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arXiv:2311.14344

Tensor-Network Formulation of the Traveling Salesman Problem and Variants

Alejandro Mata Ali, Iñigo Perez Delgado, Aitor Moreno-Fernández-de-Leceta

Published Nov 24, 2023• 1 citation
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Abstract

This work presents a tensor-network formulation of the Traveling Salesman Problem (TSP) and several of its variants. The approach represents candidate tours with tensor-network layers, weights them by Boltzmann factors, and enforces constraints through explicit counting filters. This formalism also yields an explicit tensor-network marginal formula whose zero-temperature, exact-arithmetic limit identifies an optimal feasible tour through a sequential marginal rule. At finite $τ$ and finite precision, the implemented extraction is a heuristic whose behavior depends on numerical contrast, calibration, and near-degeneracies. We adapt the construction to several generalizations of the TSP and apply it to the Job Reassignment Problem, as a representative industrial integration. The experiments are deliberately small and illustrative; they contextualize the method against exact and heuristic references but do not establish general computational superiority over specialized classical solvers.

DOI
10.48550/arxiv.2311.14344

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This work presents a tensor-network formulation of the Traveling Salesman Problem (TSP) and several of its variants. The approach represents candidate tours with tensor-network layers, weights them by Boltzmann factors, and enforces constraints through explicit counting filters. This formalism also yields an explicit tensor-network marginal formula whose zero-temperature, exact-arithmetic limit identifies an optimal feasible tour through a sequential marginal rule. At finite τττ and finite precision, the implemented extraction is a heuristic whose behavior depends on numerical contrast, calibration, and near-degeneracies. We adapt the construction to several generalizations of the TSP and apply it to the Job Reassignment Problem, as a representative industrial integration. The experiments are deliberately small and illustrative; they contextualize the method against exact and heuristic references but do not establish general computational superiority over specialized classical solvers.
We present an application of the tensor network generator-enhanced optimization (TN-GEO) framework to address the traveling salesman problem (TSP), a fundamental combinatorial optimization challenge. Our approach employs a tensor network Born machine based on automatically differentiable matrix product states (MPS) as the generative model, using the Born rule to define probability distributions over candidate solutions. Unlike approaches based on binary encoding, which require N2N^2N2 variables and penalty terms to enforce valid tour constraints, we adopt a permutation-based formulation with integer variables and use autoregressive sampling with masking to guarantee that every generated sample is a valid tour by construction. We also introduce a kkk-site MPS variant that learns distributions over kkk-grams (consecutive city subsequences) using a sliding window approach, enabling parameter-efficient modeling for larger instances. Experimental validation on TSPLIB benchmark instances with up to 52 cities demonstrates that TN-GEO can outperform classical heuristics including swap and 2-opt hill-climbing. The kkk-site variants, which put more focus on local correlations, show better results compared to the full-MPS case.
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 Travelling Salesman Problem (TSP) is a well-known NP-hard combinatorial optimisation problem with important industrial applications. Although extensively studied, quantum solutions for instances beyond a dozen locations remain uncommon. We present a hybrid penalty-free Variational Quantum Algorithm (VQA) that significantly reduces both the qubit count and the runtime through a compact penalty-free formulation scaling as O(nlog2n)\mathcal{O}(nlog_2 n)O(nlog2​n) qubits compared to conventional formulations with O(n2)\mathcal{O}(n^2)O(n2) scaling. The VQA achieves solution qualities exceeding 90\% on Rigetti quantum hardware for instances of up to 17 locations. To the best of our knowledge, this represents the largest TSP instance solved on gate-based hardware. We systematically evaluate multiple encoding and optimisation strategies, and show that warm-start initialisation improves solution quality, while runtime is reduced by nearly two orders of magnitude through the use of gradient-free optimisation methods and cost-function caching. We additionally introduce a novel VQA inspired classical machine learning model (VQAi-CML) and benchmark the VQA against VQAi-CML, a greedy heuristic, a Monte Carlo baseline, and a strong classical optimisation baseline. Across the networks evaluated, the VQA consistently achieves superior solution quality relative to VQAi-CML, greedy heuristic, and Monte Carlo baselines, but not the classical solver
The Travelling Salesman Problem (TSP) constitutes a fundamental Combinatorial Optimisation Problem (COP) with wide-ranging applications in logistics, transportation design and production scheduling. However, the best-known exact classical algorithms exhibit exponential time complexity, rendering large instances intractable in practice and motivating the exploration of alternative paradigms such as quantum computing. In this work, we implement the Quadratic Unconstrained Binary Optimisation (QUBO) formulation and its generalisation, Tensor Quadratic Unconstrained D-ary Optimisation (T-QUDO), which extends quantum circuit encoding from qubits to qudits, for a TSP variant. This variant seeks to determine a minimum-cost route while incorporating a timedependent cost function and precedence constraints, and remains largely unexplored with T-QUDO despite its relevance to dynamic travel planning. The formulations are encoded within the Quantum Approximate Optimisation Algorithm (QAOA) framework and analysed under noiseless simulations with the Cirq and CUDA-Q libraries, aiming to assess the potential advantages of qudit-based representations over qubit-based ones, particularly in terms of expressivity, resource efficiency, and solution quality.
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.
Ising formulation is important for many NP problems (Lucas, 2014). This formulation enables implementing novel quantum computing methods including Quantum Approximate Optimization Algorithm and Variational Quantum Eigensolver (VQE). Here, we investigate closely the traveling salesman problem (TSP). First, we present some non-trivial issues related to Ising model view versus a realistic salesman. Then, focusing on VQE we discuss and clarify the use of: a.-- Conventional VQE and how it is relevant as a novel SAT-solver; b.-- Qubit efficiency and its importance in the Noisy Intermediate Scale Quantum-era; and c.-- the relevance and importance of a novel approach named Discrete Quantum Exhaustive Search (Alfassi, Meirom, and Mor, 2024), for enhancing VQE and other methods using mutually unbiased bases. The approach we present here in details can potentially be extended for analyzing approximating and solving various other NP complete problems. Our approach can also be extended beyond the Ising model and beyond the class NP, for example to the class Quantum Merlin Arthur (QMA) of problems, relevant for quantum chemistry and for general spin problems.