Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

翻译：某些表示论中的重数，例如Kostka系数和Littlewood-Richardson系数，具有组合解释，这使其计算属于复杂性类#P。这一性质是否具有普遍性被认为是数学和计算机科学中的一个重要开放问题，与几何复杂性理论和量子信息领域密切相关。近期研究探讨了特定重数的量子计算复杂性，例如Kronecker系数及plethysm系数的某些特例。本文证明一大类表示论重数属于#BQP类。特别地，我们的结果意味着plethysm系数属于#BQP类——此前仅在某些特例中已知此结论。该结果还将以往研究中所有关于系数量子复杂性的已知结论作为特例包含其中，从而统一、简化并扩展了先前工作。我们通过多次应用Schur变换获得此结果。近期研究改进了其对局部维度的依赖关系，这对我们的工作至关重要。我们进一步描述了一种证明表示论重数属于#BQP类的通用框架，该框架涵盖我们的方法以及先前工作的研究路径。作为补充，我们还证明这些重数同样自然地属于GapP类，并在特定参数固定时获得多项式时间经典算法。