Plethysm is in #BQP

TL;DR

This paper demonstrates that a broad class of representation-theoretic multiplicities, including plethysm coefficients, are in the quantum complexity class #BQP, unifying and extending prior results and providing new quantum algorithms.

Contribution

The authors prove that many representation-theoretic multiplicities, such as plethysm coefficients, are in #BQP, and develop a general approach for establishing this inclusion.

Findings

Plethysm coefficients are in #BQP, previously known only in special cases.

The approach unifies quantum complexity results for various multiplicities.

Classical polynomial-time algorithms exist for fixed parameters.

Abstract

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.