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Europhysics Letters (EPL)

Solving Sudoku game using a hybrid classical-quantum algorithm

Ankur Kumar Pal, Sanghita Chandra, Vardaan Mongia, Bikash K. Behera, Prasanta Kumar Panigrahi

Published Jan 31, 2020• 15 citations
Publisher

Abstract

Sudoku is a fun combinatorial game, based on the Latin square, which has wide applications as being an efficient design in controlling multiple sources of variable nuisance simultaneously. Quantum Sudoku solver uses basics of quantum mechanics such as superposition and entanglement. Using the concept of duality quantum computing, we propose an algorithm which solves a 4 × 4 Sudoku puzzle with a space complexity [ O ( N 2 ( 4 + log ⁡ ( N ) ) ] (where N = 4). Our proposed algorithm has a primary aim to provide a theoretical framework for an application of duality quantum computing.

DOI
10.1209/0295-5075/128/40007

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Sudoku is a fun combinatorial game, based on the Latin square, which has wide applications as being an efficient design in controlling multiple sources of variable nuisance simultaneously. Quantum Sudoku solver uses basics of quantum mechanics such as superposition and entanglement. Using the concept of duality quantum computing, we propose an algorithm which solves a 4 × 4 Sudoku puzzle with a space complexity [ O ( N 2 ( 4 + log ⁡ ( N ) ) ] (where N = 4). Our proposed algorithm has a primary aim to provide a theoretical framework for an application of duality quantum computing.
Abstract: We use ability of quantum computing such as superposition and entanglement to solve the sudoku. In recent years, quantum computers have shown promise as a new technology for solving complex problems in various fields, including optimization and cryptography. In this paper, we investigate the potential of quantum computers for solving Sudoku puzzles. We present a quantum algorithm for solving Sudoku puzzles, and compare its performance to classical algorithms. Our results show that the quantum algorithm outperforms classical algorithms in terms of both speed and accuracy, and provides a new tool for solving Sudoku puzzles efficiently. Additionally, we discuss the implications of our results for the development of quantum algorithms for solving other combinatorial problems.
We use ability of quantum computing such as superposition and entanglement to solve the sudoku. In recent years, quantum computers have shown promise as a new technology for solving complex problems in various fields, including optimization and cryptography. In this paper, we investigate the potential of quantum computers for solving Sudoku puzzles. We present a quantum algorithm for solving Sudoku puzzles, and compare its performance to classical algorithms. Our results show that the quantum algorithm outperforms classical algorithms in terms of both speed and accuracy, and provides a new tool for solving Sudoku puzzles efficiently. Additionally, we discuss the implications of our results for the development of quantum algorithms for solving other combinatorial problems.
Quantum technologies are moving towards enabling real world uses and as these technologies develop. The most important thing that differs Quantum from classical computers is that Quantum Computers can do “multiple” tasks at the same time as opposed to the classical computers doing a single task at a time. This phenomenon has led to the discovery of a few algorithms that are faster than regular computers. One such algorithm is the Grover’s Algorithm[l]. This algorithm has shown a few real-world applications such as Quantum Search Algorithm, Travelling salesman, etc[2]. In this paper, Grover’s algorithm is introduced, and a 2 x 2 Sudoku example is solved using Intel Quantum Software Development Kit[3]. The Intel Quantum SDK is quite new and in this paper we show how a simple Grover’s Problem can be implemented in the SDK. Sudoku is a popular number-based puzzle game played on a grid consisting of nine squares subdivided into smaller 3x3 grids. The objective is to fill the grid with numbers from 1 to 9, ensuring that each row, column, and 3x3 grid contains all numbers exactly once.
ABSTRACT Quantum Sudoku is the generalized version of the popular game Sudoku. A quantum Sudoku square is a special kind of quantum Latin square. A genuinely quantum solution of quantum Sudoku squares has been solved completely. In this paper, we introduce the notion of orthogonality of two quantum Sudoku squares. We establish three novel constructions of mutually orthogonal quantum Sudoku squares including Lorch construction, Pedersen‐Vis construction and direct product construction. As a result, we determine the genuinely quantum solution of a pair of orthogonal Sudoku squares of size for any positive integer . In general, we show that the resulting squares naturally define an absolutely maximally entangled state . Furthermore, we also show that the mutually orthogonal quantum Sudoku squares give rise to mutually unbiased bases based on maximally entangled states in for any prime power .
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.
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.