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系数是否计数某些组合对象集是一个长期悬而未决的问题。在本工作中，我们证明给定的Kronecker系数与一个可使用量子计算机有效测量的投影算子的秩成正比。换言之，Kronecker系数计数由QMA验证器的接受证言所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近似Kronecker系数并不难于某一类自然量子近似计数问题，这类问题捕获了估算量子多体系统热力学性质的复杂性。第二个推论是，判定Kronecker系数的正性包含于QMA中，这补充了Ikenmeyer、Mulmuley和Walter近期提出的NP困难性结果。对于对称群特征表行求和的近似相关问题，我们也得到了类似结果。最后，我们讨论了一个有效的量子算法，该算法能以逆多项式加法误差近似归一化的Kronecker系数。