论文标题
两分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
论文作者
论文摘要
我们调查了由$ CP $，$ CNOT $和$ swap^α$（swap）量子量子操作产生的组产生的组。发现了标准组的同构，并使用表示理论中的技术，我们能够确定每个组的作用下$ n- $ Qubit Hilbert空间的不变子空间。对于$ cp $操作，我们发现$ n（n-1）/2 $循环订单$ 2 $的直接产品的同构，并确定$ 2^n $ 1 $ 1 $维数不变子空间，与计算状态媒介相对应。对于$ cnot $操作，我们发现在$ 2 $元素（$ gl（n，2）$的字段上，$ n $维空间的普通线性组的同构，并确定两个$ 1 $二维不变的子空间和一个$（2^n-2）$ - 尺寸 - 尺寸的不变型子空间。对于$ swap^α$操作，我们确定了一个不变子空间的复杂结构，具有不同的维度和发生的情况，并提出了构造它们的递归程序。作为我们工作应用程序的一个示例，我们建议这些不变子空间可用于构建简单的正式验证程序，以评估任意大小的量子计算机的操作。
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.
