Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Concurrency and Computation Practice and Experience

Ising Models for Solving the N‐Queens Puzzle Based on the Domain‐Wall Vectors

Shunsuke Tsukiyama, Koji Nakano, Yasuaki Ito, Takumi Kato, Yuya Kawamata

Published Jan 16, 2025• 1 citation
Publisher

Abstract

ABSTRACT An Ising model is a mathematical model defined by an objective function comprising a quadratic formula of multiple spin variables, each taking values of either or . The task of determining a spin value assignment to these variables that minimizes the resulting value of an Ising model is a challenging optimization problem. Recently, quantum annealers, consisting of qubit cells interconnected according to principles of quantum mechanics, have emerged as a solution for tackling such problems. Ising models characterized by fewer quadratic terms are preferable as they reduce the resource requirements of quantum annealers. Additionally, it is advantageous for the absolute values of coefficients associated with linear and quadratic terms to be small to facilitate the discovery of good solutions, given the inherent limitations in the resolution of quantum annealers. The primary contribution of this article lies in presenting Ising models tailored for solving the ‐Queens puzzle. The conventional Ising model for this puzzle involves quadratic terms, with the maximum absolute value of coefficients being . Our novel Ising model significantly reduces the number of quadratic terms to only , with a maximum absolute coefficient of 6. Furthermore, we provide embedding results for a quantum annealer D‐Wave Advantage utilizing a Pegasus graph . We succeeded in embedding our novel Ising model for up to the 21‐Queens puzzle, while the conventional Ising model can be embedded only for up to the 14‐Queens puzzle.

DOI
10.1002/cpe.8364

Related papers

Dual-Matrix Domain Wall: A Novel Technique for Generating Permutations by QUBO and Ising Models with Quadratic Sizes

Koji Nakano, Shunsuke Tsukiyama, Yasuaki Ito +6 more

The Ising model is defined by an objective function using a quadratic formula of qubit variables. The problem of an Ising model aims to determine the qubit values of the variables that minimize the objective function, and many optimization problems can be reduced to this problem. In this paper, we focus on optimization problems related to permutations, where the goal is to find the optimal permutation out of the n! possible permutations of n elements. To represent these problems as Ising models, a commonly employed approach is to use a kernel that applies one-hot encoding to find any one of the n! permutations as the optimal solution. However, this kernel contains a large number of quadratic terms and high absolute coefficient values. The main contribution of this paper is the introduction of a novel permutation encoding technique called the dual-matrix domain wall, which significantly reduces the number of quadratic terms and the maximum absolute coefficient values in the kernel. Surprisingly, our dual-matrix domain-wall encoding reduces the quadratic term count and maximum absolute coefficient values from n3−n2 and 2n−4 to 6n2−12n+4 and 2, respectively. We also demonstrate the applicability of our encoding technique to partial permutations and Quadratic Unconstrained Binary Optimization (QUBO) models. Furthermore, we discuss a family of permutation problems that can be efficiently implemented using Ising/QUBO models with our dual-matrix domain-wall encoding.

arXiv:2308.01024•Technologies•Oct 17, 2023•9 citations

Domain wall encoding of discrete variables for quantum annealing and QAOA

Nicholas Chancellor

Abstract In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimisation. The new method is based on the physics of domain walls in one-dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms Following on from similar results for the ‘one hot’ method of encoding discrete variables (Hadfield et al 2019 Algorithms 12 34) I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz. I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.

arXiv:1903.05068•Quantum Science and Technology•Jul 19, 2019•87 citations

Effectively encoding satisfiability problems into Ising models for quantum annealing

Xiaotian Li, Koji Nakano, Victor Parque +4 more

Ising Models, defined by quadratic objective functions (or Hamiltonians), enable to use quantum annealers to search for optimal or near-optimal solutions of satisfiability problems. However, current quantum annealers have limited resolution, meaning that small or closely-valued coefficients in the Hamiltonian may be obscured by flux noise, leading to a degradation in the performance of quantum annealing. In this paper, we propose a novel design methodology for encoding satisfiability problems into Ising models via Integer Linear Programming. Experimental results show that our method can effectively reduce the resolution requirements for quantum annealers.

