Dynamical Logical Qubits in the Bacon-Shor Code

TL;DR

This paper introduces a Floquet version of the Bacon-Shor code that hosts multiple dynamical logical qubits with scalable code distance, self-correcting properties, and potential for new dynamical quantum error correction strategies.

Contribution

It demonstrates a measurement schedule for the Bacon-Shor code that creates multiple dynamical logical qubits with scalable distance and self-correcting features, expanding Floquet code design.

Findings

Code distance scales as Θ(d/√k) with k dynamical logical qubits.

The measurement schedule preserves logical information over time.

Several errors are self-corrected by the measurement schedule.

Abstract

The Bacon-Shor code is a quantum error correcting subsystem code composed of weight 2 check operators that admits a single logical qubit, and has distance $d$ on a $d \times d$ square lattice. We show that when viewed as a Floquet code, by choosing an appropriate measurement schedule of the check operators, it can additionally host several dynamical logical qubits. Specifically, we identify a period 4 measurement schedule of the check operators that preserves logical information between the instantaneous stabilizer groups. Such a schedule measures not only the usual stabilizers of the Bacon-Shor code, but also additional stabilizers that protect the dynamical logical qubits against errors. We show that the code distance of these Floquet-Bacon-Shor codes scales as $\Theta(d/\sqrt{k})$ on an $n = d \times d$ lattice with $k$ dynamical logical qubits, along with the logical qubit of the parent subsystem code. Unlike the usual Bacon-Shor code, the Floquet-Bacon-Shor code family introduced here can therefore saturate the subsystem bound $kd = O(n)$. Moreover, several errors are shown to be self-corrected purely by the measurement schedule itself. This work provides insights into the design space for dynamical codes and expands the known approaches for constructing Floquet codes.