arXiv 2009.036422020-09-09quant-ph

两量子比特量子门的不变子空间及其在量子计算机验证中的应用

Invariant subspaces of two-qubit quantum gates and their application in the verification of quantum computers

Yordan S. Yordanov, Jacob Chevalier-Drori, Thierry Ferrus, Matthew Applegate, Crispin H. W. Barnes

摘要：

我们研究了$CP$、$CNOT$和$SWAP^α$（幂交换）量子门操作作用于$n$量子比特所产生的群.发现标准群的同构，并使用表示论的技术，我们能够确定$n-$量子比特希尔伯特空间在每个群的作用下的不变子空间。对于$CP$运算，我们找到了与$2$阶的$n(n-1)/2$循环群的直积的同构，并确定了对应于计算状态向量的$2^n$ $1$维不变子空间.对于$CNOT$运算，我们找到了$n$维空间在$2$元素域$GL(n,2)$上的一般线性群的同构，并确定了两个$1$维不变子空间和一个$(2^n-2)$维不变子空间.对于$SWAP^α$操作，我们确定了一个复杂的结构不变的子空间，不同的尺寸和出现，并提出了一个递归的过程来构建它们。作为我们工作的应用的一个例子，我们建议这些不变子空间可以用来构建简单的形式验证程序，以评估任意大小的量子计算机的操作。

英文摘要：

We investigate the groups generated by the sets of $CP$, $CNOT$ and $SWAP^α$ (power-of-SWAP) quantum gate operations acting on $n$ qubits. Isomorphisms to standard groups are found, and using techniques from representation theory, we are able to determine the invariant subspaces of the $n-$qubit Hilbert space under the action of each group. For the $CP$ operation, we find isomorphism to the direct product of $n(n-1)/2$ cyclic groups of order $2$, and determine $2^n$ $1$-dimensional invariant subspaces corresponding to the computational state-vectors. For the $CNOT$ operation, we find isomorphism to the general linear group of an $n$-dimensional space over a field of $2$ elements, $GL(n,2)$, and determine two $1$-dimensional invariant subspaces and one $(2^n-2)$-dimensional invariant subspace. For the $SWAP^α$ operation we determine a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. As an example of an application for our work, we suggest that these invariant subspaces can be used to construct simple formal verification procedures to assess the operation of quantum computers of arbitrary size.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文