Approximate QCAs in one dimension using approximate algebras

TL;DR

This paper demonstrates that in one dimension, approximate quantum cellular automata (QCAs) can be closely approximated by exact QCAs, ensuring their classification remains consistent with the exact case, even for finite systems.

Contribution

The authors develop a local approach to approximate QCAs on finite systems, showing they can be rounded to exact QCAs using a new intersection construction of subalgebras.

Findings

Approximate QCAs on finite circles can be rounded to exact QCAs.

The classification of QCAs remains unchanged for approximate cases in one dimension.

A new method for constructing exact subalgebras from approximate commuting projections.

Abstract

Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate $C^*$-algebras.