Classical and quantum algorithms for characters of the symmetric group

TL;DR

This paper introduces a quantum algorithm using Matrix Product States to efficiently compute and sample characters of the symmetric group, connecting classical and quantum approaches to a computationally hard problem.

Contribution

It presents a novel MPS-based quantum algorithm for symmetric group characters and a reduction linking classical hardness to sampling problems.

Findings

Efficient quantum circuit for MPS preparation of characters

Reduction from strong to weak simulation for sampling problems

Mapping characters to quantum spin chains

Abstract

Characters of irreducible representations are ubiquitous in group theory. However, computing characters of some groups such as the symmetric group $S_n$ is a challenging problem known to be $\#P$-hard in the worst case. Here we describe a Matrix Product State (MPS) algorithm for characters of $S_n$. The algorithm computes an MPS encoding all irreducible characters of a given permutation. It relies on a mapping from characters of $S_n$ to quantum spin chains proposed by Crichigno and Prakash. We also provide a simpler derivation of this mapping. We complement this result by presenting a $poly(n)$ size quantum circuit that prepares the corresponding MPS, obtaining an efficient quantum algorithm for certain sampling problems based on characters of $S_n$. To assess classical hardness of these problems we present a general reduction from strong simulation (computing a given probability) to weak simulation (sampling with a small error). This reduction applies to any sampling problem with a certain granularity structure and may be of independent interest.