Quantum Codes on Graphs

TL;DR

This paper explores quantum codes constructed from various graphs, including Floquet and toric codes, analyzing their properties, limitations on code distance growth, and the effects of vacancies and decoding mechanics.

Contribution

It generalizes Floquet codes to non-planar graphs, investigates the growth limits of code distance on 2-complexes, and studies the impact of vacancies and decoding in planar codes.

Findings

Code distance on 2-complexes cannot grow faster than square-root.

Floquet codes can be constructed using emergent fermions on non-planar graphs.

Vacancies affect decoding performance, with implications for error correction.

Abstract

We consider some questions related to codes constructed using various graphs, in particular focusing on graphs which are not lattices in two or three dimensions. We begin by considering Floquet codes which can be constructed using ``emergent fermions". Here, we are considering codes that in some sense generalize the honeycomb code[1] to more general, non-planar graphs. We then consider a class of these codes that is related to (generalized) toric codes on $2$-complexes. For (generalized) toric codes on $2$-complexes, the following question arises: can the distance of these codes grow faster than square-root? We answer the question negatively, and remark on recent systolic inequalities[2]. We then turn to the case that of planar codes with vacancies, or ``dead qubits", and consider the statistical mechanics of decoding in this setting. Although we do not prove a threshold, our results should be asymptotically correct for low error probability and high degree decoding graphs (high degree taken before low error probability). In an appendix, we discuss a toy model of vacancies in planar quantum codes, giving a phenomenological discussion of how errors occur when ``super-stabilizers" are not measured, and in a separate appendix we discuss a relation between Floquet codes and chain maps.