Quantum complexity of the Kronecker coefficients
Sergey Bravyi, Anirban Chowdhury, David Gosset, Vojtech Havlicek, Guanyu Zhu
TL;DR
The paper shows that the Kronecker coefficient $g_{\mu\nu\lambda}$ for the symmetric group can be expressed as $g_{\mu\nu\lambda} = \frac{1}{d_{\mu}d_{\nu}d_{\lambda}} \mathrm{Tr}(P_{\mu\nu\lambda})$, where $P_{\mu\nu\lambda}$ is a projector arising from a combination of weak Fourier sampling and generalized phase estimation using Beals' quantum Fourier transform on $S_n$. This leads to the result that $d_{\mu}d_{\nu}d_{\lambda} g_{\mu\nu\lambda}$ is in $\#\BQP$, and approximating $g_{\mu\nu\lambda}$ to relative error reduces to quantum approximate counting $\QAPC$, since the irreducible dimensions $d_{\omega}$ are efficiently computable via the hook-length formula. A corollary is that deciding positivity of $g_{\mu\nu\lambda}$ lies in $\QMA$, complementing NP-hardness results for vanishing, and analogous statements hold for row sums $R_{\lambda}$ of the symmetric group's character table via the conjugation representation. The authors also discuss an efficient quantum algorithm to approximate the normalized Kronecker coefficients $g_{\mu\nu\lambda} d_{\lambda}/(d_{\mu}d_{\nu})$ to inverse-polynomial additive error, highlighting a concrete bridge between representation theory and quantum counting complexity.
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.