Solving systems of Boolean multivariate equations with quantum annealing

TL;DR

This paper explores quantum annealing techniques for solving Boolean multivariate quadratic equations, presenting embedding methods, an iterative algorithm, and experimental validation on D-Wave quantum devices.

Contribution

It introduces new embedding strategies and an iterative algorithm for quantum annealing of Boolean systems, with practical implementation on existing quantum hardware.

Findings

Successful implementation on D-Wave devices

Different embedding methods vary in quantum resource requirements

Iterative approach improves solution accuracy

Abstract

Polynomial systems over the binary field have important applications, especially in symmetric and asymmetric cryptanalysis, multivariate-based post-quantum cryptography, coding theory, and computer algebra. In this work, we study the quantum annealing model for solving Boolean systems of multivariate equations of degree 2, usually referred to as the Multivariate Quadratic problem. We present different methodologies to embed the problem into a Hamiltonian that can be solved by available quantum annealing platforms. In particular, we provide three embedding options, and we highlight their differences in terms of quantum resources. Moreover, we design a machine-agnostic algorithm that adopts an iterative approach to better solve the problem Hamiltonian by repeatedly reducing the search space. Finally, we use D-Wave devices to successfully implement our methodologies on several instances of the Multivariate Quadratic problem.