Polynomial time classical versus quantum algorithms for representation theoretic multiplicities

TL;DR

This paper demonstrates that many representation theoretic multiplicities, previously thought to require quantum algorithms for efficient computation, can actually be computed classically in polynomial time, challenging prior assumptions about quantum speedups.

Contribution

It shows that for many cases, Kronecker and plethysm coefficients are computable in polynomial time using classical algorithms, refuting some earlier conjectures about quantum advantages.

Findings

Many cases of Kronecker and plethysm coefficients are classically polynomial-time computable.

This limits the potential for super-polynomial quantum speedups in these problems.

The results challenge previous beliefs about the necessity of quantum algorithms for these computations.

Abstract

Littlewood-Richardson, Kronecker and plethysm coefficients are fundamental multiplicities of interest in Representation Theory and Algebraic Combinatorics. Determining a combinatorial interpretation for the Kronecker and plethysm coefficients is a major open problem, and prompts the consideration of their computational complexity. Recently it was shown that they behave relatively well with respect to quantum computation, and for some large families there are polynomial time quantum algorithms [Larocca,Havlicek, arXiv:2407.17649] (also [BCGHZ,arXiv:2302.11454]). In this paper we show that for many of those cases the Kronecker and plethysm coefficients can also be computed in polynomial time via classical algorithms, thereby refuting some of the conjectures in [LH24]. This vastly limits the cases in which the desired super-polynomial quantum speedup could be achieved.