A Quantum-Inspired Algorithm for Solving Sudoku Puzzles and the MaxCut Problem

TL;DR

This paper introduces a quantum-inspired algorithm using Matrix Product States and DMRG to efficiently solve large QUBO problems, including Sudoku puzzles and MaxCut instances, demonstrating scalability and effectiveness.

Contribution

The paper presents a novel quantum-inspired algorithm employing MPS and DMRG for solving large QUBO problems, extending applicability to complex puzzles and optimization tasks.

Findings

Successfully solved Sudoku puzzles with over 200 spins

Solved MaxCut instances with up to 251 nodes and 3,265 edges

Demonstrated scalability and reliability of the heuristic algorithm

Abstract

We propose and evaluate a quantum-inspired algorithm for solving Quadratic Unconstrained Binary Optimization (QUBO) problems, which are mathematically equivalent to finding ground states of Ising spin-glass Hamiltonians. The algorithm employs Matrix Product States (MPS) to compactly represent large superpositions of spin configurations and utilizes a discrete driving schedule to guide the MPS toward the ground state. At each step, a driver Hamiltonian -- incorporating a transverse magnetic field -- is combined with the problem Hamiltonian to enable spin flips and facilitate quantum tunneling. The MPS is updated using the standard Density Matrix Renormalization Group (DMRG) method, which iteratively minimizes the system's energy via multiple sweeps across the spin chain. Despite its heuristic nature, the algorithm reliably identifies global minima, not merely near-optimal solutions, across diverse QUBO instances. We first demonstrate its effectiveness on intermediate-level Sudoku puzzles from publicly available sources, involving over $200$ Ising spins with long-range couplings dictated by constraint satisfaction. We then apply the algorithm to MaxCut problems from the Biq Mac library, successfully solving instances with up to $251$ nodes and $3,265$ edges. We discuss the advantages of this quantum-inspired approach, including its scalability, generalizability, and suitability for industrial-scale QUBO applications.