Bacon-Shor Code

Figure

Description

The Bacon-Shor code is a family of subsystem quantum error-correcting codes defined on a 2D grid of physical qubits. It encodes one logical qubit in $m\times m$ physical qubits with distance $d=m$, using only two-body gauge measurements (weight-2 XX and ZZ checks on neighboring pairs). The stabilizers are products of gauge operators and have weight $2m$, but syndrome extraction requires only weight-2 measurements — a major hardware simplification.

The Bacon-Shor code exploits the subsystem code framework: the physical Hilbert space decomposes into a logical subsystem, a gauge subsystem, and a stabilizer-fixed subsystem. Only the logical subsystem encodes information; the gauge subsystem is “don’t care” degrees of freedom.

For the $[[m^2,1,m]]$ code on an $m\times m$ grid:

• X stabilizers: Products of $XX$ along each pair of adjacent rows.
• Z stabilizers: Products of $ZZ$ along each pair of adjacent columns.
• Gauge operators: Individual $XX$ (horizontal pairs) and $ZZ$ (vertical pairs).

Syndrome extraction measures only the weight-2 gauge operators, then classically computes the stabilizer syndrome from their products. This avoids the need for ancilla-mediated multi-qubit parity measurements entirely.

Stabilizer Structure

For the $[[9,1,3]]$ Bacon-Shor code on a 3×3 grid:

Gauge generators (weight 2): $X_i{X}_j\text{ for horizontal neighbors }(i,j)$ $Z_i{Z}_j\text{ for vertical neighbors }(i,j)$

Stabilizers (weight 6, but measured via weight-2 gauge operators): $S_X^{(k)}={\prod }_{i\in \text{row }k}{\prod }_{j\in \text{row }k+1}{X}_i{X}_j$ $S_Z^{(k)}={\prod }_{i\in \text{col }k}{\prod }_{j\in \text{col }k+1}{Z}_i{Z}_j$

Logical operators: $\bar{X}={\prod }_{i\in \text{any row}}{X}_i,\bar{Z}={\prod }_{i\in \text{any column}}{Z}_i$

Motivation

• Hardware simplicity: Only weight-2 nearest-neighbor measurements required — no ancilla qubits, no multi-qubit parity checks.
• Noise-bias exploitation: Rectangular $m_1\times m_2$ variants can correct more X than Z errors (or vice versa), naturally matching biased-noise hardware like cat qubits or fluxonium.
• Concatenation-friendly: Effective when concatenated with other codes, especially for noise with strong bias.
• Pedagogical importance: Simplest subsystem code, used as a building block for understanding gauge codes and operator QEC.

Experimental Status

First demonstration on trapped ions — Egan et al. (2021):

• Implemented the $[[9,1,3]]$ Bacon-Shor code on a 13-ion trapped-ion processor (Honeywell/Quantinuum H1)
• Demonstrated fault-tolerant syndrome extraction using only weight-2 measurements
• Achieved logical error rate suppression below physical error rate

Key Metrics

Metric	Value	Notes	Fidelity reference
Physical qubits	$m^2$	For distance $d=m$	Bacon 2006
Measurement weight	2	Only nearest-neighbor two-body measurements	Bacon 2006
Encoding rate	$1/m^2$	Same as surface code	—
Threshold	~0.1% (concatenated)	Lower than surface code, but simpler measurements	—
Distance	$m$	Linear in grid dimension	—

Scaling Considerations

• Simplicity: Weight-2 measurements only, no ancilla overhead for syndrome extraction.
• Asymmetric protection: Naturally biased — can be rectangular ($m_1\times m_2$) to correct more X than Z errors or vice versa, matching hardware noise bias.
• Concatenation: Effective when concatenated with other codes (e.g., repetition code for dominant error type).
• Threshold limitation: Lower threshold than surface code makes it less competitive for depolarizing noise, but advantageous for biased noise.

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

Experimental demonstrations

• L. Egan et al., “Fault-tolerant control of an error-corrected qubit,” Nature 598, 281 (2021) — arXiv:2009.11482

Related 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

Linked Papers

• bacon-2006-bacon-shor

Related Entries

• surface-code-logical-qubit — Higher threshold but requires weight-4 measurements
• color-code-logical-qubit — Alternative topological code with transversal gates
• kerr-cat-qubit — Biased-noise qubit that pairs well with Bacon-Shor