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Freeze and Conquer: Reusable Ansatz for Solving the Traveling Salesman Problem

Fabrizio Fagiolo, Nicolò Vescera

TL;DR

This work tackles solving the Traveling Salesman Problem with variational quantum algorithms by introducing a compact permutation encoding that reduces qubit requirements to $O(n \log n)$. It then implements a two-phase optimize-freeze-reuse workflow: an SA-driven optimization over both the Ansatz topology and parameters on a training instance, followed by freezing the topology and re-optimizing only the parameters for new instances. Empirical results on 44 symmetric TSP instances (4–7 cities) show perfect performance for 4 cities, high success for 5 and 6 cities, and a substantial drop for 7 cities, highlighting strong generalization at moderate sizes and scalability limits. The study emphasizes practical applicability on NISQ devices, while outlining future directions such as hardware experiments, error mitigation, gradient-based optimization, warm-start strategies, and extensions to VRP/JSS.

Abstract

In this paper we present a variational algorithm for the Traveling Salesman Problem (TSP) that combines (i) a compact encoding of permutations, which reduces the qubit requirement too, (ii) an optimize-freeze-reuse strategy: where the circuit topology (``Ansatz'') is first optimized on a training instance by Simulated Annealing (SA), then ``frozen'' and re-used on novel instances, limited to a rapid re-optimization of only the circuit parameters. This pipeline eliminates costly structural research in testing, making the procedure immediately implementable on NISQ hardware. On a set of $40$ randomly generated symmetric instances that span $4 - 7$ cities, the resulting Ansatz achieves an average optimal trip sampling probability of $100\%$ for 4 city cases, $90\%$ for 5 city cases and $80\%$ for 6 city cases. With 7 cities the success rate drops markedly to an average of $\sim 20\%$, revealing the onset of scalability limitations of the proposed method. The results show robust generalization ability for moderate problem sizes and indicate how freezing the Ansatz can dramatically reduce time-to-solution without degrading solution quality. The paper also discusses scalability limitations, the impact of ``warm-start'' initialization of parameters, and prospects for extension to more complex problems, such as Vehicle Routing and Job-Shop Scheduling.

Freeze and Conquer: Reusable Ansatz for Solving the Traveling Salesman Problem

TL;DR

This work tackles solving the Traveling Salesman Problem with variational quantum algorithms by introducing a compact permutation encoding that reduces qubit requirements to . It then implements a two-phase optimize-freeze-reuse workflow: an SA-driven optimization over both the Ansatz topology and parameters on a training instance, followed by freezing the topology and re-optimizing only the parameters for new instances. Empirical results on 44 symmetric TSP instances (4–7 cities) show perfect performance for 4 cities, high success for 5 and 6 cities, and a substantial drop for 7 cities, highlighting strong generalization at moderate sizes and scalability limits. The study emphasizes practical applicability on NISQ devices, while outlining future directions such as hardware experiments, error mitigation, gradient-based optimization, warm-start strategies, and extensions to VRP/JSS.

Abstract

In this paper we present a variational algorithm for the Traveling Salesman Problem (TSP) that combines (i) a compact encoding of permutations, which reduces the qubit requirement too, (ii) an optimize-freeze-reuse strategy: where the circuit topology (``Ansatz'') is first optimized on a training instance by Simulated Annealing (SA), then ``frozen'' and re-used on novel instances, limited to a rapid re-optimization of only the circuit parameters. This pipeline eliminates costly structural research in testing, making the procedure immediately implementable on NISQ hardware. On a set of randomly generated symmetric instances that span cities, the resulting Ansatz achieves an average optimal trip sampling probability of for 4 city cases, for 5 city cases and for 6 city cases. With 7 cities the success rate drops markedly to an average of , revealing the onset of scalability limitations of the proposed method. The results show robust generalization ability for moderate problem sizes and indicate how freezing the Ansatz can dramatically reduce time-to-solution without degrading solution quality. The paper also discusses scalability limitations, the impact of ``warm-start'' initialization of parameters, and prospects for extension to more complex problems, such as Vehicle Routing and Job-Shop Scheduling.

Paper Structure

This paper contains 13 sections, 8 equations, 4 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Proposed solution
  4. Ansatz Optimization
  5. Reusing the Optimized Ansatz
  6. Experiments
  7. Problem setup
  8. Experimental Setup
  9. Results
  10. Convergence to the optimal solution
  11. Best Ansatz found
  12. Reusing
  13. Conclusions and Future Work

Figures (4)

  • Figure 1: Quantum Circuit associated to example in (\ref{['eq:ansatzexample']})
  • Figure 2: Convergence of SA toward the optimal solution probability for TSP instances with 4, 5, 6 and 7 cities (index 0, training set). The fitness evolves by 500 steps.
  • Figure 3: Best Ansatz found for training instance with 5 nodes
  • Figure 4: Distribution of solution accuracy across the four problem‑size (city‑count) scenarios.