Topological subsystem bivariate bicycle codes with four-qubit check operators

TL;DR

This paper introduces subsystem bivariate bicycle codes with local weight-4 checks, enabling high-rate quantum memories with low-overhead syndrome extraction and topological properties.

Contribution

It presents a new CSS subsystem construction for BB codes that uses local gauge measurements, enabling efficient syndrome extraction and topological features.

Findings

Low-overhead examples include [[27,6,3]], [[75,10,5]], and [[108,12,6]] codes.

The codes encode six times more logical qubits than subsystem surface codes at similar parameters.

Gauge degrees of freedom enable compatibility of high-rate BB codes with local syndrome extraction.

Abstract

High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight $6$ or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-$4$ gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including $[[27,6,3]]$, $[[75,10,5]]$, and $[[108,12,6]]$; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-$4$ syndrome extraction.