Bacon-Shor Code

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Description

The Bacon-Shor code is a family of subsystem CSS quantum error-correcting codes defined on an $m_1\times m_2$ grid of physical qubits, with parameters $[[m_1{m}_2,1,\min (m_1,m_2)]]$. In the symmetric square case this reduces to $[[m^2,1,m]]$. Its defining practical feature is that syndrome extraction is built from weight-2 gauge measurements rather than direct high-weight stabilizer measurements.

A convenient convention is to use X-type gauge operators on vertical nearest-neighbor pairs and Z-type gauge operators on horizontal nearest-neighbor pairs. The high-weight stabilizers are products of those weight-2 gauge operators across adjacent rows or adjacent columns. Because the gauge subsystem does not store logical information, multiple gauge configurations correspond to the same encoded logical state.

Operationally, Bacon-Shor is attractive when high-weight parity checks are difficult but repeated low-weight checks are natural, or when the physical noise is strongly biased so rectangular Bacon-Shor blocks can preferentially protect the dominant error channel.

Hamiltonian and Gauge Structure

Let qubits be indexed by row $r\in \{1,\dots ,m_1\}$ and column $c\in \{1,\dots ,m_2\}$. A standard gauge-generator choice is

$g_{r,c}^X=X_{r,c}{X}_{r+1,c},g_{r,c}^Z=Z_{r,c}{Z}_{r,c+1},$

for $r=1,\dots ,m_1-1$ and $c=1,\dots ,m_2-1$ where defined. The commuting stabilizers are products of these gauge generators:

$S_r^X={\prod }_{c=1}^{m_2}{X}_{r,c}{X}_{r+1,c},S_c^Z={\prod }_{r=1}^{m_1}{Z}_{r,c}{Z}_{r,c+1}.$

A representative commuting code Hamiltonian is therefore

$H_{\mathrm{code}}=-{\sum }_{r=1}^{m_1-1}{S}_r^X-{\sum }_{c=1}^{m_2-1}{S}_c^Z,$

whose code space is the simultaneous $+1$ eigenspace of all stabilizers. Logical operators can be chosen as a full X string along any row and a full Z string along any column,

$\bar{X}={\prod }_{c=1}^{m_2}{X}_{r,c},\bar{Z}={\prod }_{r=1}^{m_1}{Z}_{r,c},$

with the choice of row or column equivalent up to multiplication by gauge operators. At the operator level one can also write a compass-model-like gauge Hamiltonian $H_{\mathrm{gauge}}=-J_X\sum g^X-J_Z\sum g^Z$, but that noncommuting weight-2 model is distinct from the commuting stabilizer Hamiltonian defining the code space.

Motivation

• Low-weight check extraction: Only weight-2 nearest-neighbor gauge measurements are required, avoiding direct measurement of high-weight stabilizers.
• Bias-friendly geometry: Rectangular $m_1\times m_2$ variants can protect X and Z errors asymmetrically, matching hardware with strongly biased noise.
• Subsystem flexibility: Gauge degrees of freedom simplify measurement design and make the code a useful bridge between Shor-style constructions and modern subsystem/floquet ideas.
• Historical importance: Bacon-Shor remains one of the cleanest pedagogical examples of subsystem fault tolerance and gauge-based QEC.

Experimental Status

Fault-tolerant trapped-ion demonstration, Egan et al. (2021):

• Implemented a $[[9,1,3]]$ Bacon-Shor logical qubit on a 13-ion trapped-ion processor.
• Demonstrated fault-tolerant preparation, measurement, rotation, and stabilizer measurement using weight-2 gauge checks.
• Reported an average state-preparation-and-measurement error of $0.6\%$ and an average logical Clifford-gate error of $0.3\%$ after error correction.
• Demonstrated magic-state preparation above the distillation threshold, but did not yet realize a long-lived repeatedly stabilized logical memory.

Recent Bacon-Shor-family developments (2024-2025):

• Alam and Rieffel showed that a period-4 measurement schedule can turn Bacon-Shor into a Floquet code hosting additional dynamical logical qubits while keeping low-weight checks.
• Sun et al. experimentally realized a 3×3 Floquet-Bacon-Shor code on a superconducting processor, including repeated error detection, logical gates on the dynamical qubit, and an error-detected logical Bell state with $75.9\%$ fidelity.

