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

International Conference on Parallel Processing Workshops (ICPPW)

Constraint Solving by Quantum Annealing

Philippe Codognet

Published Aug 9, 2021• 8 citations
Publisher

Abstract

We present experiments in solving combinatorial optimization and constraint satisfaction problems by means of Quantum Annealing. We describe how to model classic constraint problems such as N-queens and Magic Square as well as hard combinatorial problems such as the Costas Array Problem or the Quadratic Assignment Problem in terms of QUBO (Quadratic Unconstrained Binary Optimization). QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the ”quantum-inspired” but classical devices such as Fujitsu’s Digital Annealing Unit or Hitachi’s CMOS Annealing Machine. We present preliminary results for solving these combinatorial optimization and constraint satisfaction problems by implementation on the D-Wave quantum computer.

DOI
10.1145/3458744.3473364

Related papers

A Constraint Programming Approach for QUBO Solving and Quantum Annealing

Philippe Codognet

Quantum Annealing is an alternative type of computation in which problems are encoded in quantum Hamiltonians (energy functions) and quantum dynamics is used to find solutions (ground states of minimal energy). Quantum computers such as the D-Wave systems are indeed implementing those ideas in hardware, as well as "quantum-inspired" devices based on classical electronics such as Fujitsu’s Digital Annealing Unit. All those systems use the same modeling language: Quadratic Unconstrained Binary Optimization (QUBO). However, QUBO is a low-level language and for modeling combinatorial problems such as constraint satisfaction and constrained optimization problems, we need to introduce higher-level abstractions in order to define complex constraints.

International Conference on Art, Science, and Engineering of Programming (<Programming>)•Mar 11, 2024•2 citations

Encoding the At-Most-One Constraint for QUBO and Quantum Annealing: Experiments with the N-Queens problem

Philippe Codognet

We present experiments in solving constrained optimization and constraint satisfaction problems by means of Quantum Annealing in terms of QUBO (Quadratic Unconstrained Binary Optimization). We investigate different QUBO formulations for the encoding of integers into Booleans and for the encoding of the At-Most-One constraint, which is used in combinatorial problems requiring capacity constraints. We compare four different QUBO models and report experiments done on the D-Wave annealing computer and the "quantum-inspired" Fixstars Amplify Annealing Engine. For this, we consider a simple example taken from the domain of Constraint Satisfaction Problems (CSP), the N-queens problem, as its formulation involves multiple instances of the At-Most-One constraints.

Genetic and Evolutionary Computation Conference Companion (GECCO Companion)•Jul 15, 2023•9 citations

Quantum and Digital Annealing for the Quadratic Assignment Problem

Philippe Codognet, Daniel Díaz, Salvador Pinto Abreu

The Quadratic Assignment Problem is a a classical constrained optimization problem used to model many real-life applications. We present experiments in solving the Quadratic Assignment Problem by means of Quantum Annealing and Quantum-inspired Annealing. We describe how to model this classical combinatorial problem in terms of QUBO (Quadratic Unconstrained Binary optimization) for implementing it on hardware solvers based on quantum or quantum-inspired annealing (D-Wave, Fujitsu Digital Annealing Unit and Fixstars Amplify Annealing Engine). We present performance result for these implementations and compare them with well established metaheuristic solvers on classical hardware, such as Robust Tabu Search and External Optimization.

IEEE International Conference on Quantum Software (QSW)•Jul 1, 2022•12 citations

On good encodings for quantum annealer and digital optimization solvers

Alberto Ceselli, Marco Premoli

Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.

Scientific Reports•Apr 6, 2023•10 citations

Benchmarking Quantum Computing for Combinatorial Optimization

Nathan Kittichaikoonkij, Nutthapat Pongtanyavichai, Poopha Suwananek +2 more

Quantum computers can be much faster than any classical computers in solving a certain class of problems. One of the interesting problems is the combinatorial optimization problem, which is a challenge for classical computer systems. We encode these problems into the Quadratic Unconstrained Binary Optimization (QUBO) format and then solve them with quantum solvers. In this study, we compare the performance of D-Wave’s quantum annealing system with classical solvers, namely, Gurobi and Fixstars. We demonstrate the current capabilities of the D-Wave system.

International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON)•May 20, 2025•0 citations

Quantum Annealing versus Digital Computing

Michael Jünger, Elisabeth Lobe, Petra Mutzel +4 more

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.

arXiv:1904.11965•ACM Journal of Experimental Algorithmics•Jul 9, 2021•48 citations
We present experiments in solving combinatorial optimization and constraint satisfaction problems by means of Quantum Annealing. We describe how to model classic constraint problems such as N-queens and Magic Square as well as hard combinatorial problems such as the Costas Array Problem or the Quadratic Assignment Problem in terms of QUBO (Quadratic Unconstrained Binary Optimization). QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the ”quantum-inspired” but classical devices such as Fujitsu’s Digital Annealing Unit or Hitachi’s CMOS Annealing Machine. We present preliminary results for solving these combinatorial optimization and constraint satisfaction problems by implementation on the D-Wave quantum computer.
Quantum Annealing is an alternative type of computation in which problems are encoded in quantum Hamiltonians (energy functions) and quantum dynamics is used to find solutions (ground states of minimal energy). Quantum computers such as the D-Wave systems are indeed implementing those ideas in hardware, as well as "quantum-inspired" devices based on classical electronics such as Fujitsu’s Digital Annealing Unit. All those systems use the same modeling language: Quadratic Unconstrained Binary Optimization (QUBO). However, QUBO is a low-level language and for modeling combinatorial problems such as constraint satisfaction and constrained optimization problems, we need to introduce higher-level abstractions in order to define complex constraints.
We present experiments in solving constrained optimization and constraint satisfaction problems by means of Quantum Annealing in terms of QUBO (Quadratic Unconstrained Binary Optimization). We investigate different QUBO formulations for the encoding of integers into Booleans and for the encoding of the At-Most-One constraint, which is used in combinatorial problems requiring capacity constraints. We compare four different QUBO models and report experiments done on the D-Wave annealing computer and the "quantum-inspired" Fixstars Amplify Annealing Engine. For this, we consider a simple example taken from the domain of Constraint Satisfaction Problems (CSP), the N-queens problem, as its formulation involves multiple instances of the At-Most-One constraints.
The Quadratic Assignment Problem is a a classical constrained optimization problem used to model many real-life applications. We present experiments in solving the Quadratic Assignment Problem by means of Quantum Annealing and Quantum-inspired Annealing. We describe how to model this classical combinatorial problem in terms of QUBO (Quadratic Unconstrained Binary optimization) for implementing it on hardware solvers based on quantum or quantum-inspired annealing (D-Wave, Fujitsu Digital Annealing Unit and Fixstars Amplify Annealing Engine). We present performance result for these implementations and compare them with well established metaheuristic solvers on classical hardware, such as Robust Tabu Search and External Optimization.
Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.
Quantum computers can be much faster than any classical computers in solving a certain class of problems. One of the interesting problems is the combinatorial optimization problem, which is a challenge for classical computer systems. We encode these problems into the Quadratic Unconstrained Binary Optimization (QUBO) format and then solve them with quantum solvers. In this study, we compare the performance of D-Wave’s quantum annealing system with classical solvers, namely, Gurobi and Fixstars. We demonstrate the current capabilities of the D-Wave system.
Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.