Fermionic quantum cellular automata and generalized matrix-product unitaries

In this paper, we study matrix-product unitary operators (MPUs) for fermionic one-dimensional chains. In stark contrast to the case of 1D qudit systems, we show that (i) fermionic MPUs (fMPUs) do not necessarily feature a strict causal cone and (ii) not all fermionic quantum cellular automata (QCA) can be represented as fMPUs. 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 (fQCA) and vice versa. Finally, we prove an index theorem for generalized MPUs, recovering the recently derived classification of fQCA 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.

Automated summary

In this paper, the study focuses on matrix-product unitary operators (MPUs) for fermionic one-dimensional chains. A natural generalization of fermionic MPUs is introduced, which preserves locality and is equivalent to fermionic quantum cellular automata (fQCA). An index theorem for generalized MPUs is proven, recovering the classification of fQCA in one dimension.