Quantum complexity of the Kronecker coefficients

TL;DR

This paper demonstrates that Kronecker coefficients can be efficiently approximated using quantum algorithms, linking their complexity to quantum verification and counting problems, and explores implications for positivity and related computations.

Contribution

It establishes a quantum complexity framework for Kronecker coefficients, showing they relate to quantum verification and can be approximated efficiently on quantum computers.

Findings

Kronecker coefficients are proportional to the rank of a quantum-measurable projector.

Approximating Kronecker coefficients is as hard as certain quantum counting problems.

Deciding positivity of Kronecker coefficients is in QMA, a quantum complexity class.

Abstract

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a QMA verifier, where QMA is the quantum analogue of NP. This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of quantum approximate counting problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in QMA, complementing a recent NP-hardness result of Ikenmeyer, Mulmuley and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error.