Bacon–Shor subsystem

The Bacon-Shor code is a CSS subsystem code on an m1-by-m2 qubit array with parameters [[m1m2,1,(m1-1)(m2-1),min(m1,m2)]]. Local weight-two gauge measurements replace direct measurement of Shor-like high-weight stabilizers.

Nearest-neighbor XX operators across columns and ZZ operators across rows generate the gauge group; products along strips form stabilizers in its center. Both gauge generators and stabilizers must be recorded because stabilizers alone hide the locality of the measured operators.

Logical information is protected against worst-case Pauli logical operators below weight min(m1,m2), while the gauge subsystem need not be restored to a fixed state. Rectangular aspect ratio can target X/Z bias, but directional effective distances must be reported separately.

A square m-by-m code uses m squared data qubits, encodes one logical qubit, has distance m, and contains (m-1) squared gauge qubits. Measured gauge operators have weight two, while stabilizer weight grows with m, so it is not a qLDPC family under bounded stabilizer weight.

Logical Paulis can be represented by full rows or columns, and the CSS structure supports useful blockwise Clifford operations. Gauge fixing moves between related stabilizer presentations but does not automatically provide a fault-tolerant universal gate set.

Decode by reconstructing strip stabilizer syndromes from redundant weight-two gauge outcomes and applying repetition-code majority or maximum-likelihood inference in each direction. With measurement noise, the redundant gauge spacetime record should be used directly rather than losslessly assumed to be one stabilizer bit.

Low-weight gauge measurements limit single-fault propagation and are the main engineering motivation. Nevertheless, the static local two-dimensional Bacon-Shor family has no nonzero asymptotic threshold in standard settings; finite-size pseudothresholds must not be called thresholds.

The strip structure is attractive for trapped ions, superconducting arrays, and biased noise, but measurement count, parallel depth, and crosstalk depend on layout. The 3-by-3 instance shares Shor-code parameters yet can use a very different ancilla schedule.

Fixing gauges yields a stabilizer presentation closely related to the Shor code, while leaving them free gives the Bacon-Shor subsystem structure. Equivalence must state whether it compares logical subspaces, stabilizer centers, or gauge codes with measurement schemes.

Parameters are often written [[n,k,r,d]], where r counts gauge qubits rather than additional protected logical qubits. A database storing only [[n,k,d]] loses essential subsystem information.