Color code off-the-hook: avoiding hook errors with a single auxiliary per plaquette

TL;DR

This paper introduces a minimal-depth, single-auxiliary qubit syndrome extraction circuit for the color code that avoids hook errors, maintaining full circuit-level distance and improving error correction performance.

Contribution

It presents a novel single-auxiliary, color-dependent gate schedule that eliminates all bulk hook errors and enhances temporal distance, applicable to XYZ color codes.

Findings

The proposed circuit avoids all bulk hook errors, preserving circuit-level distance.

It achieves minimal depth with stabilizer measurements in six parallel steps.

Monte Carlo simulations show it outperforms previous methods under various noise models.

Abstract

Syndrome extraction in the planar color code is complicated by high weight stabilizers and hook errors that can reduce the circuit-level distance. With a single auxiliary qubit per plaquette, any spatially uniform circuit halves the circuit-level distance. We propose a single-auxiliary syndrome extraction circuit with color-dependent gate schedules that avoids all malign hook errors in the bulk, thereby preserving the full circuit-level distance. The circuit has minimal depth: all stabilizers of the same Pauli type are measured in parallel in six time steps. Furthermore, this schedule can be readily applied to the XYZ color code circuit, yielding an improved temporal distance. We find that at the boundary, no single hook error alone reduces the distance; instead, only certain combinations of hook errors do, which we call fractional hook errors. We demonstrate through Monte Carlo simulations over a range of circuit-level noise models and physical error rates that our circuit outperforms the previous state of the art.