Homotopy Classification of loops of Clifford unitaries

• URL: http://arxiv.org/abs/2306.09903v3
• Date: Wed, 29 Nov 2023 05:38:45 GMT
• Title: Homotopy Classification of loops of Clifford unitaries
• Authors: Roman Geiko and Yichen Hu
• Abstract summary: We study Clifford quantum circuits acting on $mathsfd$-dimensional lattices of prime $p$-dimensional qudits. We calculate homotopy classes of such loops for any odd $p$ and $mathsfd=0,1,2,3$, and $4$. We observe that the homotopy classes of loops of Clifford circuits in $(mathsfd+1)$-dimensions coincide with the quotient of the group of Clifford Quantum Cellular Automata modulo shallow circuits and lattice translations in $mathsfd$-dimensions
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathsf{d}$-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 $\mathsf{d}=0,1,2,3$, 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 $(\mathsf{d}+1)$-dimensions coincide with the quotient of the group of Clifford Quantum Cellular Automata modulo shallow circuits and lattice translations in $\mathsf{d}$-dimensions.

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