International Journal of Parallel Emergent and Distributed Systems•Dec 12, 2024•1 citation

Solving the N-Queens Puzzle by a QUBO Model with Quadratic Size

Shunsuke Tsukiyama, Koji Nakano, Yasuaki Ito +6 more

A quadratic unconstrained binary optimization (QUBO) model is defined by an objective function that consists of a quadratic formula involving multiple binary variables. The goal of the QUBO problem is to find an optimal assignment of values to the variables that minimizes the objective function. One of the most well-known problems that can be reduced to a QUBO problem is the N-Queens problem. The N-Queens problem seeks to find a configuration of n queens on an $n\times n$ chessboard such that no two queens can attack each other. It is widely recognized that a conventional technique can be used to design a QUBO model with an $n\times n$ matrix of binary variables representing the chessboard, where a queen is placed in a particular position if and only if the corresponding binary variable is set to 1. This technique involves incorporating terms in the objective function to ensure that the sum of each row and column is 1, and the sum of each diagonal and anti-diagonal is either 0 or 1. The resulting reduced QUBO model contains $\displaystyle \frac{5}{3}n^{3}-2n^{2}+\frac{1}{3}n$ quadratic terms. Due to the simplicity and straightforwardness of designing such a QUBO model, there are numerous resources available that explain this model as an easy example of QUBO models. However, to the best of our knowledge, no fundamentally different approach to designing a QUBO model for the N-Queens problem has been discovered. The prevailing view within the community is that this approach is the only method for designing a QUBO model for the N-Queens problem, and no superior alternative has been found. In this paper, we will present a counterintuitive result that demonstrates a more efficient QUBO model for the N-Queens problem, requiring fewer quadratic terms. Surprisingly, this alternative model involves only $12n^{2}-24n+12$ quadratic terms, which is smaller than the conventional QUBO model whenever n is greater than or equal to 7. This discovery marks a significant breakthrough in the theoretical challenge of minimizing a QUBO model for solving the N-Queens problem.

International Symposium on Computing and Networking•Nov 28, 2023•4 citations

Designing unit Ising models for logic gate simulation through integer linear programming

Shunsuke Tsukiyama, Koji Nakano, Victor Parque +4 more

Ising models minimizing a quadratic objective function with spin variables of either −1 or +1 are instrumental in tackling combinatorial optimization problems by programmable Quantum Annealers. This paper introduces unit Ising models, where non-zero coefficients are restricted to −1 or +1. Due to the limited resolution of quantum annealers, unit Ising models are more suitable for quantum annealers. A fixed unit Ising model for logic circuits could lead to Application-Specific Unit Quantum Annealers (ASUQAs) for inverse function computation, similar to ASICs. Our findings suggest a powerful new method for compromising the RSA cryptosystem by leveraging ASUQAs for factorization.

arXiv:2406.18130•International Journal of Parallel Emergent and Distributed Systems•Dec 5, 2024•3 citations

Improving the Solving of Optimization Problems: A Comprehensive Review of Quantum Approaches

Deborah Volpe, Giacomo Orlandi, Giovanna Turvani

Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.

