Quantum search algorithm and method

Abstract

To provide a quantum search algorithm, when searching for feasible solutions to combinatorial optimization problems with constraints on a quantum computer, using a divide-and-conquer method to search for all feasible solutions that satisfy the constraints with a small amount of computation.SOLUTION: An optimization problem solving algorithm 1 primarily consists of an initialization unit 10, a space division unit 20, and an integration unit 30. The initialization unit 10 prepares operators for an entire space (initial state) that represents all combination states relating to a feasible solution. The space division unit 20 divides the operators representing the entire space of the search range into a plurality of subspaces for every constraint condition in order to divide the constraint conditions of the combinatorial optimization problem into a plurality of subproblems. The integration unit 30 embeds each divided operator an appropriate number of times in the same way, executes calculation processing, and executes measurement.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a quantum search algorithm and method for searching for feasible solutions to combinatorial optimization problems on a quantum computer, and more particularly to a quantum search algorithm and method for quickly searching for all feasible solutions that satisfy constraints of a combinatorial optimization problem using a divide-and-conquer method. [Background technology]

[0002] The divide-and-conquer algorithm is an approximation algorithm that divides a problem into multiple small subproblems, solves each subproblem independently, and then combines the results to find a solution to the original problem. It is one of the techniques used for large-scale problems such as combinatorial optimization problems.

[0003] Non-Patent Publication 1 proposes a method for solving the Traveling Salesman Problem (TSP), a representative combinatorial optimization problem, using a divide-and-conquer algorithm. Specifically, the set of cities is first divided into non-overlapping subsets, and each set is considered a subproblem that can be solved independently of the other divided sets. A two-stage approach is used to solve each subproblem.

[0004] In the first stage, an initial solution is obtained by applying a local search algorithm to each set. In the second stage, the initial solution is refined using an iterative method to obtain an optimal or near-optimal solution. After solving each subproblem, the solutions of each subproblem are combined into the original problem using K-opt. This two-stage approach, combined with the K-opt method, allows the efficient solving of larger traveling salesman problems.

[0005] On the other hand, quantum computers are expected to be able to quickly solve large-scale problems such as combinatorial optimization problems. A quantum search algorithm is an algorithm that uses a quantum computer to reduce the amount of calculation required to search for a solution compared to a classical computer.

[0006] As a method for quickly solving combinatorial optimization problems using a quantum search algorithm, Non-Patent Publication 2 presents a method for solving the optimal solution to the traveling salesman problem, which is a combinatorial optimization problem, using a classical hybrid quantum search algorithm.

[0007] This algorithm can calculate the corresponding path cost by inputting the path information of any one feasible solution as an input value. In the quantum search algorithm of Non-Patent Document 2, the Grover algorithm, which uses an oracle operator that rotates the phase of any solution by π, is extended, and a cost oracle operator that rotates by the path cost and Grover's diffusion matrix are used, and all feasible solutions are input into the input state, thereby amplifying the path state with the minimum cost. [Prior art documents] [Non-patent literature]

[0008] [Non-Patent Document 1] K. Srinivasan et al. "Solving for large-scale traveling salesman problem with divide-and-conquer strategy", SCIS & ICIS (2010). [Non-Patent Document 2] D. Koch et al. "Gaussian Amplitude Amplification for Quantum Pathfinding", Entropy 24, no. 7: 963 (2022). Summary of the Invention [Problem to be solved by the invention]

[0009] It is not easy to implement a divide-and-conquer algorithm that runs on a classical computer on a quantum computer. This is because classical computers and quantum computers work completely differently. In a classical computer, a given problem is solved by executing instructions in a sequence. For example, when adding two numbers on a classical computer, the addition is performed by executing each operation one by one.

[0010] In contrast, a quantum computer uses qubits to process multiple input states simultaneously, so instead of a series of operations to perform an addition as in a classical computer, it performs a single operation to obtain the result of the addition.

[0011] In the divide and conquer method, a problem is divided into small subproblems, each of which is solved independently, and then the results are combined to find a solution to the original problem. However, since a quantum computer only prepares and executes operators to solve a given problem, it is difficult to carry out the process of solving the results of multiple independent subproblems individually and combining them efficiently. Therefore, a specific method for implementing the divide and conquer method in a quantum algorithm is not known.

[0012] An object of the present invention is to provide a quantum search algorithm and method that can solve the above technical problems and significantly reduce the amount of calculation required when searching for all solutions that satisfy the constraints of a combinatorial optimization problem using a divide-and-conquer method. [Means for solving the problem]

[0013] In order to achieve the above object, the present invention is characterized in that a quantum search algorithm for searching for a feasible solution to a combinatorial optimization problem on a quantum computer using a divide-and-conquer method has the following configuration.

