Quantum Algorithms for Representation-Theoretic Multiplicities

TL;DR

This paper develops quantum algorithms for computing key representation-theoretic multiplicities, explores their classical and quantum complexities, and discusses potential superpolynomial quantum speedups for certain coefficients.

Contribution

It introduces quantum algorithms for Kostka, Littlewood-Richardson, Plethysm, and Kronecker coefficients and analyzes their computational complexity and potential advantages over classical methods.

Findings

Classical polynomial-time algorithm for Kostka numbers under certain conditions

Quantum algorithms potentially offer superpolynomial speedups for Plethysm and Kronecker coefficients

Disproof of a conjecture regarding Kronecker coefficients complexity by Greta Panova

Abstract

Kostka, Littlewood-Richardson, Plethysm and Kronecker coefficients are the multiplicities of irreducible representations in the decomposition of representations of the symmetric group that play an important role in representation theory, geometric complexity and algebraic combinatorics. We give quantum algorithms for computing these coefficients whenever the ratio of dimensions of the representations is polynomial and study the computational complexity of this problem. We show that there is an efficient classical algorithm for computing the Kostka numbers under this restriction and conjecture the existence of an analogous algorithm for the Littlewood-Richardson coefficients. We argue why such classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients and conjecture that our quantum algorithms lead to superpolynomial speedups for these problems. The conjecture about Kronecker coefficients was disproved by Greta Panova in [arXiv:2502.20253] with a classical solution which, if optimal, points to a $\OC(n^{4+2k})$ vs $\tilde{\Omega}(n^{4k^2+1})$ polynomial gap in quantum vs classical computational complexity for an integer parameter $k$.