# Clifford Quantum Cellular Automata: Trivial group in 2D and Witt group in 3D

## TL;DR

This paper classifies translation-invariant Clifford quantum cellular automata (QCA) in various dimensions, proving triviality in 2D, establishing that the fourth power is always trivial, and providing explicit nontrivial examples in 3D related to topological order.

## Contribution

It introduces a classification framework for TI Clifford QCA using antihermitian forms, proves triviality in 2D, and constructs explicit 3D examples linked to Witt groups and topological order.

## Key findings

- 2D TI Clifford QCA are always trivial.
- The fourth power of any TI Clifford QCA is trivial.
- Explicit nontrivial 3D TI Clifford QCA examples are provided.

## Abstract

We study locality preserving automorphisms of operator algebras on $D$-dimensional uniform lattices of prime $p$-dimensional qudits (QCA), specializing in those that are translation invariant (TI) and map every prime $p$-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries, and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this case is always trivial. Further, we prove that for any $D$ the fourth power of any TI Clifford QCA is trivial. We present explicit examples of nontrivial TI Clifford QCA for $D=3$ and any odd prime $p$, and show that the Witt group of the finite field $\mathbb F_p$ is a subgroup of the group $\mathfrak C(D = 3, p)$ of all TI Clifford QCA modulo trivial ones. That is, $\mathfrak C(D = 3, p \equiv 1 \mod 4) \supseteq \mathbb Z_2 \times \mathbb Z_2$ and $\mathfrak C(D = 3, p \equiv 3 \mod 4) \supseteq \mathbb Z_4$. The examples are found by disentangling the ground state of a commuting Pauli Hamiltonian which is constructed by coupling layers of prime dimensional toric codes such that an exposed surface has an anomalous topological order that is not realizable by commuting Pauli Hamiltonians strictly in two dimensions. In an appendix independent of the main body of the paper, we revisit a recent theorem of Freedman and Hastings that any two-dimensional QCA, which is not necessarily Clifford or translation invariant, is a constant depth quantum circuit followed by a shift. We give a more direct proof of the theorem without using any ancillas.