Topological phases of unitary dynamics: Classification in Clifford category

Understand

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.