Long-time efficacy of the surface code in the presence of a super-Ohmic environment

We study the long-time evolution of a quantum memory coupled to a bosonic environment on which quantum error correction (QEC) is performed using the surface code. The memory's evolution encompasses $N$ QEC cycles, each of them yielding a nonerror syndrome. This assumption makes our analysis independent of the recovery process. We map the expression for the time evolution of the memory onto the partition function of an equivalent statistical-mechanical spin system. In the super-Ohmic dissipation case the long-time evolution of the memory has the same behavior as the time evolution for just one QEC cycle. For this case we find analytical expressions for the critical parameters of the order-disorder phase transition of an equivalent spin system. These critical parameters determine the threshold value for the system-environment coupling below which it is possible to preserve the memory's state.

Figure 1

The surface code. Physical qubits are shown as black dots. The stabilizer operators are products of four Pauli operators, $A_S$ and $B_P$. Logical operators correspond to a string of Pauli operators crossing the qubit lattice from one side to the other, $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$.

Figure 2

Illustration of mass field variables in the bulk and in the boundary for the spin variable $\sigma$.