Reversible quantum cellular automata

TL;DR

This paper characterizes quantum cellular automata as reversible, local, and finite-propagation quantum lattice systems, providing explicit rules, structural theorems, and various construction methods, including connections to quantum walks.

Contribution

It offers a comprehensive structural theorem for quantum cellular automata, showing their reversibility and providing explicit construction methods, extending classical automata concepts to the quantum domain.

Findings

Quantum cellular automata are structurally reversible.

All local rules generate global automata with periodic boundary conditions.

Quantum walks can be viewed as special cases of cellular automata.

Abstract

We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.