| Bacon–Shor code | |
|---|---|
| Name | Bacon–Shor code |
| Type | Subsystem quantum error-correcting code |
| Introduced | 2006 |
| Authors | Dave Bacon; inspired by Peter Shor |
| Related | Shor code; stabilizer code; subsystem code; surface code |
Bacon–Shor code
The Bacon–Shor code is a family of quantum error correction codes that encode logical qubits into arrays of physical qubits using a subsystem code construction. It combines features of the Shor code and stabilizer formalism to reduce the weight of measured gauge operators while retaining protection against quantum noise such as bit-flip errors and phase-flip errors. The construction matters in Quantum Physics and quantum computing because it offers experimentally attractive syndrome extraction and resource trade-offs for near-term quantum error-correcting hardware.
The Bacon–Shor code was proposed as a low-overhead subsystem code variant of the Shor code and is formally described within the stabilizer code framework. It belongs to a class of operator stabilizer codes that exploit gauge degrees of freedom to simplify syndrome measurement. The approach is relevant to groups working on superconducting qubits at institutions such as IBM and Google Quantum AI, and to teams building ion trap processors at places like Honeywell/Quantinuum and IonQ, because gauge measurements can be implemented with lower-weight circuits than full stabilizer measurements. The code also connects to theoretical developments in fault-tolerant quantum computation and threshold estimates developed in the Eastin–Knill theorem era.
A common Bacon–Shor instantiation is defined on an r × s rectangular lattice of physical qubits. The code is constructed by combining two classical repetition codes—one protecting against X errors and one against Z errors—into a CSS code-like architecture. The full stabilizer group is generated by products of nearest-neighbor two-qubit gauge operators (weight-two X⊗X and Z⊗Z terms) whose commutant yields higher-weight logical stabilizers. The subsystem perspective treats some Pauli operators as gauge operators that need not be stabilized, allowing measurement of lower-weight operators to infer the eigenvalues of the logical stabilizers. This construction is often explained using the language of Pauli matrices and the commutator/centralizer within the Pauli group.
Logical qubits of the Bacon–Shor code are encoded nonlocally across rows and columns of the lattice. Logical Pauli X and Z operators correspond to long strings of single-qubit Paulis across a full row or column, similarly to the Shor code where logical operators span the code blocks. The code distance is determined by the smaller of the two lattice dimensions (min(r,s)): a single logical operator requires that many physical errors to enact an undetectable logical fault. For square r = s arrays the distance scales as r, while asymmetric choices allow trade-offs between distance for X-type and Z-type errors. Logical operator implementation and transversal gates are constrained by Eastin–Knill theorem limitations; nevertheless some logical Clifford gates can be implemented transversally or via gauge-fixing techniques.
Syndrome extraction in Bacon–Shor codes measures the gauge operators (nearest-neighbor X⊗X and Z⊗Z) rather than the full high-weight stabilizers. The measured gauge outcomes are classically processed to reconstruct the values of stabilizer syndromes and locate probable error chains. Typical decoding strategies borrow from minimum-weight perfect matching or from tailored decoders for concatenated repetition codes; maximum-likelihood decoding and tensor-network approaches have also been applied in theoretical studies. Because gauge measurements are lower-weight, circuits can have reduced depth and lower correlated error rates, which affects the performance of quantum error mitigation and active correction routines used in real-time feedback systems.
Fault-tolerant designs using Bacon–Shor codes exploit gauge measurement locality to reduce ancillary resource requirements and circuit complexity. Threshold estimates depend on noise models and decoder performance; while the Bacon–Shor code typically does not achieve thresholds as high as optimized surface code implementations under some noise models, it can outperform in regimes with biased noise or limited qubit connectivity. Experimental proposals and demonstrations have focused on small-scale realizations in superconducting circuit and trapped ion platforms where two-qubit gates implement the gauge measurements. The code also informs architectures for logical qubit layouts in quantum processors built by academic research groups and industrial labs pursuing near-term fault tolerance.
Compared with the original Shor code, the Bacon–Shor code reduces the weight of measured operators via gauge fixing, at the cost of introducing gauge degrees of freedom managed classically. Relative to the surface code, Bacon–Shor offers simpler syndrome extraction for particular geometries but typically requires more physical qubits per logical qubit to reach comparable thresholds. It sits within the broader class of subsystem codes alongside the Kitaev honeycomb model-inspired codes and gauge color codes, sharing the strategy of converting some stabilizers into gauge operators to simplify operations. Trade-offs among these codes guide choices in quantum error correction stack design depending on constraints like qubit connectivity, gate fidelity, and dominantly biased noise (e.g., dephasing-dominated channels).
Category:Quantum error correction Category:Quantum information theory