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Quantum Information and Computation

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

Max Zhao, Fei Li

Published Dec 1, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.2478/qic-2025-0023

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

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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.

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A Resource Efficient Ising Model‐Based Quantum Sudoku Solver

Wen-Li Wang, Mei-Huei Tang, Kevin Wang +3 more

ABSTRACT Background Quantum algorithms exploit superposition and parallelism to address complex combinatorial problems, many of which fall into the non‐polynomial (NP) class. Sudoku, a widely known logic‐based puzzle, is proven to be NP‐complete and thus presents a suitable testbed for exploring quantum optimization approaches. The Ising model—originally introduced for NP‐hard Ising spin glass problems—provides a natural mathematical framework for expressing constraints in a form amenable to quantum computation. Objective This work aims to develop a quantum Sudoku solver inspired by the Ising model that minimizes the number of logical qubits required, making it suitable for today's resource‐limited quantum hardware. The broader goal is to demonstrate a modeling strategy that may generalize to other NP optimization problems. Methods The solver construction begins by translating Sudoku constraints into mathematical expressions represented through couplings of atomic spins. These constraints are formulated into observable operators using Pauli operators, enabling the calculation of expectation values over candidate quantum states. Individual quantum algorithmic components are then integrated into a global optimization pipeline. The performance and correctness of this solver are evaluated through the Quantum Approximate Optimization Algorithm (QAOA) combined with the COBYLA classical optimizer within the IBM Qiskit SDK. A code example illustrates the implementation of multiple puzzle constraints and the verification of the resulting quantum circuits. Results The modeling approach successfully encodes Sudoku rules into an Ising Hamiltonian with reduced qubit requirements. Preliminary evaluations using QAOA and COBYLA demonstrate that the solver can identify puzzle‐consistent solutions while maintaining low resource consumption. The quantum circuit construction matches theoretical expectations, and the code snippet confirms successful constraint enforcement within the Qiskit environment. Conclusions The proposed quantum Sudoku solver highlights the potential of Ising‐based formulations to address NP optimization problems on near‐term quantum devices. By reducing logical qubit usage without sacrificing algorithmic integrity, this strategy may support broader applications in constrained optimization. The implementation serves as both a proof of concept and a practical guide for extending Ising‐inspired quantum modeling to other NP‐hard domains.

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Hybrid Quantum Branch-and-Bound Method for Quadratic Unconstrained Binary Optimization

Zedong Peng, Daniel de Roux, David E. Bernal

Quantum algorithms have shown promise in solving Quadratic Unconstrained Binary Optimization (QUBO) problems, benefiting from their connection to the transverse field Ising model.Various Ising solvers, both classical and quantum, have emerged to tackle such problems efficiently but lack global optimality guarantees and oftentimes sufer from hardware limitations such as limited qubit availability.In this work, we propose a hybrid branch-andbound (B&B) framework that integrates Ising solvers as heuristics within a classical B&B algorithm.Unlike prior theoretical studies, our work presents a practical implementation, available as open-source at GitHub.We explore when and where to apply Ising solvers in the search tree, and introduce a custom branching rule optimized QUBO embedding.Our method is evaluated on hundreds of QUBO instances from QUBOLib.jl using Gurobi and the D-Wave quantum annealer.Our Results show up to 11% less solution time and 17% fewer nodes compared to default Gurobi, an off-the-shelf commercial optimization solver.These findings demonstrate the value of hybrid quantum-classical strategies for enhancing exact optimization.

