Understanding the Nature of Depth-1 Equivariant Quantum Circuit

TL;DR

This paper introduces SIGS, an efficient optimization method that enables simulation of depth-1 equivariant quantum circuits for large TSP instances, significantly reducing computation time and providing insights into QRL performance at scale.

Contribution

The paper proposes SIGS, a novel size-invariant optimization technique, and demonstrates its effectiveness in simulating large TSP problems with quantum circuits, extending the capabilities of QRL analysis.

Findings

Simulation time reduced by 96.4% for TSP-100

Achieved mean optimality gap within 0.005 of trained models

Provided theoretical basis for size-invariant properties

Abstract

The Equivariant Quantum Circuit (EQC) for the Travelling Salesman Problem (TSP) has been shown to achieve near-optimal performance in solving small TSP problems (up to 20 nodes) using only two parameters at depth 1. However, extending EQCs to larger TSP problem sizes remains challenging due to the exponential time and memory for quantum circuit simulation, as well as increasing noise and decoherence when running on actual quantum hardware. In this work, we propose the Size-Invariant Grid Search (SIGS), an efficient training optimization for Quantum Reinforcement Learning (QRL), and use it to simulate the outputs of a trained Depth-1 EQC up to 350-node TSP instances - well beyond previously tractable limits. At TSP with 100 nodes, we reduce total simulation times by 96.4%, when comparing to RL simulations with the analytical expression (151 minutes using RL to under 6 minutes using SIGS on TSP-100), while achieving a mean optimality gap within 0.005 of the RL trained model on the test set. SIGS provides a practical benchmarking tool for the QRL community, allowing us to efficiently analyze the performance of QRL algorithms on larger problem sizes. We provide a theoretical explanation for SIGS called the Size-Invariant Properties that goes beyond the concept of equivariance discussed in prior literature.