The travelling salesperson problem and the challenges of near-term quantum advantage

Abstract Over the last two decades, the travelling salesperson problem (TSP) has been cited as a benchmark problem to demonstrate the advantage of quantum computers over conventional computers. Its advantage is that it is a well-studied NP-hard optimisation problem that can be easily communicated to highlight the challenges of searching through an exponentially growing number of possible solutions to find the optimal solution. It is therefore a tempting problem to choose to explore quantum advantage. At what point, however, is a call made that quantum advantage is not likely, and efforts should be focused on other problems? This article challenges the continued use of the TSP as a benchmark for quantum optimisation methods—such as quantum annealing and gate-based quantum computing—that require the TSP to be formulated as a quadratic unconstrained binary optimisation (QUBO) problem. We offer explanations for why such quantum approaches are not well suited, nor competitive against state-of-the-art classical methods, for tackling the challenges of the TSP landscape, and we draw parallels with similar observations made almost four decades ago when QUBO-based neural networks proved to be uncompetitive for solving the TSP. After critically reviewing two decades of research effort to solve TSPs using QUBO-based quantum methods, we note a gradual shift in focus: from initial attempts to solve small sized TSPs with general-purpose QUBO-based quantum approaches, to growing evidence that competitiveness is only enhanced where TSP domain knowledge is integrated, via either modified formulations or hybridisation with TSP classical heuristics. We discuss the numerous challenges that must be overcome before QUBO-based quantum optimisers could ever be competitive with classical state-of-the-art TSP solvers. Acknowledging that there may be more promise for non-QUBO-based hybrid approaches, where quantum search accelerates components of conventional algorithms, we offer recommendations for how future studies should be conducted to compare fairly and rigorously any proposed quantum methods against state-of-the-art TSP solvers, or any classical optimisation method, when seeking to establish quantum advantage.