The Bacon–Shor code is a subsystem error correcting code.
[1] In a subsystem code, information is encoded in a subsystem of a Hilbert space.
Subsystem codes lend to simplified error correcting procedures unlike codes which encode information in the subspace of a Hilbert space.
[2] This simplicity led to the first claim of fault tolerant circuit demonstration on a quantum computer.
[3] It is named after Dave Bacon and Peter Shor.
Given the stabilizer generators of Shor's code:
⟨
0
1
2
3
4
0
1
2
6
7
1
, 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get:
[4] Error correction is now simplified because 4 stabilizers are needed to measure errors instead of 8.
A gauge group can be created from the stabilizer generators:
[4] Given that the Bacon–Shor code is defined on a square lattice where the qubits are placed on the vertices; laying the qubits on a grid in a way that corresponds to the gauge group shows how only 2 qubit nearest-neighbor measurements are needed to infer the error syndromes.
The simplicity of deducing the syndromes reduces the overhead for fault tolerant error correction.