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系数正比于一个可在量子计算机上高效测量的投影算子的秩。换言之，Kronecker系数计数了由QMA验证器的接受证据张成的向量空间的维数，其中QMA是NP的量子类比。这意味着在给定相对误差内逼近Kronecker系数的难度不高于某类自然的量子近似计数问题，这类问题刻画了估计量子多体系统热力学性质的复杂性。第二个推论是，判定Kronecker系数的正性属于QMA，这补充了Ikenmeyer、Mulmuley和Walter最近关于NP困难性的结果。我们针对对称群特征表行和近似这一相关问题得到了类似结果。最后，我们讨论了一种高效的量子算法，能以逆多项式加法误差逼近归一化Kronecker系数。