论文标题
执行量子Schur变换的多编码方法
A Multigraph Approach for Performing the Quantum Schur Transform
论文作者
论文摘要
我们从Okounkov-Vershik的方法中汲取灵感到对称群体的表示理论，以开发一种新的方法来理解如何使用Schur-Weyl双重性来执行量子Schur变换。量子Schur变换是$（\ Mathbb {C}^d）^{\ otimes n} $与$（\ MathBb {C}^d）^{\ otimes n} $之间的基础变换的单位变化。我们描述了一个新的多编码，我们称为Schur-Weyl-Young图，该图代表了同一图中标准的Weyl Tableaux和标准的年轻Tableaux。我们建议对Louck的公式进行重大改进，以计算两个标准的Weyl Tableaux之间的过渡幅度，仅通过查看两个tableaux的条目，以$ d = 2 $出现在Schur-weyl-Young图中的相邻水平。支撑我们结果的主要理论组成部分是发现Schur-Weyl状态的分支规则，我们称之为Schur-Weyl分支规则。该分支规则使我们能够为任何$ n $和$ d $以直接的方式执行上述基础转换的更改。
We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case $d=2$, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any $n$ and $d$.