Key Metrics

Metric	Value	Notes	Fidelity reference
Code parameters	$[[m_1{m}_2,1,\min (m_1,m_2)]]$	Square case is $[[m^2,1,m]]$	Bacon 2006
Gauge measurement weight	2	Vertical XX and horizontal ZZ nearest-neighbor checks	Bacon 2006
Stabilizer weight	$2m_2$ for X, $2m_1$ for Z	Inferred from products of gauge outcomes rather than measured directly	Bacon 2006
Demonstrated FT SPAM error	$0.6\%$	Average state-preparation and measurement error after error correction in the 13-ion $[[9,1,3]]$ demo	Egan et al. 2021
Demonstrated FT logical Clifford error	$0.3\%$	Average logical Clifford-gate error after error correction in the same demo	Egan et al. 2021
Floquet-Bacon-Shor Bell-state fidelity	$75.9\%$	Error-detected Bell state between static and dynamical logical qubits on a 3×3 superconducting lattice	Sun et al. 2025

Note: For QEC code entries, metric rows should stay source-backed and should distinguish static Bacon-Shor results from Floquet-Bacon-Shor extensions.

Scaling Considerations

• Hardware simplicity: The main win is the low-weight gauge-check schedule, not a best-in-class unbiased-noise threshold.
• Asymmetric protection: Rectangular blocks naturally trade X-versus-Z protection, making Bacon-Shor especially relevant when one error channel dominates.
• Fault-tolerance caveat: Because stabilizer values are reconstructed from products of noisy gauge measurements, measurement faults need careful circuit design and decoding.
• Threshold landscape: The basic 2D Bacon-Shor family is not the leading choice under generic depolarizing noise, but concatenated, biased-noise, and Floquet variants recover useful operating regimes.

References

Original proposal

• D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories,” Phys. Rev. A 73, 012340 (2006) — arXiv:quant-ph/0506023

Fault-tolerance theory

• P. Aliferis and A. W. Cross, “Subsystem fault tolerance with the Bacon-Shor code,” Phys. Rev. Lett. 98, 220502 (2007) — arXiv:quant-ph/0610063

Experimental demonstrations

• L. Egan et al., “Fault-tolerant control of an error-corrected qubit,” Nature 598, 281 (2021) — arXiv:2009.11482
• M. S. Alam and E. Rieffel, “Dynamical logical qubits in the Bacon-Shor code,” Phys. Rev. A 112, 022436 (2025) — arXiv:2403.03291
• X. Sun et al., “Logical Operations with a Dynamical Qubit in Floquet-Bacon-Shor Code,” Phys. Rev. Lett. 135, 220601 (2025) — arXiv:2503.03867

Linked Papers

• bacon-2006-bacon-shor
• aliferis-2007-subsystem-fault-tolerance
• egan-2021-fault-tolerant-control
• alam-2025-dynamical-logical-qubits
• sun-2025-logical-operations-floquet-bacon-shor

Evergreen context

• threshold-theorem — the big-picture reason Bacon-Shor matters is not that it beats the surface code everywhere, but that it reshuffles the hardware tradeoff around what kinds of checks you can execute reliably.
• noise-bias-and-asymmetric-error-channels — Bacon-Shor becomes much more compelling when the physical noise is strongly asymmetric and rectangular layouts can lean into that bias.
• decoherence-free-subspace — useful contrast for the word “subsystem”: Bacon-Shor is an active subsystem code, not passive symmetry protection.

Related Entries

• surface-code-logical-qubit — higher threshold under generic depolarizing noise, but a more demanding stabilizer-measurement stack
• color-code-logical-qubit — alternative logical-code family with different transversal-gate tradeoffs
• floquet-codes — dynamical-code viewpoint that extends Bacon-Shor with additional logical qubits
• trapped-ion-qubit — platform of the first fault-tolerant Bacon-Shor demonstration
• kerr-cat-qubit — a biased-noise hardware modality where asymmetric Bacon-Shor ideas are especially natural