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arXiv:2410.12181

Grover Adaptive Search for the Higher-Order Formulation of Quadratic Assignment Problems

Taku Mikuriya, Shintaro Fujiwara, Kein Yukiyoshi, Giuseppe Thadeu Freitas de Abreu, Naoki Ishikawa

Published Oct 16, 2024• 0 citations
arXivHTMLPDF

Abstract

We demonstrate that the search space of the quadratic assignment problem (QAP), known as an NP-hard combinatorial optimization problem, can be reduced using Grover adaptive search (GAS) with permutation preparation operator (PPO). To that end, we first revise the traditional quadratic unconstrained binary optimization (QUBO) formulation of the QAP into a higher-order unconstrained binary optimization (HUBO) formulation, introducing a binary encoding method. Algebraic analyses in terms of the number of qubits, quantum gates, circuit depth, and query complexity are performed, which indicate that our proposed approach significantly reduces the search space size, improving convergence performance to the optimal solution compared to the conventional one. Furthermore, although the PPO for HUBO has a greater circuit depth than the PPO for QUBO, when the analysis is extended to the entire state preparation operator, both HUBO and QUBO exhibit comparable depths. Therefore, owing to its smaller number of variables, HUBO can be concluded to be more effective.

DOI
10.48550/arxiv.2410.12181

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Abstract The Grover Adaptive Search (GAS) is an innovative approach offering solutions to optimization problems thanks to quadratic speedup enabled by the Grover Search Algorithm. However, the need for more qubits, depending on the number of variables in optimization problems, creates problems in providing optimal solutions. The central premise of this study is the principle that d -dimension is more efficient than 2-dimension could lead to more advantageous results when the GAS algorithm is adapted to d -dimension. To that end, this study adapts the GAS algorithm to the d -dimension drawing on quantum computing in the d -dimension, allowing computing using fewer quantum resources, namely, qudits. The proposed algorithm is validated against the Quadratic and Higher-Order Unconstrained Binary Optimization (QUBO-HUBO). This study concludes the d -dimensional GAS algorithm achieves solutions using fewer qudits for these problems in the QUBO and the HUBO. This study underscores the progress achieved in the advancement of high-dimensional quantum computing.
Quantum approaches to combinatorial optimization problems (COPs) are often limited by the resource demands of Quadratic Unconstrained Binary Optimization (QUBO) encodings, which enlarge circuits through penalty terms and increase qubit and gate counts. We show that Higher-Order Unconstrained Binary Optimization (HUBO) enables a more resource-efficient formulation. Our method systematically constructs HUBO Hamiltonians and, compared to a QUBO formulation in benchmarks on Gate Assignment (GAP), Maximum k-Colorable Subgraph (MkCS), and Integer Programming (IP) problems, significantly reduces qubit requirements and decreases total CNOT gate counts by at least 89.6% for all tested instances. These results highlight HUBO as a practical alternative for quantum optimization on near-term devices. To promote adoption, we release an open-source Python library that automates HUBO model construction, extends beyond the examples presented in this work, and broadens access to resource-efficient quantum optimization.
Grover adaptive search (GAS) is a quantum exhaustive search algorithm designed to solve binary optimization problems. In this article, we propose higher order binary formulations that can simultaneously reduce the numbers of qubits and gates required for GAS. Specifically, we consider two novel strategies: one that reduces the number of gates through polynomial factorization, and the other that halves the order of the objective function, subsequently decreasing circuit runtime and implementation cost. Our analysis demonstrates that the proposed higher order formulations improve the convergence performance of GAS by reducing both the search space size and the number of quantum gates. Our strategies are also beneficial for general combinatorial optimization problems using one-hot encoding.
High-order binary optimization (HUBO) formulation can represent many real world problems. This mathematical approach is particularly relevant within the quantum computation domain because quantum hardware allows to directly map and solve these problems. The thesis explores the implementation of Grover Adaptive Search (GAS) by varying the different free parameters to evaluate the quality of the solution and the success rate of the algorithm, along with a possible optimization procedure. In particular, GAS is an algorithm that is compliant with HUBO and theoretically offer a quadratic speed up over classical computation. This thesis implements a quantum circuit for GAS algorithm to solve a 4-city traveling salesman problem along with two strategies to reduce the gate counts. The results include the output quality of the value of the objective function and the success rate of reaching the true minimum of 7, where the optimal configuration achieves an average value of 9.4 and a success rate of 36%. This configuration also maintains a 7% lower average value and a 10% higher success rate compared to randomly generated parameters.
In this article, we propose new formulations of max-sum and max-min dispersion problems that enable solutions via the Grover adaptive search (GAS) quantum algorithm, offering quadratic speedup. Dispersion problems are combinatorial optimization problems classified as NP-hard, which appear often in coding theory and wireless communications applications involving optimal codebook design. In turn, GAS is a quantum exhaustive search algorithm that can be used to implement full-fledged maximum-likelihood optimal solutions. In conventional naive formulations, however, it is typical to rely on a binary vector spaces, resulting in search space sizes prohibitive even for GAS. To circumvent this challenge, we instead formulate the search of optimal dispersion problem over Dicke states, an equal superposition of binary vectors with equal Hamming weights, which significantly reduces the search space leading to a simplification of the quantum circuit via the elimination of penalty terms. In addition, we propose a method to replace distance coefficients with their ranks, contributing to the reduction of the number of qubits. Our analysis demonstrates that as a result of the proposed techniques, a reduction in query complexity compared to the conventional GAS using the Hadamard transform is achieved, enhancing the feasibility of the quantum-based solution of the dispersion problem.