© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1810.05582 · Quantum

Applying quantum algorithms to constraint satisfaction problems

Earl T. Campbell, Ankur Khurana, Ashley Montanaro

Published Jul 18, 2019• 76 citations

Abstract

Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over 105 relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over 103 . However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.

DOI: 10.22331/q-2019-07-18-167

Related papers

Quantum-Enhanced Constraint Satisfaction Problem Solving

Jincheng Zhang

This paper explores the potential of quantum algorithms to revolutionize the field of constraint satisfaction problems (CSPs). CSPs represent a significant challenge in artificial intelligence, particularly in areas like puzzle-solving and optimization. Traditional computational approaches struggle with the vast search space inherent in many CSPs, leading to computationally intensive and time-consuming solutions. This work proposes a novel quantum approach, utilizing Grover's algorithm on quantum state representations, to accelerate the search process and enhance the efficiency of solving these problems. We will detail the theoretical underpinnings of the proposed algorithm, its expected advantages over classical methods, and preliminary experimental results demonstrating its potential for improved performance. The core claim is that quantum-enhanced CSP solving, specifically through Grover's algorithm, can significantly reduce the time required to find solutions within a reasonable computational constraint.

•Sep 3, 2026•0 citations

Quantum-Inspired Algorithms for Constraint Satisfaction Problems

Jincheng Zhang

Constraint satisfaction problems (CSPs) are ubiquitous in various domains, including artificial intelligence, scheduling, and optimization. Traditional CSP solvers often struggle with large and complex problems, demanding significant computational resources. This paper investigates the potential of quantum-inspired algorithms to accelerate the solution of CSPs. We propose an algorithm that utilizes concepts reminiscent of quantum mechanics, specifically amplitude amplification and a modified annealing process, to efficiently explore the solution space. The core claim is that by intelligently mimicking quantum phenomena on classical computers, substantial speedups can be achieved compared to conventional CSP solvers. Our approach focuses on the direct application of quantum-inspired techniques without relying on the availability of quantum hardware, providing a viable strategy for tackling computationally demanding CSP instances. The algorithm's performance is evaluated theoretically, with a focus on the key aspects of search space reduction and convergence rates. The results suggest a promising direction for future research in hybrid quantum-classical algorithms for CSPs.

•Sep 5, 2026•0 citations

Solving Boolean Satisfiability Problems With The Quantum Approximate Optimization Algorithm

Sami Boulebnane, Ashley Montanaro

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k -SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n , which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024

arXiv:2208.06909•PRX Quantum•Sep 10, 2024•57 citations

A Benchmarking Study of Grover’s Algorithm for Solving Boolean SAT with Quantum Circuits

Jirapas U. Jipipob, Prabhas Chongstitvatana

This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.

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

Quantum-Walk Speedup of Backtracking Algorithms

Ashley Montanaro

$ $ We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on $n$ variables, or outputs that none exists, and whose corresponding tree contains $T$ vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using $O(\sqrt{T} n^{3/2} \log n)$ tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree.

Theory of Computing•Jan 1, 2018•70 citations

Quantum walk speedup of backtracking algorithms

Ashley Montanaro

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.

arXiv:1509.02374•Sep 8, 2015•28 citations

Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over 105 relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over 103 . However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.

This paper explores the potential of quantum algorithms to revolutionize the field of constraint satisfaction problems (CSPs). CSPs represent a significant challenge in artificial intelligence, particularly in areas like puzzle-solving and optimization. Traditional computational approaches struggle with the vast search space inherent in many CSPs, leading to computationally intensive and time-consuming solutions. This work proposes a novel quantum approach, utilizing Grover's algorithm on quantum state representations, to accelerate the search process and enhance the efficiency of solving these problems. We will detail the theoretical underpinnings of the proposed algorithm, its expected advantages over classical methods, and preliminary experimental results demonstrating its potential for improved performance. The core claim is that quantum-enhanced CSP solving, specifically through Grover's algorithm, can significantly reduce the time required to find solutions within a reasonable computational constraint.

Constraint satisfaction problems (CSPs) are ubiquitous in various domains, including artificial intelligence, scheduling, and optimization. Traditional CSP solvers often struggle with large and complex problems, demanding significant computational resources. This paper investigates the potential of quantum-inspired algorithms to accelerate the solution of CSPs. We propose an algorithm that utilizes concepts reminiscent of quantum mechanics, specifically amplitude amplification and a modified annealing process, to efficiently explore the solution space. The core claim is that by intelligently mimicking quantum phenomena on classical computers, substantial speedups can be achieved compared to conventional CSP solvers. Our approach focuses on the direct application of quantum-inspired techniques without relying on the availability of quantum hardware, providing a viable strategy for tackling computationally demanding CSP instances. The algorithm's performance is evaluated theoretically, with a focus on the key aspects of search space reduction and convergence rates. The results suggest a promising direction for future research in hybrid quantum-classical algorithms for CSPs.

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k -SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n , which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024

This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.

$ $ We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on

n

variables, or outputs that none exists, and whose corresponding tree contains

T

vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using

O(\sqrt{T} n^{3/2} \log n)

tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree.

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.