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Quantum Information and Computation

A benchmark for quantum optimization: the traveling salesman

Richard H. Warren

Published Apr 27, 2021• 8 citations
Publisher

Abstract

We present compelling reasons for symmetric traveling salesman problems (TSPs) to be the benchmark for quantum computing of combinatorial optimization problems for all types of quantum hardware. There are seven reasons for endorsing these TSPs to be the benchmark and no shortcomings.
DOI
10.26421/qic21.7-8-2

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We study small traveling salesman problems (TSPs) because current quantum computers can find optional solutions for TSPs with up to 14 cities. Also, we study small TSPs because TSPs have been recommended to be benchmarks to measure quantum optimization on all types of quantum hardware. This means comparisons of quantum data about small TSPs. We extent previous numerical results that were reported in “Small Traveling Salesman Problems” for 6, 8 and 10 cities. The new results in this paper are for 10 – 14 cities in symmetric TSPs. The data for this new range of cities is consistent with the previous data and can be the basis for estimates of results from quantum computers that are upgraded to handle more than 14 cities. The work and analysis suggest two conjectures that we discuss. The paper also contains an annotated survey of recent publications about TSPs.
The Traveling Salesman Problem (TSP) serves as a standard benchmark for assessing potential quantum advantage in combinatorial optimization. This paper provides a critical comparative analysis of quantum annealing, gate-based approaches based on quantum phase estimation, and hybrid quantum–classical dynamic programming methods within a unified complexity and implementation framework. While theoretical results indicate reductions of the classical O ( 2 n ) scaling to approximately O ( 1.728 n ) for general instances and to O ( 1.110 n ) for structured cases, these improvements remain largely asymptotic. In practice, all current quantum approaches are strongly constrained by hardware limitations. Quantum annealing suffers from embedding overhead and noise, hybrid methods rely extensively on classical decomposition without guaranteed optimality, and gate-based implementations are limited by circuit depth and error accumulation. As a result, no clear quantum advantage over classical algorithms has been demonstrated at practically relevant problem scales. By systematically separating intrinsic algorithmic scaling from hardware-induced resource constraints, this work clarifies the origin of the gap between theoretical quantum speedups and experimental performance, providing a critical assessment of the current status and realistic prospects of quantum TSP methods.
The Travelling Salesman Problem (TSP) asks for the shortest tour that visits each city once and returns to the start. It is a classic NP-hard problem and a benchmark for optimization, logistics, and network design. This paper surveys classical, quantum, and hybrid algorithms for the TSP using clear, step-by-step explanations. We summarize how each method works, what it guarantees, and how it scales with the number of cities. We also discuss practical limits of current quantum hardware and outline open problems and promising research directions. While quantum techniques can speed up certain subroutines or offer strong heuristics, no known method solves general TSP in polynomial time. Our goal is to offer a readable yet rigorous map of the landscape for students and practitioners.
The Traveling Salesman Problem (TSP) is one of the most often-used NP-Hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also being used heavily as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this work, we formulate the symmetric TSP as an optimization problem, which we solve using the quantum approximate optimization algorithm (QAOA) approach. By adopting an improved qubit encoding strategy and a layerwise learning optimization protocol, we obtain numerical results on the gate-based digital quantum simulator for the 3-, 4-, and 5-city TSPs. Specifically, we focus on three QAOA mixer designs to evaluate their performances in terms of numerical accuracy and optimization cost. Based on our results, we propose that a well-balanced QAOA mixer design is more prominent on gate-based simulators or realistic quantum devices in the near future. In addition, we study the sensitivity of TSP graph properties such as graph skewness and penalty weight in the TSP-QAOA simulation. Overall, our results prove digital quantum simulation is a powerful candidate for obtaining the optimal solution to the TSP.
At land, sea and in the air mobility and traffic management offer a vast amount of problems with a large potential of optimization with quantum computers, e.g. service scheduling, route planning, or path optimization. Many of these problems can be described at a fundamental level by the traveling salesman problem (TSP), in which the shortest route while visiting each point exactly once is to be found [1]. The TSP has already received a lot of attention in the quantum computing community, for example, implementations for adiabatic quantum annealers exist and have been tested [2,3]. We investigate the TSP with a focus on going beyond qubits by employing continuous quantum systems. Using bosonic Qiskit we simulate potential algorithms for solving the TSP and compare their performance. [1] Flood, M. M., The traveling-salesman problem. Operations research, 4(1), 61-75, (1956). [2] Martoňák, Roman, Giuseppe E. Santoro, and Erio Tosatti., Quantum annealing of the traveling-salesman problem. Physical Review E 70.5: 057701, (2004). [3] Jain, S., Solving the traveling salesman problem on the d-wave quantum computer. Frontiers in Physics, 646, (2021).