[0014] (1) The present invention is characterized by including an initialization unit that prepares an entire space of quantum states that represent all combination states of a combinatorial optimization problem, a space division unit that divides the entire space into a plurality of subspaces for each constraint condition of the combinatorial optimization problem, and an integration unit that integrates the results of simultaneously executing a quantum search for each quantum state of the subspaces.

[0015] (2) The quantum search is a Grover search using the Grover algorithm, in which an oracle operator and a Grover diffusion operator are repeatedly applied alternately, and when the space division unit divides the entire space into N, the integration unit divides the oracle operator and the Grover diffusion operator into N each, and a search is performed simultaneously on each of the N subspaces.

[0016] The present invention can be realized not only as a quantum search algorithm having such a characteristic configuration, but also as a quantum search method having such characteristic processing steps. Effect of the Invention

[0017] According to the present invention, the following effects can be achieved.

[0018] (1) By dividing the quantum state for each constraint of the combinatorial optimization problem, applying the Grover algorithm to the divided states, and using an integrated quantum search algorithm, the amount of calculations required to calculate a feasible solution can be significantly reduced.

[0019] (2) By splitting the oracle operators that determine the constraints and applying them simultaneously, the search space is doubled compared to Grover's algorithm. n ^ 2 From 2 n , which can reduce the optimal number of iterations Q. [Brief description of the drawings]

[0020] [Figure 1]FIG. 1 is a functional block diagram illustrating a configuration of a quantum search algorithm for solving a combinatorial optimization problem according to an embodiment of the present invention. [Diagram 2] FIG. 1 is a diagram showing an example of a quantum circuit of this embodiment implemented on a quantum computer to search for a feasible solution of a TSP. [Diagram 3] FIG. 1 shows an example of a conventional quantum circuit implemented on a quantum computer to search for a feasible solution to the TSP. [Figure 4] FIG. 1 is a diagram comparing the effects of the present invention with those of the prior art. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0021] Hereinafter, an embodiment of the present invention will be described in detail with reference to the drawings. Fig. 1 is a functional block diagram showing the configuration of a quantum search algorithm 1 for solving a combinatorial optimization problem according to an embodiment of the present invention, and the main components thereof are an initialization unit 10, a space division unit 20, and a merging unit 30.

[0022] Such an algorithm 1 is generated on a classical computer such as a general-purpose computer or server equipped with a CPU, ROM, RAM, bus, interface, etc., and implemented on a quantum computer, and searches for all feasible solutions that satisfy the constraint conditions of the combinatorial optimization problem on the quantum computer by a divide-and-conquer method. The search results for feasible solutions are provided to an optimal solution search algorithm (e.g., Non-Patent Document 2) as necessary. The optimal solution search algorithm further searches for an optimal solution from the search results for feasible solutions.

[0023] The initialization unit 10 prepares an operator of the entire space (initial state) that expresses all combination states related to a feasible solution. In this embodiment, the Hadamard operator is applied to the number of quantum bits q required to express the solution to the optimization problem, thereby generating all input patterns that satisfy / do not satisfy the constraint conditions of the combinatorial optimization problem as the entire space. Therefore, the input form is a binary variable of length q equal to the number of quantum bits.

[0024] The space dividing unit 20 divides the operators expressing the entire space of the search range into a plurality of subspaces for each constraint condition in order to divide the constraint conditions of the combinatorial optimization problem into a plurality of subproblems.

[0025] Here, while the well-known divide and conquer method divides a problem spatially, this embodiment is characterized in that it divides a combinatorial optimization problem into multiple sub-problems for each constraint and then implements them. Therefore, in the following description, dividing an optimization problem into multiple sub-problems and then implementing them may be referred to as the "divide and conquer method."

[0026] The integrator 30 executes a quantum search for each quantum state in each of the spatially divided subspaces, and outputs all feasible solutions by integrating the search results.

[0027] The basic mechanism of quantum search is to start with a uniform superposition of all basis states, and then apply an oracle operator consisting of a Boolean oracle function that gives an output of 1 if the solution condition is satisfied, and returns an output of 0 if it is not satisfied, to only the basis of the solution. In addition, the Grover diffusion operator is applied to amplify the solution by repeating the coefficients of the basis state corresponding to the solution an appropriate number of times.

[0028] In this embodiment, for the entire space created by the quantum states created by the initialization unit 10, a quantum search algorithm that is an improvement over conventional quantum search is implemented for each constraint condition of an arbitrary combinatorial optimization problem.

[0029] In other words, while conventional quantum search prepares one oracle operator and Grover diffusion operator equal to the scale of the total number of quantum bits q, in this embodiment, an oracle operator and Grover diffusion operator are prepared that are divided into d for each constraint, i.e., with a size equal to the number of quantum bits q / d.

