Unitary equivalence classes of split-step quantum walks
Unitary equivalence classes of split-step quantum walks
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DOI：
10.1007/s11128-021-03323-6
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发表时间：
2021-04
影响因子：
2.5
通讯作者：
Akihiro Narimatsu;Hiromichi Ohno;K. Wada
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文献类型：
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作者：
Akihiro Narimatsu;Hiromichi Ohno;K. Wada
在这项研究中，我们研究的幺正等价的分裂步量子行走（SSQW）。SSQW是由Kitagawa提出并由Suzuki推广的一维晶格上的两态离散时间量子行走。在这项工作中，我们定义了一类新的量子行走，它包括所有的SSQW通过使用秩条件，其中规定（1）表示从顶点xto顶点行走的运算符的秩小于或等于1，以及（2）从顶点xtox的秩（即，其本身）小于或等于2。首先，我们抽象地定义量子漫步在这个类中，然后通过提出这种量子漫步的显式形式来澄清。我们还计算了它们的酉等价类，并证明了每个顶点需要四个真实的参数来参数化酉等价类。进一步，我们考虑了Suzuki的SSQW的酉等价性，并证明了Suzuki的SSQW的所有参数都可以是真实的。最后，我们讨论了单量子阱的手征对称性。
In this study, we investigate the unitary equivalence of split-step quantum walks (SSQWs). Introduced by Kitagawa and generalized by Suzuki, SSQWs are two-state discrete-time quantum walks on a one-dimensional lattice. In this work, we define a new class of quantum walks that includes all SSQWs by making use of the rank conditions, which state that (1) the rank of operators representing the walk from vertexxto vertexis less than or equal to 1 and that (2) the rank from vertexxtox(i.e., to itself) is less than or equal to 2. Initially, we abstractly define quantum walks in this class, then clarify by presenting the explicit form of such quantum walks. We also calculate their unitary equivalence classes and show that four real parameters are required for each vertex to parameterize the unitary equivalence classes. Further, we consider the unitary equivalence of Suzuki’s SSQWs and prove that all parameters of Suzuki’s SSQWs can be real. Finally, we discuss the chiral symmetry of SSQWs.