Practical framework for simulating permutation-equivariant quantum circuits

TL;DR

This paper presents a practical, efficient algorithm for simulating permutation-equivariant quantum circuits, significantly reducing computational complexity and enabling simulations of larger systems on standard hardware.

Contribution

The authors develop a new algorithm that simulates $S_n$-equivariant circuits with $k$-local generators in $O(n^{ ext{ω}+1})$ time, improving over previous methods.

Findings

Algorithm runs in polynomial time with lower degree than existing methods.

Simulation of 512-spin Lipkin-Meshkov-Glick model completed in under two minutes.

Numerical validation confirms scalability and efficiency of the proposed approach.

Abstract

Understanding which subclasses of quantum circuits are efficiently classically simulable is fundamental to delineating the boundary between classical and quantum computation. In this context, it is well known that certain tasks based on permutation-equivariant unitaries-i.e., $n$-qubit circuits whose action commutes with the qubit-permuting representation of the symmetric group $S_n$-can be simulated in polynomial time. However, existing approaches scale as $O(n^7)$, and can rapidly become prohibitively expensive. In this work, we introduce a practical algorithm for simulating $S_n$-equivariant circuits under the assumption that the gate generators are at most $k$-local, with $k\in O(1)$. The resulting method runs in $O(n^{\omega+1})$ time for constant depth, where $\omega$ is the matrix multiplication exponent, significantly lowering the polynomial degree compared to existing techniques. Finally, we numerically validate this scaling by simulating the dynamical evolution of the Lipkin-Meshkov-Glick model, and show that for $n=512$ spins, a standard laptop can compute the concurrence of the evolved state in under two minutes.