Classical and quantum algorithms for characters of the symmetric group

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.