[0030] The integration unit 30 embeds each divided operator an appropriate number of times in the same way, performs calculation processing, and performs measurement. In the quantum search of this embodiment, a calculation is performed to search for a partial solution corresponding to each input state, but it is designed so that a binary variable equal to the total number of quantum bits q can be obtained as an integrated solution so that all feasible solutions, which are the overall solution, can be extracted by performing an overall measurement.

[0031] In this way, the division in the divide-and-conquer method of this embodiment corresponds to dividing the constraints of a combinatorial optimization problem into small problems (sub-problems) and embedding them. "Conquer" means performing parallel implementation on a single circuit.

[0032] Next, the operation of this embodiment will be specifically described using the Traveling Salesman Problem (TSP) as an example of a typical combinatorial optimization problem with constraints.

[0033] In the traveling salesman problem, where a salesman visits multiple cities in order, if the number of cities is n, then n×n quantum bits are prepared. The total space in this case is 2 n×n In this embodiment, an example in which the number of cities to be visited is n=3 will be described.

[0034] 2 and 3 show examples of quantum circuits implemented on a quantum computer to find a feasible solution |T> for the TSP. In particular, FIG. 2 shows the configuration of this embodiment, and FIG. 3 shows the configuration of the prior art for comparison.

[0035] Here, the 3 × 3 input quantum bits are labeled in ascending order with the variable i (1≦i≦3) representing the city number and the variable t (1≦t≦3) representing the visiting order as subscripts. In either configuration, the quantum state is input from the Hadamard gate (H) on the left side of the figure, and a measurement is performed on the output of the Grover diffusion operator (W) on the right side.

[0036] The traveling salesman problem (TSP) is mathematically proven to be equivalent to the Ising model of a quantum computer if its formulation uses quadratic unconstrained binary optimization (QUBO). Therefore, in this embodiment, the formulation using QUBO is considered.

[0037] In the formulation of the TSP using QUBO, first, when the i-th city is visited at the t-th time, the binary variable α ti is set to "1" (α ti = 1). Therefore, first, quantum bits for all binary variables α ti where 1 < t < n and 1 < i < n are prepared in the quantum circuit. In the examples of FIGS. 2 and 3, since n = 3, a total of n 2 = 9 binary variables α ti corresponding quantum bits |α 11 > to |α 33 > are prepared.

[0038] Next, in order to obtain an executable solution, the constraint conditions are embedded in the quantum circuit. The TSP has the following two constraints.

[0039] Constraint 1: Exactly one city must be visited at each time step t. Constraint 2: All cities must be visited exactly once.

[0040] When the above two constraints 1 and 2 are expressed in QUBO notation, the following two equations (1) and (2) become the constraint conditions to be solved.

[0041]

Number

[0042]

Number

[0043] Each constraint condition H1 QUBO , H2 QUBOTo represent this, we use n for each of the n cities. 2 qubits are required.

[0044] TSP H1 QUBO ,H2 QUBO In order to find a feasible solution that satisfies the above condition by quantum search, the conventional quantization method shown in Fig. 3 requires finding all feasible solutions of the TSP using the Grover algorithm. Here, the Grover algorithm is an algorithm for efficiently finding a specific solution.

[0045] The Grover algorithm is based on two constraints: H1 QUBO ,H2 QUBO It consists of an oracle operator R that determines the input that satisfies and a Grover diffusion operator W. In constructing the Grover algorithm, we first prepare a quantum state in which all possible solutions are selected with equal probability. This corresponds to the first black box H (Hadamard gate).

[0046] Then, the second black box R has two constraints H1 QUBO ,H2 QUBO The oracle operator R is used to determine solutions that satisfy the following: The oracle operator R marks feasible solutions of the TSP.

[0047] The third black box, W, is a Grover diffusion operator (matrix) that amplifies the probability of only the marked feasible solution. Then, by repeating the operation of alternating between the oracle operator R and the Grover diffusion operator W an optimal number of times Q, we can finally obtain the marked feasible solution with a high probability.

[0048] However, the conventional quantum search algorithm has two n^2 Therefore, the optimal number of iterations Q required to find all feasible solutions grows exponentially with the number of cities n to be visited.

[0049] Therefore, in this embodiment, the two constraints 1 and 2 are divided into subproblems, and the size of the oracle operator R and the diffusion operator W is reduced, thereby reducing the number of iterations Q required to extract a feasible solution.

[0050] In this embodiment, as shown in FIG. 2, each constraint H1 QUBO ,H2 QUBO Divide into multiple subproblems h(t) and h(i) and apply quantum search to each subproblem. For example, if the TSP has three cities, divide into a total of six subproblems: h(t=1), h(t=2), h(t=3), h(i=1), h(i=2), and h(i=3).

[0051] Here, h(t=1) is the subproblem with order t fixed to 1, h(t=2) is the subproblem with order t fixed to 2, h(t=3) is the subproblem with order t fixed to 3, h(i=1) is the subproblem with city i fixed to 1, h(i=2) is the subproblem with city i fixed to 2, and h(i=3) is the subproblem with city i fixed to 3.

