{"title":"Homotopy Classification of Loops of Clifford Unitaries","authors":"Roman Geiko, Yichen Hu","doi":"10.1007/s00220-024-05066-8","DOIUrl":null,"url":null,"abstract":"<p>Clifford quantum circuits are elementary invertible transformations of quantum systems that map Pauli operators to Pauli operators. We study periodic one-parameter families of Clifford circuits, called loops of Clifford circuits, acting on <span>\\({\\textsf{d}}\\)</span>-dimensional lattices of prime <i>p</i>-dimensional qudits. We propose to use the notion of algebraic homotopy to identify topologically equivalent loops. We calculate homotopy classes of such loops for any odd <i>p</i> and <span>\\({\\textsf{d}}=0,1,2,3\\)</span>, and 4. Our main tool is the Hermitian K-theory, particularly a generalization of the Maslov index from symplectic geometry. We observe that the homotopy classes of loops of Clifford circuits in <span>\\(({\\textsf{d}}+1)\\)</span>-dimensions coincide with the quotient of the group of Clifford Quantum Cellular Automata modulo shallow circuits and lattice translations in <span>\\({\\textsf{d}}\\)</span>-dimensions.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-05066-8","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Clifford quantum circuits are elementary invertible transformations of quantum systems that map Pauli operators to Pauli operators. We study periodic one-parameter families of Clifford circuits, called loops of Clifford circuits, acting on \({\textsf{d}}\)-dimensional lattices of prime p-dimensional qudits. We propose to use the notion of algebraic homotopy to identify topologically equivalent loops. We calculate homotopy classes of such loops for any odd p and \({\textsf{d}}=0,1,2,3\), and 4. Our main tool is the Hermitian K-theory, particularly a generalization of the Maslov index from symplectic geometry. We observe that the homotopy classes of loops of Clifford circuits in \(({\textsf{d}}+1)\)-dimensions coincide with the quotient of the group of Clifford Quantum Cellular Automata modulo shallow circuits and lattice translations in \({\textsf{d}}\)-dimensions.
克利福德量子回路是量子系统的基本可逆变换,它将泡利算子映射为泡利算子。我们研究作用于质点 p 维网格的克利福德电路周期性单参数族,称为克利福德电路环。我们建议使用代数同调的概念来识别拓扑上等价的回路。我们计算了在任意奇数 p 和 \({\textsf{d}}=0,1,2,3/),以及 4 的情况下这些环的同调类。我们的主要工具是赫米蒂 K 理论,特别是来自交映几何的马斯洛夫指数的广义化。我们观察到在(({\textsf{d}}+1)\)-维度中克利福德回路的同调类与(({\textsf{d}}+1)\)-维度中克利福德量子蜂窝自动机调制浅回路和晶格平移的商重合。
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.