When is a Quantum Cellular Automaton (QCA) a Quantum Lattice Gas Automaton (QLGA)?

TL;DR

This paper establishes clear criteria to distinguish quantum cellular automata that are equivalent to quantum lattice gas automata, clarifying the relationship between these models in quantum computation and simulation.

Contribution

It provides necessary and sufficient conditions for finite, unbounded QCA to be classified as QLGA, using advanced mathematical tools from functional analysis and algebra.

Findings

Identified a local classification condition for QLGA within QCA

Proved that some QCA are not QLGA

Developed a mathematical framework for analyzing QCA and QLGA

Abstract

Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently partitioned quantum cellular automata) represent an interesting subclass of QCA. QLGA have been more deeply analyzed than QCA, whereas general QCA are likely to capture a wider range of quantum behavior. Discriminating between QLGA and QCA is therefore an important question. In spite of much prior work, classifying which QCA are QLGA has remained an open problem. In the present paper we establish necessary and sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA) (finitely many active cells in a quiescent background) to be Quantum Lattice Gas Automata (QLGA). We define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA. We use a number of tools from functional analysis of separable Hilbert spaces and representation theory of associative algebras that enable us to treat QCA on finite but unbounded configurations in full detail.