Bacon-Shor Board Games

TL;DR

This work demonstrates that the Bacon-Shor code can achieve a fault-tolerance threshold by switching to a four-round, constant-weight detector measurement schedule derived from a board-game coloring construction, avoiding concatenation. The schedule yields a steady sequence of instantaneous stabilizer groups and dynamical detectors whose weights remain $O(1)$ per round, enabling threshold behavior under uniform circuit-level noise when decoded with minimum-weight perfect matching. A generalization theorem shows these properties extend to larger lattices by stacking, with detectors remaining bounded in weight (e.g., at most 20 in constructed examples). Numerically, the modified schedule achieves exponential suppression of logical errors with code distance and suggests a threshold near $p_{th}\approx 3\times 10^{-3}$, marking a significant improvement over prior Bacon-Shor implementations and highlighting the crucial role of measurement order in subsystem codes.

Abstract

We identify a period-4 measurement schedule for the checks of the Bacon-Shor code that fully covers spacetime with constant-weight detectors, and is numerically observed to provide the code with a threshold. Unlike previous approaches, our method does not rely on code concatenation and instead arises as the solution to a coloring game on a square grid. Under a uniform circuit-level noise model, we observe a threshold of approximately $0.3\%$ when decoding with minimum weight perfect matching, and we conjecture that this could be improved using a more tailored decoder.

Figures (12)

• Figure 1: Virtual qubit operators for the Bacon-Shor code. Red denotes $X$, while blue denotes $Z$. The $\overline{X}$ and the $\overline{Z}$ operators for gauge qubits are respectively the products of $XX$ checks above, and the product of $ZZ$ checks on the right of the associated square plaquette, or box. The vertical and horizontal stabilizers are associated with the $\overline{Z}$ operators of their respective stabilizer qubits, with the corresponding $\overline{X}$ operators represented respectively as a horizontal string of physical $Z$ and vertical string of physical $X$ operators. Finally, the $\overline{X}$ and $\overline{Z}$ operators of the single logical qubit is shown respectively as a string of physical $X$ and $Z$ operators.
• Figure 2: (Top) Period $4$ measurement schedule on a $5 \times 5$ lattice, where horizontal $XX$ checks are drawn in red, and vertical $ZZ$ checks in blue. (Bottom) Induced ISG sequence after one complete measurement cycle. This periodic sequence of colorings solves the game described in Sect. \ref{['subsecn:board-game']}. The details of the moves involved are described in Appendix \ref{['appdx:game']}.
• Figure 3: An example detector provided by the measurement schedule of Fig. \ref{['fig:meas-ISGs']}. The detector is given by (top) the product $\overline{X}_{A}^{(0)} \cdot \left( \overline{X}_{A}^{(2)} \cdot \overline{X}_{B}^{(2)} \right) \cdot \overline{X}_{B}^{(4)}$, or equivalently (bottom) the product of 8 different nearest-neighbor $XX$ checks over 8 qubits, measured across 3 different rounds, yielding a detector with weight $16$.
• Figure 4: A $9 \times 9$ example generalization of Fig. \ref{['fig:meas-ISGs']} providing the (top) measurement schedule and (bottom) steady state ISG sequence that results in detectors with weight $\leq 10$ (see Appendix \ref{['appdx:detector-examples']}). White dotted lines are drawn around each coloring in the sub-solution for the $5 \times 5$ lattice, which are stacked together to form the solution to the larger $9 \times 9$ lattice.
• Figure 5: Comparison of the logical error rate (per round) for three implementations of the Bacon-Shor code, our modified version (blue), the standard (unmodified) code (orange), and the fractal-pitch variants from gidney2023baconthreshold (pitches 5, 7, 9 with surgery hold factor = 1). The grid diameter d ranges from 3 to 35, and each data point indicates the probability of a logical Z error occurring in the encoded qubit (initialized in the X basis) per round. The uniform circuit-level noise strength is $p=0.001$, and each simulation runs for $4d$ rounds. Shaded regions represent statistical confidence intervals, based on up to $10^{8}$ Monte Carlo trials or 1000 logical errors, whichever occurs first.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Theorem 3
• proof : Proof of Theorem \ref{['thm:coloring-ISG']}.
• proof : Proof of Theorem \ref{['thm:generalization']}.