Quantum Speedup of the Dispersion and Codebook Design Problems

TL;DR

This paper introduces quantum algorithms leveraging Grover adaptive search for dispersion and codebook design problems, achieving quadratic speedup by reformulating the problem over Dicke states and reducing search space and qubit requirements.

Contribution

It presents novel formulations of dispersion problems suitable for quantum algorithms, including techniques to reduce search space and qubit count, enabling more feasible quantum solutions.

Findings

Quadratic speedup over classical methods.

Reduced search space via Dicke states.

Lower qubit requirements and query complexity.

Abstract

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. Additionally, 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 Hadamard transform is achieved, enhancing the feasibility of the quantum-based solution of the dispersion problem.