# Nontrivial Quantum Cellular Automata in Higher Dimensions

## TL;DR

This paper constructs a 3D quantum cellular automaton (QCA) that disentangles a complex topological ground state and explores its properties, providing evidence that certain QCAs are inherently nontrivial and cannot be realized by constant-depth quantum circuits.

## Contribution

It introduces a novel 3D QCA related to the Walker-Wang model, and demonstrates its nontriviality and implications for topological order and quantum circuit complexity.

## Key findings

- The 3D QCA maps local Pauli operators to local Pauli operators.
- The QCA is not a constant-depth Clifford circuit.
- The square of the QCA can be realized by a constant-depth quantum circuit.

## Abstract

We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $\mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.