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arXiv:2306.17363 · New Journal of Physics

Quantum optimization algorithm based on multistep quantum computation

Hefeng Wang, Hua Xiang

Published Jul 1, 2024• 4 citations
arXivHTMLPDF

Abstract

Abstract We present a quantum algorithm for finding the minimum of a function based on multistep quantum computation, and apply the algorithm for solving optimization problems with continuous variables. We construct the state space of the problem by discretizing the variables of the problem, and divide the state space according to the function values of the vectors of the state space. By comparing the function values of the vectors with a series of threshold values in decreasing order, we construct a sequence of Hamiltonians where the search space of a Hamiltonian is nested in that of the previous one. By applying a multistep quantum computation process for finding the ground state of the last Hamiltonian, the optimal vector of the state space of the problem is located in a small search space and can be determined efficiently. One of the most difficult problems in optimization algorithms is that a trial vector is trapped in a deep local minimum while the global minimum is missed, this problem can be alleviated in our algorithm and the run time is proportional to the number of the steps of the algorithm, provided that the reduction rate of the search spaces is polynomial large. We discuss the implementation of the algorithm and test the algorithm for some test functions.

DOI
10.1088/1367-2630/ad60ed

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Abstract We present a quantum algorithm for finding the minimum of a function based on multistep quantum computation, and apply the algorithm for solving optimization problems with continuous variables. We construct the state space of the problem by discretizing the variables of the problem, and divide the state space according to the function values of the vectors of the state space. By comparing the function values of the vectors with a series of threshold values in decreasing order, we construct a sequence of Hamiltonians where the search space of a Hamiltonian is nested in that of the previous one. By applying a multistep quantum computation process for finding the ground state of the last Hamiltonian, the optimal vector of the state space of the problem is located in a small search space and can be determined efficiently. One of the most difficult problems in optimization algorithms is that a trial vector is trapped in a deep local minimum while the global minimum is missed, this problem can be alleviated in our algorithm and the run time is proportional to the number of the steps of the algorithm, provided that the reduction rate of the search spaces is polynomial large. We discuss the implementation of the algorithm and test the algorithm for some test functions.
We introduce a quantum approximate optimization algorithm (QAOA) for continuous optimization. The algorithm is based on the dynamics of a quantum system moving in an energy potential which encodes the objective function. By approximating the dynamics at finite time steps, the algorithm can be expressed as alternating evolution under two non-commuting Hamiltonians. We show that each step of the algorithm updates the wavefunction in the direction of its local gradient, with an additional momentum-dependent displacement. For initial states in a superposition over many points, this method can therefore be interpreted as a coherent version of gradient descent, i.e., 'gradient descent in superposition.' This approach can be used for both constrained and unconstrained optimization. In terms of computational complexity, we show how variants of the algorithm can recover continuous-variable Grover search, and how a single iteration can replicate continuous-variable instantaneous quantum polynomial circuits. We also discuss how the algorithm can be adapted to solve discrete optimization problems. Finally, we test the algorithm through numerical simulation in optimizing the Styblinski-Tang function.
We present a quantum algorithm for obtaining a quantum state that encodes the PageRank vector of the Google matrix through multistep quantum resonant transition (mQRT). In the algorithm, the PageRank vector is encoded in the ground state of a problem Hamiltonian associated with the Google matrix. By dividing the web graph corresponding to the Google matrix into a series of subgraphs with nested structure, we construct a sequence of Hamiltonians based on the subgraphs to form a Hamiltonian evolution path from a simple initial Hamiltonian to the problem Hamiltonian. The ground state of the problem Hamiltonian is obtained by going through ground states of the intermediate Hamiltonians via QRT step by step. This algorithm requires only one ancillary qubit, and the runtime of the algorithm is proportional to the number of steps. It provides a new way for efficiently obtaining the quantum state of the PageRank vector of large-scale networks.
Quantum optimization algorithms hold the promise of solving classically hard, discrete optimization problems in practice. The requirement of encoding such problems in a Hamiltonian realized with a finite (and currently small) number of qubits, however, poses the risk of finding only the optimum within the restricted space supported by this Hamiltonian. We describe an iterative algorithm in which a solution obtained with such a restricted problem Hamiltonian is used to define a new problem Hamiltonian that is better suited than the previous one. In numerical examples of the shortest vector problem, we show that the algorithm with a sequence of improved problem Hamiltonians converges to the desired solution.
Problems of optimization are very common in several areas like engineering, physics, economics and mathematics and usually, they are very difficult to solve. Basically, one has to find the minimum or maximum of an objective function. In this work, we chose two optimization problems and we present quantum algorithms to solve them. The problems are: (1) to find the best decomposition of a unitary operation achievable by a programmable quantum circuit; (2) to find the minimal distance of a linear code. The quantum algorithms used are asymptotic quantum search and their main property is the fact that only one measurement is required.
This paper presents a novel algorithm optimization method leveraging the unique capabilities of quantum computing. Traditional optimization algorithms often struggle with complex systems, requiring significant computational resources and time. We propose a quantum-enhanced optimization approach designed to overcome these limitations by utilizing quantum parallelism and superposition. The core mechanism involves mapping the optimization problem onto a quantum state representation, enabling parallel processing and enhanced exploration of the solution space. The paper details the design of the quantum algorithm, its potential advantages over classical approaches, and preliminary experimental results demonstrating its effectiveness in specific optimization scenarios.
This paper presents a novel approach to multi-variable quantum optimization, aiming to leverage the potential of quantum computing for solving complex optimization problems involving multiple variables. The core of the proposed algorithm is a framework based on either Variational Quantum Eigensolver (VQE) or Quantum Annealing, adapted for multi-variable scenarios. We explore the challenges and opportunities inherent in mapping multi-variable optimization problems onto quantum systems and discuss potential strategies for efficient solution. The algorithm's efficacy is evaluated through theoretical analysis and conceptual design, focusing on the key elements of quantum state preparation, Hamiltonian design, and measurement strategies. While experimental validation is deferred to future research, this work provides a foundational outline for the development of practical multi-variable quantum optimization algorithms. The primary goal is to demonstrate the feasibility of applying quantum techniques to complex, real-world problems, contributing to the growing field of quantum optimization.