Topological phases of unitary dynamics: Classification in Clifford category

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 $\mathfrak{C}(\mathsf{d},p)\cong \tilde{\mathfrak{W}}({\mathbb{F}}_p)$, the classical Witt group of nonsingular quadratic forms over the finite field ${\mathbb{F}}_p$. It is well known that $\tilde{\mathfrak{W}}({\mathbb{F}}_2)\cong \mathbb{Z}/2\mathbb{Z}$, $\tilde{\mathfrak{W}}({\mathbb{F}}_p)\cong \mathbb{Z}/4\mathbb{Z}$ if $p=3\mathrm{mod}4$, and $\tilde{\mathfrak{W}}({\mathbb{F}}_p)\cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ if $p=1\mathrm{mod}4$. 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.

arXiv comment48 pages (v2) minor revision in Sec.5.7 (v3) typo correction

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Cited by in corpus (4)

• A QCA for every SPT
• Categorifying Clifford QCA
• Short remarks on shallow unitary circuits
• Classification of qubit cellular automata on hypercubic lattices