantum gates includes a parameterized quantum gate (14), and the method further includes optimizing the cost function by means of the parameters of the parameterized quantum gate (14), in particular by iteratively applying the parameterized quantum gates in multiple layers (22a, 22b) to the qubit for different values ​​of the parameters.

11. The method according to any one of claims 7 to 10, wherein, The number of qubits is logarithmically proportional to the number of binary transformation variables, and / or logarithmically proportional to the number of selected subsets.

12. The method of claim 8, optionally combined with any one of claims 9 to 11, further comprising: The optimal set of continuous variables is determined from the measurements of the output state; as well as The optimal set of the continuous variables is transformed into the optimal set of the binary optimization variables.

13. The method according to claim 12, wherein, Transforming the optimal set of the continuous variables into the optimal set of the binary optimized variables includes multiplying the continuous variables, in particular according to their respective selected subsets.

14. A computer program comprising computer-readable instructions, wherein, When the computer-readable instructions are executed on a computer system, the method according to any one of the preceding claims is implemented on the computer system.

15. A quantum computing network (10, 20), comprising: A quantum register suitable for storing multiple qubits (12); Multiple quantum gates (14) suitable for acting on the qubit; A measurement unit (16) suitable for measuring the output state of the quantum computing network (10, 20); and Control unit (18), wherein the control unit (18) is coupled to the plurality of qubits, the plurality of quantum gates (14) and the measurement unit (16), and is adapted to control the plurality of qubits and the plurality of quantum gates (14). The control unit (18) is adapted to implement the method according to any one of claims 1 to 13.

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