Quantum Reports•Jan 7, 2025•28 citations
ABSTRACT An Ising model is a mathematical model defined by an objective function comprising a quadratic formula of multiple spin variables, each taking values of either or . The task of determining a spin value assignment to these variables that minimizes the resulting value of an Ising model is a challenging optimization problem. Recently, quantum annealers, consisting of qubit cells interconnected according to principles of quantum mechanics, have emerged as a solution for tackling such problems. Ising models characterized by fewer quadratic terms are preferable as they reduce the resource requirements of quantum annealers. Additionally, it is advantageous for the absolute values of coefficients associated with linear and quadratic terms to be small to facilitate the discovery of good solutions, given the inherent limitations in the resolution of quantum annealers. The primary contribution of this article lies in presenting Ising models tailored for solving the ‐Queens puzzle. The conventional Ising model for this puzzle involves quadratic terms, with the maximum absolute value of coefficients being . Our novel Ising model significantly reduces the number of quadratic terms to only , with a maximum absolute coefficient of 6. Furthermore, we provide embedding results for a quantum annealer D‐Wave Advantage utilizing a Pegasus graph . We succeeded in embedding our novel Ising model for up to the 21‐Queens puzzle, while the conventional Ising model can be embedded only for up to the 14‐Queens puzzle.
The Ising model is defined by an objective function using a quadratic formula of qubit variables. The problem of an Ising model aims to determine the qubit values of the variables that minimize the objective function, and many optimization problems can be reduced to this problem. In this paper, we focus on optimization problems related to permutations, where the goal is to find the optimal permutation out of the n! possible permutations of n elements. To represent these problems as Ising models, a commonly employed approach is to use a kernel that applies one-hot encoding to find any one of the n! permutations as the optimal solution. However, this kernel contains a large number of quadratic terms and high absolute coefficient values. The main contribution of this paper is the introduction of a novel permutation encoding technique called the dual-matrix domain wall, which significantly reduces the number of quadratic terms and the maximum absolute coefficient values in the kernel. Surprisingly, our dual-matrix domain-wall encoding reduces the quadratic term count and maximum absolute coefficient values from n3−n2 and 2n−4 to 6n2−12n+4 and 2, respectively. We also demonstrate the applicability of our encoding technique to partial permutations and Quadratic Unconstrained Binary Optimization (QUBO) models. Furthermore, we discuss a family of permutation problems that can be efficiently implemented using Ising/QUBO models with our dual-matrix domain-wall encoding.
Abstract In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimisation. The new method is based on the physics of domain walls in one-dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms Following on from similar results for the ‘one hot’ method of encoding discrete variables (Hadfield et al 2019 Algorithms 12 34) I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz. I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.
Ising Models, defined by quadratic objective functions (or Hamiltonians), enable to use quantum annealers to search for optimal or near-optimal solutions of satisfiability problems. However, current quantum annealers have limited resolution, meaning that small or closely-valued coefficients in the Hamiltonian may be obscured by flux noise, leading to a degradation in the performance of quantum annealing. In this paper, we propose a novel design methodology for encoding satisfiability problems into Ising models via Integer Linear Programming. Experimental results show that our method can effectively reduce the resolution requirements for quantum annealers.
A quadratic unconstrained binary optimization (QUBO) model is defined by an objective function that consists of a quadratic formula involving multiple binary variables. The goal of the QUBO problem is to find an optimal assignment of values to the variables that minimizes the objective function. One of the most well-known problems that can be reduced to a QUBO problem is the N-Queens problem. The N-Queens problem seeks to find a configuration of n queens on an n×nn\times nn×n chessboard such that no two queens can attack each other. It is widely recognized that a conventional technique can be used to design a QUBO model with an n×nn\times nn×n matrix of binary variables representing the chessboard, where a queen is placed in a particular position if and only if the corresponding binary variable is set to 1. This technique involves incorporating terms in the objective function to ensure that the sum of each row and column is 1, and the sum of each diagonal and anti-diagonal is either 0 or 1. The resulting reduced QUBO model contains 53n3−2n2+13n\displaystyle \frac{5}{3}n^{3}-2n^{2}+\frac{1}{3}n35​n3−2n2+31​n quadratic terms. Due to the simplicity and straightforwardness of designing such a QUBO model, there are numerous resources available that explain this model as an easy example of QUBO models. However, to the best of our knowledge, no fundamentally different approach to designing a QUBO model for the N-Queens problem has been discovered. The prevailing view within the community is that this approach is the only method for designing a QUBO model for the N-Queens problem, and no superior alternative has been found. In this paper, we will present a counterintuitive result that demonstrates a more efficient QUBO model for the N-Queens problem, requiring fewer quadratic terms. Surprisingly, this alternative model involves only 12n2−24n+1212n^{2}-24n+1212n2−24n+12 quadratic terms, which is smaller than the conventional QUBO model whenever n is greater than or equal to 7. This discovery marks a significant breakthrough in the theoretical challenge of minimizing a QUBO model for solving the N-Queens problem.
Ising models minimizing a quadratic objective function with spin variables of either −1 or +1 are instrumental in tackling combinatorial optimization problems by programmable Quantum Annealers. This paper introduces unit Ising models, where non-zero coefficients are restricted to −1 or +1. Due to the limited resolution of quantum annealers, unit Ising models are more suitable for quantum annealers. A fixed unit Ising model for logic circuits could lead to Application-Specific Unit Quantum Annealers (ASUQAs) for inverse function computation, similar to ASICs. Our findings suggest a powerful new method for compromising the RSA cryptosystem by leveraging ASUQAs for factorization.
Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.