A Bravyi-K\"onig theorem for Floquet codes generated by locally conjugate instantaneous stabiliser groups

TL;DR

This paper extends the Bravyi-K"onig theorem to Floquet codes, a new class of quantum error correcting codes, demonstrating that logical operations in these codes are constrained similarly to topological stabiliser codes.

Contribution

It proves a Bravyi-K"onig type theorem for Floquet codes based on locally conjugate stabiliser groups and introduces a class of generalized unitaries with a canonical form.

Findings

BK theorem applies to Floquet codes with locally conjugate stabiliser groups

Generalized unitaries in Floquet codes can be characterized by a canonical form

Logical operations are constrained by the BK theorem in this new setting

Abstract

The Bravyi-K\"onig (BK) theorem is an important no-go theorem for the dynamics of topological stabiliser quantum error correcting codes. It states that any logical operation on a $D$-dimensional topological stabiliser code that can be implemented by a short-depth circuit acts on the codespace as an element of the $D$-th level of the Clifford hierarchy. In recent years, a new type of quantum error correcting codes based on Pauli stabilisers, dubbed Floquet codes, has been introduced. In Floquet codes, syndrome measurements are arranged such that they dynamically generate a codespace at each time step. Here, we show that the BK theorem holds for a definition of Floquet codes based on locally conjugate stabiliser groups. Moreover, we introduce and define a class of generalised unitaries in Floquet codes that need not preserve the codespace at each time step, but that combined with the measurements constitute a valid logical operation. We derive a canonical form of these generalised unitaries and show that the BK theorem holds for them too.