Topological phases of unitary dynamics: Classification in Clifford category

TL;DR

This paper classifies translation-invariant Clifford quantum cellular automata in various dimensions and prime dimensions, revealing their structure through algebraic Witt groups and topological methods.

Contribution

It provides a complete classification of Clifford QCA groups in all dimensions and prime fields, connecting quantum automata with algebraic topology and Witt groups.

Findings

Nontrivial groups only in specific dimensions related to parity and prime mod 4.

Groups are isomorphic to classical Witt groups of quadratic forms over finite fields.

Classification achieved via a dimensional descent using algebraic surgery theory.

Abstract

A quantum cellular automaton (QCA) or a causal unitary is by definition an automorphism of local operator algebra, by which local operators are mapped to local operators. Quantum circuits of small depth, local Hamiltonian evolutions for short time, and translations (shifts) are examples. A Clifford QCA is one that maps any Pauli operator to a finite tensor product of Pauli operators. Here, we obtain a complete table of groups $\mathfrak C(\mathsf d,p)$ of translation invariant Clifford QCA in any spatial dimension $\mathsf d \ge 0$ modulo Clifford quantum circuits and shifts over prime $p$-dimensional qudits, where the circuits and shifts are allowed to obey only coarser translation invariance. The group $\mathfrak C(\mathsf d,p)$ is nonzero only for $\mathsf d = 2k+3$ if $p=2$ and $\mathsf d = 4k+3$ if $p$ is odd where~$k \ge 0$ is any integer, in which case $\mathfrak C(\mathsf d,p) \cong \widetilde{\mathfrak W}(\mathbb F_p)$, the classical Witt group of nonsingular quadratic forms over the finite field $\mathbb F_p$. It is well known that $\widetilde{\mathfrak W}(\mathbb F_2) \cong \mathbb Z/2\mathbb Z$, $\widetilde{\mathfrak W}(\mathbb F_p) \cong \mathbb Z/4\mathbb Z$ if $p = 3 \bmod 4$, and $\widetilde{\mathfrak W}(\mathbb F_p)\cong \mathbb Z/2\mathbb Z \oplus \mathbb Z/2\mathbb Z$ if $p = 1 \bmod 4$. The classification is achieved by a dimensional descent, which is a reduction of Laurent extension theorems for algebraic $L$-groups of surgery theory in topology.