Bacon–Shor code

The Bacon–Shor code is a subsystem error correcting code. 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. This simplicity led to the first claim of fault tolerant circuit demonstration on a quantum computer. It is named after Dave Bacon and Peter Shor.

Given the stabilizer generators of Shor's code: $⟨X_0{X}_1{X}_2{X}_3{X}_4{X}_5,X_0{X}_1{X}_2{X}_6{X}_7{X}_8,Z_0{Z}_1,Z_1{Z}_2,Z_3{Z}_4,Z_4{Z}_5,Z_6{Z}_7,Z_7{Z}_8⟩$, 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: $⟨X_0{X}_1{X}_2{X}_3{X}_4{X}_5,X_0{X}_1{X}_2{X}_6{X}_7{X}_8,Z_0{Z}_1{Z}_3{Z}_4{Z}_6{Z}_7,Z_1{Z}_2{Z}_4{Z}_5{Z}_7{Z}_8⟩$. 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:$⟨Z_1{Z}_2,X_2{X}_8,Z_4{Z}_5,X_5{X}_8,Z_0{Z}_1,X_0{X}_6,Z_3{Z}_4,X_3{X}_6,X_1{X}_7,X_4{X}_7,Z_6{Z}_7,Z_7{Z}_8⟩$. 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.

Geometry

q0	ZZ	q1	ZZ	q2
XX		XX		XX
q6	ZZ	q7	ZZ	q8
XX		XX		XX
q3	ZZ	q4	ZZ	q5

See also

• Five-qubit error correcting code