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Logical and Inequality Implications for Reducing the Size and Complexity of Quadratic Unconstrained Binary Optimization Problems

Fred Glover, Mark W. Lewis, Gary A. Kochenberger

The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.

arXiv:1705.09545•May 26, 2017•0 citations
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.
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 200200200 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 251251251 nodes and 3,2653,2653,265 edges. We discuss the advantages of this quantum-inspired approach, including its scalability, generalizability, and suitability for industrial-scale QUBO applications.
This article describes how to solve Sudoku puzzles using Quadratic Unconstrained Binary Optimization (QUBO). To this end, a QUBO instance with 729 variables is constructed, encoding a Sudoku grid with all constraints in place, which is then partially assigned to account for clues. The resulting instance can be solved with a Quantum Annealer, or any other strategy, to obtain the fully filled-out Sudoku grid. Moreover, as all valid solutions have the same energy, the QUBO instance can be used to sample uniformly from the space of valid Sudoku grids. We demonstrate the described method using both a heuristic solver and a Quantum Annealer.
ABSTRACT Background Quantum algorithms exploit superposition and parallelism to address complex combinatorial problems, many of which fall into the non‐polynomial (NP) class. Sudoku, a widely known logic‐based puzzle, is proven to be NP‐complete and thus presents a suitable testbed for exploring quantum optimization approaches. The Ising model—originally introduced for NP‐hard Ising spin glass problems—provides a natural mathematical framework for expressing constraints in a form amenable to quantum computation. Objective This work aims to develop a quantum Sudoku solver inspired by the Ising model that minimizes the number of logical qubits required, making it suitable for today's resource‐limited quantum hardware. The broader goal is to demonstrate a modeling strategy that may generalize to other NP optimization problems. Methods The solver construction begins by translating Sudoku constraints into mathematical expressions represented through couplings of atomic spins. These constraints are formulated into observable operators using Pauli operators, enabling the calculation of expectation values over candidate quantum states. Individual quantum algorithmic components are then integrated into a global optimization pipeline. The performance and correctness of this solver are evaluated through the Quantum Approximate Optimization Algorithm (QAOA) combined with the COBYLA classical optimizer within the IBM Qiskit SDK. A code example illustrates the implementation of multiple puzzle constraints and the verification of the resulting quantum circuits. Results The modeling approach successfully encodes Sudoku rules into an Ising Hamiltonian with reduced qubit requirements. Preliminary evaluations using QAOA and COBYLA demonstrate that the solver can identify puzzle‐consistent solutions while maintaining low resource consumption. The quantum circuit construction matches theoretical expectations, and the code snippet confirms successful constraint enforcement within the Qiskit environment. Conclusions The proposed quantum Sudoku solver highlights the potential of Ising‐based formulations to address NP optimization problems on near‐term quantum devices. By reducing logical qubit usage without sacrificing algorithmic integrity, this strategy may support broader applications in constrained optimization. The implementation serves as both a proof of concept and a practical guide for extending Ising‐inspired quantum modeling to other NP‐hard domains.
The LogQ algorithm encodes Quadratic Unconstrained Binary Optimization (QUBO) problems with exponentially fewer qubits than the Quantum Approximate Optimization Algorithm (QAOA). The advantages of conventional LogQ are accompanied by a challenge related to the optimization of its free parameters, which requires the usage of resourceintensive evolutionary or even global optimization algorithms. We propose a newlog⁡Q\log QlogQparameterization that can be optimized with a gradient-inspired method, which is less resource-intensive and thus strengthens the advantage of LogQ over QAOA for large/industrial problems. We illustrate the features of our method on an analytical model and present larger scale numerical results on MaxCut problems.
Quantum algorithms have shown promise in solving Quadratic Unconstrained Binary Optimization (QUBO) problems, benefiting from their connection to the transverse field Ising model.Various Ising solvers, both classical and quantum, have emerged to tackle such problems efficiently but lack global optimality guarantees and oftentimes sufer from hardware limitations such as limited qubit availability.In this work, we propose a hybrid branch-andbound (B&B) framework that integrates Ising solvers as heuristics within a classical B&B algorithm.Unlike prior theoretical studies, our work presents a practical implementation, available as open-source at GitHub.We explore when and where to apply Ising solvers in the search tree, and introduce a custom branching rule optimized QUBO embedding.Our method is evaluated on hundreds of QUBO instances from QUBOLib.jl using Gurobi and the D-Wave quantum annealer.Our Results show up to 11% less solution time and 17% fewer nodes compared to default Gurobi, an off-the-shelf commercial optimization solver.These findings demonstrate the value of hybrid quantum-classical strategies for enhancing exact optimization.
The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.