plethysm 属于 #BQP
Plethysm is in #BQP
AI总结：
该研究证明了plethysm系数属于#BQP，并通过Schur变换等方法统一和扩展了先前关于表示论多重性的量子复杂性结果。
摘要：
一些表示论中的多重性，如Kostka系数和Littlewood-Richardson系数，允许一种组合解释，将它们的计算置于#P复杂性类中。是否更一般地成立，是数学和计算机科学中的重要开放问题，对几何复杂性理论和量子信息有相关性。最近的工作已研究了特定多重性的量子复杂性，如Kronecker系数和某些plethysm系数的特殊情况。在这里，我们证明了一类广泛的表示论多重性属于#BQP。特别是，我们的结果表明plethysm系数属于#BQP，这之前只在特殊情况下已知。它还表明所有已知的先前研究系数的量子复杂性结果都是特殊情况，统一、简化并扩展了先前的工作。我们通过多次应用Schur变换获得结果。最近的工作改进了其对局部维度的依赖性，这对我们的工作至关重要。我们进一步描述了一种一般方法，以证明表示论多重性属于#BQP，该方法涵盖了我们的方法以及先前工作的方法。我们还补充说明，这些相同的多重性也自然属于GapP，并在某些参数固定时获得多项式时间的经典算法。
英文摘要：
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.
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