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Quantum speedup of the Travelling Salesman Problem for bounded-degree graphs

Dom J Moylett, Noah Linden, Ashley Montanaro

Published Apr 4, 2017• 4 citations
OpenAlex

Abstract

The Travelling Salesman Problem is one of the most famous problems in graph theory. However, little is currently known about the extent to which quantum computers could speed up algorithms for the problem. In this paper, we prove a quadratic quantum speedup when the degree of each vertex is at most 3 by applying a quantum backtracking algorithm to a classical algorithm by Xiao and Nagamochi. We then use similar techniques to accelerate a classical algorithm for when the degree of each vertex is at most 4, before speeding up higher-degree graphs via reductions to these instances.

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Quantum speedup of the traveling-salesman problem for bounded-degree graphs

Dominic J. Moylett, Noah Linden, Ashley Montanaro

The traveling-salesman problem is an iconic route-finding task, with applications from chip design to planning and logistics. Here the authors show that if the graph of cities to be visited is of low degree, a traveling salesman armed with a quantum satnav can find the best route quadratically faster than using any known classical method.

arXiv:1612.06203•Physical Review A•Mar 22, 2017•43 citations

An Efficient Quantum Algorithm for the Traveling Salesman Problem

Anant Sharma, Nupur Deshpande, Sanchita Purohit Ghosh +2 more

Abstract The traveling salesman problem is the problem of finding out the shortest route in a network of cities, that a salesman needs to travel to cover all the cities, without visiting the same city more than once. This problem is known to be N P -hard with a brute-force complexity of O ( N N ) or O ( N 2 N ) for N number of cities. This problem is equivalent to finding out the shortest Hamiltonian cycle in a given graph, if at least one Hamiltonian cycle exists in it. Quantum algorithms for this problem typically provide with a quadratic speedup only, using Grover’s search, thereby having a complexity of O ( N N /2 ) or O ( N N ). We present a bounded-error quantum polynomial-time (BQP) algorithm for solving the problem, providing with an exponential speedup. The overall complexity of our algorithm is O ( N 3 log( N ) κ / ϵ + 1/ ϵ 3 ), where the errors ϵ are O (1/poly( N )), and κ is the not-too-large condition number of the matrix encoding all Hamiltonian cycles.

Quantum Information and Computation•Feb 22, 2025•1 citation

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.

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Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

Caroppo, Susanna, Da Lozzo, Giordano, Giuseppe Di Battista +2 more

We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the dependency digraph GP(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V (GP(I))|√δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√nm) time, which improves the best known classical upper bound when m ∈ Ω(n1.4).

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Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista +2 more

We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem $\mathcal P$ and an instance $I$ of $\mathcal P$, such a speedup can be expressed in terms of the average degree $δ$ of the dependency digraph $G_{\mathcal{P}}(I)$ of $I$, determined by a recursive formulation of $\mathcal P$. The nodes of this graph are the subproblems of $\mathcal P$ induced by $I$ and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in $\tilde{O}(|V(G_{\mathcal{P}}(I))| \sqrtδ)$ time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted $n$-vertex digraph with $m$ edges that runs in $\tilde{O}(n\sqrt{nm})$ time, which improves the best known classical upper bound when $m \in Ω(n^{1.4})$.

arXiv:2507.00823•Jul 1, 2025•0 citations

Efficient quantum algorithm for solving travelling salesman problem: An IBM quantum experience

Karthik Srinivasan, Saipriya Satyajit, Bikash K. Behera +1 more

The famous Travelling Salesman Problem (TSP) is an important category of optimization problems that is mostly encountered in various areas of science and engineering. Studying optimization problems motivates to develop advanced techniques more suited to contemporary practical problems. Among those, especially the NP hard problems provide an apt platform to demonstrate supremacy of quantum over classical technologies in terms of resources and time. TSP is one such NP hard problem in combinatorial optimization which takes exponential time order for solving by brute force method. Here we propose a quantum algorithm to solve the travelling salesman problem using phase estimation technique. We approach the problem by encoding the given distances between the cities as phases. We construct unitary operators whose eigenvectors are the computational basis states and eigenvalues are various combinations of these phases. Then we apply phase estimation algorithm to certain eigenstates which give us all the total distances possible for all the routes. After obtaining the distances we can search through this information using the quantum search algorithm for finding the minimum to find the least possible distance as well the route taken. This provides us a quadratic speedup over the classical brute force method for a large number of cities. In this paper, we illustrate an example of the travelling salesman problem by taking four cities and present the results by simulating the codes in the IBM's quantum simulator.

