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  • Open Access

Classical and Quantum Algorithms for Characters of the Symmetric Group

Sergey Bravyi1, David Gosset2,3,*, Vojtech Havlicek1, and Louis Schatzki4

  • *Contact author: dgosset@uwaterloo.ca
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PRX Quantum 6, 030323 – Published 7 August, 2025

DOI: https://doi.org/10.1103/bq28-r2r7

Abstract

Characters of irreducible representations are ubiquitous in group theory. However, computing characters of some groups such as the symmetric group Sn 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 Sn. The algorithm computes an MPS encoding all irreducible characters of a given permutation. It relies on a mapping from characters of Sn 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 Sn. 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.

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Popular Summary

Representation theory is the backbone of modern quantum mechanics and provides a powerful set of tools for analyzing systems invariant under the action of some group, such as the molecular group in chemistry or gauge groups in particle physics. One of the most challenging computational problems in this area is computing irreducible characters of the symmetric group Sn, the group of permutations of n elements. Here we show that this problem can be tackled using tensor-network algorithms developed by physicists for simulating time evolution of quantum spin chains.

Firstly, we provide a new classical algorithm that computes a matrix product state (MPS) of 2n qubits encoding all irreducible characters of a given permutation. This algorithm relies on a mapping from the characters of Sn to quantum spin chains recently proposed by Crichigno and Prakash. We numerically benchmark the new algorithm and find that its runtime is on par with the leading computer algebra systems.

We complement this result with a construction of a quantum circuit that prepares the MPS encoding characters of a given permutation. The quantum circuit has a size polynomial in n, even though the bond dimension of the considered MPS may be exponentially large. This allows us to obtain efficient quantum algorithms for two natural sampling problems based on characters of the symmetric group.

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PRX Quantum 6, 030323– Published 7 August, 2025
  • Received 20 March 2025
  • Accepted 25 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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