Bacon–Shor code

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: ${\displaystyle \langle X_{0}X_{1}X_{2}X_{3}X_{4}X_{5},X_{0}X_{1}X_{2}X_{6}X_{7}X_{8},Z_{0}Z_{1},Z_{1}Z_{2},Z_{3}Z_{4},Z_{4}Z_{5},Z_{6}Z_{7},Z_{7}Z_{8}\rangle }$, 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: ${\displaystyle \langle X_{0}X_{1}X_{2}X_{3}X_{4}X_{5},X_{0}X_{1}X_{2}X_{6}X_{7}X_{8},Z_{0}Z_{1}Z_{3}Z_{4}Z_{6}Z_{7},Z_{1}Z_{2}Z_{4}Z_{5}Z_{7}Z_{8}\rangle }$.^[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:${\displaystyle \langle Z_{1}Z_{2},X_{2}X_{8},Z_{4}Z_{5},X_{5}X_{8},Z_{0}Z_{1},X_{0}X_{6},Z_{3}Z_{4},X_{3}X_{6},X_{1}X_{7},X_{4}X_{7},Z_{6}Z_{7},Z_{7}Z_{8}\rangle }$.^[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 overheard for fault tolerant error correction.^[5]

    ZZ   ZZ   q0---q1--q2XX|  XX|   |XX  |  ZZ| ZZ|  q6--q7--q8XX|  XX|   |XX  |    |   |  q3--q4--q5   ZZ   ZZ