Four-qubit states generated by Clifford gates

IF 0.7 4区物理与天体物理 Q3 COMPUTER SCIENCE, THEORY & METHODS

International Journal of Quantum Information Pub Date : 2022-02-26 DOI:10.1142/s0219749922500022

F. Latour, O. Perdomo

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引用次数: 1

Abstract

The Clifford group is the set of gates generated by controlled not (CNOT) gates and the two local gates [Formula: see text] and [Formula: see text]. We will say that an [Formula: see text]-qubit state is a Clifford state if it can be prepared using Clifford gates, this is, [Formula: see text] is Clifford if [Formula: see text] where [Formula: see text] is a Clifford gate. In this paper, we study the set of all 4-qubit Clifford states. We prove that there are 293760 states, each of which has entanglement entropy equal to 0, [Formula: see text], 1, [Formula: see text], or [Formula: see text]. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We also study the Clifford states with real entries under the action of the subgroup [Formula: see text] of Clifford gates with real entries. This time we show that every pair of Clifford states with real entries can be connected with at most 5 CNOT gates and local gates in [Formula: see text]. To understand the action of the 12 different CNOT gates, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We label each orbit in such a way that it is easy to see the effect of the CNOT gates. Diagrams and tables explaining the action of the CNOT gates on all the orbits are presented in the paper. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.

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Clifford gates生成的四量子位态

Clifford群是由可控非(CNOT)门和两个局部门[公式:见文]和[公式:见文]生成的门的集合。我们说，如果一个[公式:见文]-量子比特状态可以用Clifford门来准备，那么它就是Clifford态，也就是说，如果[公式:见文]是Clifford门，那么[公式:见文]就是Clifford态，其中[公式:见文]是Clifford门。本文研究了所有4量子位Clifford态的集合。我们证明有293760个状态，每个状态的纠缠熵等于0，[公式:见文]，1，[公式:见文]，或[公式:见文]。我们还证明了任何一对这些状态都可以使用本地门和最多3个CNOT门连接。我们还研究了实数Clifford门在实数Clifford门子群作用下的实数Clifford状态[公式:见文]。这一次我们证明了在[公式:见文]中，每一对有实数条目的Clifford状态可以与最多5个CNOT门和局部门相连。为了理解12个不同的CNOT门的作用，我们使用等价关系将Clifford状态划分为轨道:如果两个状态相差一个局部Clifford门，则它们是等价的。我们以这样一种方式标记每个轨道，这样很容易看到CNOT门的影响。文中给出了CNOT门在所有轨道上的作用的图表和表格。链接https://youtu.be/42MI6ks2_eU指向一个YouTube视频，该视频解释了本文中最重要的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Quantum Information 物理-计算机：理论方法

CiteScore

2.20

自引率

8.30%

发文量

审稿时长

10 months

期刊介绍： The International Journal of Quantum Information (IJQI) provides a forum for the interdisciplinary field of Quantum Information Science. In particular, we welcome contributions in these areas of experimental and theoretical research: Quantum Cryptography Quantum Computation Quantum Communication Fundamentals of Quantum Mechanics Authors are welcome to submit quality research and review papers as well as short correspondences in both theoretical and experimental areas. Submitted articles will be refereed prior to acceptance for publication in the Journal.