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Traveling Salesman Problem from a Tensor Networks Perspective

Alejandro Mata Ali, Iñigo Perez Delgado, Aitor Moreno Fdez. de Leceta

TL;DR

This work introduces a quantum-inspired tensor-network algorithm for the Traveling Salesman Problem (TSP) that uses uniform superposition, imaginary time evolution, projection, and partial tracing to obtain approximate solutions with reduced cost. It extends the approach to generalized TSP variants (DNSNN, NMTSP, BTSP, PTSP, TSPP) and demonstrates a proof-of-concept on the ONCE Job Reassignment Problem, including an approximate, scalable option via constraint-layer reduction and MPS compression. The method relies on non-unitary operations and MPO-based constraint layers to enforce the one-visit-per-node rule and other problem-specific constraints, with a detailed contraction scheme and complexity analysis showing exponential scaling in the number of nodes, yet practical advantages through reuse and approximations. The results on industrially relevant instances indicate competitive performance against specialized hardware and highlight the framework's potential for hybrid classical-quantum or quantum-inspired optimization workflows.

Abstract

We present a novel quantum-inspired algorithm for solving the Traveling Salesman Problem (TSP) and some of its variations using tensor networks. This approach consists on the simulated initialization of a quantum system with superposition of all possible combinations, an imaginary time evolution, a projection, and lastly a partial trace to search for solutions. This is a heuristically approximable algorithm to obtain approximate solutions with a more affordable computational cost. We adapt it to different generalizations of the TSP and apply it to the job reassignment problem, a real productive industrial case.

Traveling Salesman Problem from a Tensor Networks Perspective

TL;DR

This work introduces a quantum-inspired tensor-network algorithm for the Traveling Salesman Problem (TSP) that uses uniform superposition, imaginary time evolution, projection, and partial tracing to obtain approximate solutions with reduced cost. It extends the approach to generalized TSP variants (DNSNN, NMTSP, BTSP, PTSP, TSPP) and demonstrates a proof-of-concept on the ONCE Job Reassignment Problem, including an approximate, scalable option via constraint-layer reduction and MPS compression. The method relies on non-unitary operations and MPO-based constraint layers to enforce the one-visit-per-node rule and other problem-specific constraints, with a detailed contraction scheme and complexity analysis showing exponential scaling in the number of nodes, yet practical advantages through reuse and approximations. The results on industrially relevant instances indicate competitive performance against specialized hardware and highlight the framework's potential for hybrid classical-quantum or quantum-inspired optimization workflows.

Abstract

We present a novel quantum-inspired algorithm for solving the Traveling Salesman Problem (TSP) and some of its variations using tensor networks. This approach consists on the simulated initialization of a quantum system with superposition of all possible combinations, an imaginary time evolution, a projection, and lastly a partial trace to search for solutions. This is a heuristically approximable algorithm to obtain approximate solutions with a more affordable computational cost. We adapt it to different generalizations of the TSP and apply it to the job reassignment problem, a real productive industrial case.
Paper Structure (18 sections, 23 equations, 5 figures, 1 table)

This paper contains 18 sections, 23 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Description of the problem
  3. Tensor network solver
  4. Tensor network construction
  5. Iterative resolution and contraction
  6. Computational complexity analysis
  7. Generalized TSP cases
  8. Different number of steps than number of nodes (DNSNN)
  9. Non-markovian TSP (NMTSP)
  10. Bottleneck TSP (BTSP)
  11. Politician TSP (PTSP)
  12. TSP with precedence (TSPP)
  13. Proof of Concept for the ONCE
  14. Problem description
  15. Results
  16. ...and 3 more sections

Figures (5)

  • Figure 1: TSP graph with the solution $\vec{x} = (0,3,1,6,4,2,9,8,13,11,7,12,10,5)$. The blue edges represent possible paths that were not taken, and arrowed red edges represent the path taken by the solution. Edges with weight considered infinite are not represented.
  • Figure 2: TSP tensor network, with the initialization and tracing '+' tensor layer, the '$S$' optimization layer and the '$F$' projection layers.
  • Figure 3: Notation for all indexes used in all tensors in this paper. a), b), c) Tensors with 1 bond index, d), e), f) Tensors with 2 bond index, g), h), i) Tensors with 3 bond index.
  • Figure 4: Contraction scheme for 3 variables with the cost of its steps for $\hat{N}$ variables.
  • Figure 5: MPO layer for 3 multiple bond index tensors.