[0052] Then, for each of these six subproblems, six oracle operators R are simultaneously implemented on a single circuit to correspond to the divide-and-conquer method.

[0053] In this embodiment, inside the oracle operator R shown in FIG. 2, only the solution state of each subproblem is marked as in the following equation (3).

[0054]

number

[0055] In TSP, the solution state |x0> n corresponds to One Hot Label. -iθThe θ value is the phase when rotating the solution in the oracle operator, and in the case of normal quantum search, θ=-π or θ=π, but in this embodiment, since TSP has two constraints, θ=-π / 2 was experimentally determined. When this θ value is used, all feasible solutions are amplified simultaneously at the iteration count Q. The Grover diffusion operator W has the same configuration as the conventional method, although its size is different.

[0056] According to this embodiment, the search space is expanded by 2 compared to the conventional Grover algorithm shown in FIG. n^2 From 2 n Therefore, the optimal number of iterations Q can be reduced.

[0057] FIG. 4 is a diagram comparing data when feasible solutions of the TSP according to the prior art of FIG. 3 and the present embodiment of FIG. 2 are obtained, and the definitions of each symbol are as shown below the diagram.

[0058] According to this embodiment, the amount of calculation Q is significantly smaller than that of the conventional technology, but the accuracy is reduced. However, the overall accuracy comparison Q / P 1 / 2 From the comparison results, the effects of the present invention are as follows.

[0059] (1) By dividing the quantum state for each constraint of the combinatorial optimization problem, applying the Grover algorithm to the divided states, and using an integrated quantum search algorithm, the amount of calculations required to calculate a feasible solution can be significantly reduced.

[0060] (2) By splitting the oracle operators that determine the constraints and applying them simultaneously, the search space is doubled compared to Grover's algorithm. n^2 From 2 n , which can reduce the optimal number of iterations Q.

[0061] In the above embodiment, the present invention has been described using an example of its application to the search for a feasible solution to the traveling salesman problem. However, the present invention is not limited to this, and can be similarly applied to the search for any solution as long as the problem has a search space that can be divided according to constraints.

[0062] Furthermore, according to the above embodiment, the amount of calculation required to search for a feasible solution to a combinatorial optimization problem can be reduced, making it possible to contribute to Goal 9 "Build resilient infrastructure and promote inclusive and sustainable industrialization" and Goal 11 "Make cities inclusive, safe, resilient and sustainable" of the Sustainable Development Goals (SDGs) led by the United Nations. [Explanation of symbols]

[0063] 1...Quantum search algorithm for solving optimization problems, 10...Initialization section, 20...Space division section, 30...Integration section

Claims

1. In a quantum search algorithm that searches for feasible solutions to combinatorial optimization problems on a quantum computer using a divide-and-conquer method, An initialization unit that prepares an entire space of quantum states that represent all combination states of a combinatorial optimization problem; a space division unit that divides the entire space into a plurality of subspaces for each constraint condition of a combinatorial optimization problem; and a merging unit that merges results of simultaneously executing a quantum search for each quantum state of the subspace.

2. The quantum search is a Grover search using a Grover algorithm, and an oracle operator and a Grover diffusion operator are repeatedly applied alternately; The quantum search algorithm according to claim 1, characterized in that when the space division unit divides the entire space into N, the integration unit divides the oracle operator and the Grover diffusion operator into N, and searches are performed simultaneously for each of the N subspaces.

3. The quantum search algorithm according to claim 2, characterized in that when the combinatorial optimization problem is a traveling salesman problem with two constraints, the phase θ when rotating the solution in the oracle operator is set to -π / 2.

4. the combinatorial optimization problem is a traveling salesman problem, The quantum search algorithm according to any one of claims 1 to 3, characterized in that each constraint condition is formulated in QUBO and is equivalent to an Ising model of a quantum computer.

5. A quantum search method in which a quantum computer searches for a feasible solution to a combinatorial optimization problem on the quantum computer by a divide-and-conquer method, Divide the combinatorial optimization problem into multiple subproblems for each constraint, A quantum search method comprising: integrating results of simultaneously performing a quantum search for each of the subproblems.

6. The quantum search is a Grover search using a Grover algorithm, and an oracle operator and a Grover diffusion operator are repeatedly applied alternately; The quantum search method according to claim 5, characterized in that when the combinatorial optimization problem is divided into N subproblems for each constraint condition, the oracle operator and the Grover diffusion operator are each divided into N, and searches are performed simultaneously on the N subproblems.

7. The quantum search method according to claim 6, characterized in that, when the combinatorial optimization problem is a traveling salesman problem with two constraints, the phase θ when rotating the solution in the oracle operator is set to -π / 2.

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