Fermionic quantum cellular automata and generalized matrix product unitaries

TL;DR

This paper explores fermionic matrix product unitaries and quantum cellular automata in 1D, revealing their differences, introducing a generalized class, and establishing a classification theorem.

Contribution

It introduces a generalized class of fermionic MPUs, characterizes their locality-preserving properties, and proves an index theorem linking them to fermionic QCA classifications.

Findings

Fermionic MPUs lack a strict causal cone unlike qudit systems.

Not all fermionic QCA can be represented as fermionic MPUs.

A classification index theorem for generalized MPUs is established.

Abstract

We study matrix product unitary operators (MPUs) for fermionic one-dimensional (1D) chains. In stark contrast with the case of 1D qudit systems, we show that (i) fermionic MPUs do not necessarily feature a strict causal cone and (ii) not all fermionic Quantum Cellular Automata (QCA) can be represented as fermionic MPUs. We then introduce a natural generalization of the latter, obtained by allowing for an additional operator acting on their auxiliary space. We characterize a family of such generalized MPUs that are locality-preserving, and show that, up to appending inert ancillary fermionic degrees of freedom, any representative of this family is a fermionic QCA and viceversa. Finally, we prove an index theorem for generalized MPUs, recovering the recently derived classification of fermionic QCA in one dimension. As a technical tool for our analysis, we also introduce a graded canonical form for fermionic matrix product states, proving its uniqueness up to similarity transformations.