Subsystem Fault Tolerance with the Bacon-Shor Code

We discuss how the presence of gauge subsystems in the Bacon-Shor code [D. Bacon , Phys. Rev. A 73, 012340 (2006)] leads to remarkably simple and efficient methods for fault-tolerant error correction (FTEC). Most notably, FTEC does not require entangled ancillary states, and it can be implemented with nearest-neighbor two-qubit measurements. By using these methods, we prove a lower bound on the quantum accuracy threshold, $1.94\times {10}^{-4}$ for adversarial stochastic noise, that improves previous lower bounds by nearly an order of magnitude.

Figure 1

Qubits in the ${\mathcal{C}}_{\mathrm{BS}}^{(n)}$ block sit on the vertices of an $n\times n$ square lattice. An element of the code stabilizer, the operator $X_{2,*}{X}_{3,*}$ applies $X$ on all qubits in the second and third rows.

Figure 2

(a) A circuit for measuring the operator $X_{j,k}{X}_{j+1,k}$ using one ancillary qubit. (b) A similar circuit for measuring $Z_{k,j}{Z}_{k,j+1}$. $|+⟩\propto |0⟩+|1⟩$ is the $+1$ eigenstate of $X$.