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Transferable Equivariant Quantum Circuits for TSP: Generalization Bounds and Empirical Validation

Monit Sharma, Hoong Chuin Lau

TL;DR

The paper tackles generalization in quantum reinforcement learning for the Traveling Salesman Problem by leveraging permutation-equivariant quantum circuits (EQCs) to enable zero-shot transfer from small to larger problem instances. It derives a formal transfer bound that decomposes risk into a source-generalization term and a task-dissimilarity penalty $\mathcal{D}_{n \rightarrow m}$, which itself splits into parametric and structural components, both scaling with the size gap $m-n$. The theory is complemented by empirical validation on Euclidean TSP benchmarks, where policies trained on small graphs exhibit strong zero-shot transfer and can be further improved through targeted fine-tuning, with permutation-equivariant models consistently outperforming non-equivariant baselines. The work demonstrates that embedding symmetry into quantum models yields scalable, transferable QRL solutions for symmetric combinatorial tasks and outlines practical guidelines for curriculum-style training and symmetry-preserving architectures.

Abstract

In this work, we address the challenge of generalization in quantum reinforcement learning (QRL) for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Training quantum policies on large TSP instances is often infeasible, so existing QRL approaches are limited to small-scale problems. To mitigate this, we employed Equivariant Quantum Circuits (EQCs) that respect the permutation symmetry of the TSP graph. This symmetry-aware ansatz enabled zero-shot transfer of trained parameters from $n-$city training instances to larger m-city problems. Building on recent theory showing that equivariant architectures avoid barren plateaus and generalize well, we derived novel generalization bounds for the transfer setting. Our analysis introduces a term quantifying the structural dissimilarity between $n-$ and $m-$node TSPs, yielding an upper bound on performance loss under transfer. Empirically, we trained EQC-based policies on small $n-$city TSPs and evaluated them on larger instances, finding that they retained strong performance zero-shot and further improved with fine-tuning, consistent with classical observations of positive transfer between scales. These results demonstrate that embedding permutation symmetry into quantum models yields scalable QRL solutions for combinatorial tasks, highlighting the crucial role of equivariance in transferable quantum learning.

Transferable Equivariant Quantum Circuits for TSP: Generalization Bounds and Empirical Validation

TL;DR

The paper tackles generalization in quantum reinforcement learning for the Traveling Salesman Problem by leveraging permutation-equivariant quantum circuits (EQCs) to enable zero-shot transfer from small to larger problem instances. It derives a formal transfer bound that decomposes risk into a source-generalization term and a task-dissimilarity penalty , which itself splits into parametric and structural components, both scaling with the size gap . The theory is complemented by empirical validation on Euclidean TSP benchmarks, where policies trained on small graphs exhibit strong zero-shot transfer and can be further improved through targeted fine-tuning, with permutation-equivariant models consistently outperforming non-equivariant baselines. The work demonstrates that embedding symmetry into quantum models yields scalable, transferable QRL solutions for symmetric combinatorial tasks and outlines practical guidelines for curriculum-style training and symmetry-preserving architectures.

Abstract

In this work, we address the challenge of generalization in quantum reinforcement learning (QRL) for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Training quantum policies on large TSP instances is often infeasible, so existing QRL approaches are limited to small-scale problems. To mitigate this, we employed Equivariant Quantum Circuits (EQCs) that respect the permutation symmetry of the TSP graph. This symmetry-aware ansatz enabled zero-shot transfer of trained parameters from city training instances to larger m-city problems. Building on recent theory showing that equivariant architectures avoid barren plateaus and generalize well, we derived novel generalization bounds for the transfer setting. Our analysis introduces a term quantifying the structural dissimilarity between and node TSPs, yielding an upper bound on performance loss under transfer. Empirically, we trained EQC-based policies on small city TSPs and evaluated them on larger instances, finding that they retained strong performance zero-shot and further improved with fine-tuning, consistent with classical observations of positive transfer between scales. These results demonstrate that embedding permutation symmetry into quantum models yields scalable QRL solutions for combinatorial tasks, highlighting the crucial role of equivariance in transferable quantum learning.
Paper Structure (69 sections, 1 theorem, 50 equations, 3 figures)

This paper contains 69 sections, 1 theorem, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions:
  3. Prior Approaches:
  4. Paper Structure.
  5. Foundations and Related Work
  6. Symmetry
  7. Locality
  8. Generalization
  9. Theoretical Framework for Zero-Shot Transfer
  10. Background and Motivation
  11. Problem Statement
  12. Approach and Contribution
  13. Notation and Preliminaries
  14. Assumption on Source-Task Training Quality.
  15. Problem Formulation: Zero-Shot Transfer for TSP with QRL
  16. ...and 54 more sections

Key Result

Theorem 1

Let $\theta_n^*$ denote the parameters trained on $M_n$ i.i.d. episodes from the $n$-city task. Under Assumptions(A1)–(A7), the performance on the $m$-city target task satisfies, with probability at least $1 - \delta$: where: and:

Figures (3)

  • Figure 1: Comparison of circuit architectures used in this study. (a) EQC uses globally entangling $ZZ$ interactions; (b) Efficient SU(2) uses local SU(2) rotations and linear entanglement.
  • Figure 2: Comparison of transfer performance: Permutation Equivariant (left column) vs. Efficient SU(2) (right column) for 4-node, 6-node, 8-node and 10-node source graphs.
  • Figure 3: Empirical Validation of the Theoretical Transfer Performance Bound. Each panel shows the performance of a model trained on a source TSP of size $n$ and transferred to larger target TSPs of size $m$. The $y$-axis represents the normalized performance metric (higher is better, with 1 being optimal). The black star indicates the model's performance on its original source task ($m = n$). Blue circles show the empirical zero-shot transfer performance, with error bars indicating one standard deviation. The red dashed line is the theoretical lower bound on zero-shot performance derived from our analysis, and the green shaded region represents the theoretically allowed performance space. Green squares show the improved performance achieved after fine-tuning the transferred model on the target task. The results demonstrate that empirical zero-shot performance consistently remains above the theoretical bound, and that fine-tuning substantially closes the performance gap, validating its role as a practical and effective transfer strategy.

Theorems & Definitions (2)

  • Theorem 1: Zero-Shot Transfer Performance Bound
  • Remark 1: On Notation Simplification