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.

翻译：对称群的Kronecker系数是否计数某类组合对象，是一个长期悬而未决的问题。本文证明，给定Kronecker系数与一个可在量子计算机上高效测量的投影算子的秩成正比。换言之，Kronecker系数计数量子验证器QMA（量子版NP）接受证据张成的向量空间的维数。这意味着在给定相对误差内近似Kronecker系数，不复杂于某类自然量子近似计数问题，后者捕捉了估算量子多体系统热力学性质的复杂度。第二个推论是：判定Kronecker系数的正性包含于QMA中，这补充了Ikenmeyer、Mulmuley和Walter最近的NP困难性结果。对于对称群特征表行求和的近似问题，我们获得了类似结果。最后，我们讨论了一个高效量子算法，能以多项式倒数加法误差近似归一化Kronecker系数。