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

## TL;DR

This paper introduces a reusable variational quantum algorithm for the Traveling Salesman Problem that reduces qubit requirements and computational effort by freezing optimized circuit structures for new instances, demonstrating strong results on small problem sizes.

## Contribution

The paper proposes a novel optimize-freeze-reuse strategy for variational algorithms, enabling rapid re-optimization on new TSP instances with minimal structural adjustments.

## Key findings

- Achieves near 100% success on 4-city TSP instances
- Maintains high success rates (~80%) on 6-city instances
- Shows scalability limitations beyond 7 cities

## 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.