arXiv:1805.10928•May 28, 2018•62 citations
The Travelling Salesman Problem is one of the most famous problems in graph theory. However, little is currently known about the extent to which quantum computers could speed up algorithms for the problem. In this paper, we prove a quadratic quantum speedup when the degree of each vertex is at most 3 by applying a quantum backtracking algorithm to a classical algorithm by Xiao and Nagamochi. We then use similar techniques to accelerate a classical algorithm for when the degree of each vertex is at most 4, before speeding up higher-degree graphs via reductions to these instances.
Abstract The traveling salesman problem is the problem of finding out the shortest route in a network of cities, that a salesman needs to travel to cover all the cities, without visiting the same city more than once. This problem is known to be N P -hard with a brute-force complexity of O ( N N ) or O ( N 2 N ) for N number of cities. This problem is equivalent to finding out the shortest Hamiltonian cycle in a given graph, if at least one Hamiltonian cycle exists in it. Quantum algorithms for this problem typically provide with a quadratic speedup only, using Grover’s search, thereby having a complexity of O ( N N /2 ) or O ( N N ). We present a bounded-error quantum polynomial-time (BQP) algorithm for solving the problem, providing with an exponential speedup. The overall complexity of our algorithm is O ( N 3 log( N ) κ / ϵ + 1/ ϵ 3 ), where the errors ϵ are O (1/poly( N )), and κ is the not-too-large condition number of the matrix encoding all Hamiltonian cycles.
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.
We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the dependency digraph GP(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V (GP(I))|√δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√nm) time, which improves the best known classical upper bound when m ∈ Ω(n1.4).
We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P\mathcal PP and an instance III of P\mathcal PP, such a speedup can be expressed in terms of the average degree δδδ of the dependency digraph GP(I)G_{\mathcal{P}}(I)GP​(I) of III, determined by a recursive formulation of P\mathcal PP. The nodes of this graph are the subproblems of P\mathcal PP induced by III and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in O~(∣V(GP(I))∣δ)\tilde{O}(|V(G_{\mathcal{P}}(I))| \sqrtδ)O~(∣V(GP​(I))∣δ​) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted nnn-vertex digraph with mmm edges that runs in O~(nnm)\tilde{O}(n\sqrt{nm})O~(nnm​) time, which improves the best known classical upper bound when m∈Ω(n1.4)m \in Ω(n^{1.4})m∈Ω(n1.4).
The famous Travelling Salesman Problem (TSP) is an important category of optimization problems that is mostly encountered in various areas of science and engineering. Studying optimization problems motivates to develop advanced techniques more suited to contemporary practical problems. Among those, especially the NP hard problems provide an apt platform to demonstrate supremacy of quantum over classical technologies in terms of resources and time. TSP is one such NP hard problem in combinatorial optimization which takes exponential time order for solving by brute force method. Here we propose a quantum algorithm to solve the travelling salesman problem using phase estimation technique. We approach the problem by encoding the given distances between the cities as phases. We construct unitary operators whose eigenvectors are the computational basis states and eigenvalues are various combinations of these phases. Then we apply phase estimation algorithm to certain eigenstates which give us all the total distances possible for all the routes. After obtaining the distances we can search through this information using the quantum search algorithm for finding the minimum to find the least possible distance as well the route taken. This provides us a quadratic speedup over the classical brute force method for a large number of cities. In this paper, we illustrate an example of the travelling salesman problem by taking four cities and present the results by simulating the codes in the IBM's quantum simulator.