Homotopy Classification of Loops of Clifford Unitaries

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.