Surface code fidelity at finite temperatures

We study the dependence of the fidelity of the surface code in the presence of a single finite-temperature massless bosonic environment after a quantum error correction cycle. The three standard types of environment are considered: super-Ohmic, Ohmic, and sub-Ohmic. Our results show that, for regimes relevant to current experiments, quantum error correction works well even in the presence of environment-induced, long-range interqubit interactions. A threshold always exists at finite temperatures, although its temperature dependence is very sensitive to the type of environment. For the super-Ohmic case, the critical coupling constant separating high from low fidelity decreases with increasing temperature. For both Ohmic and super-Ohmic cases, the dependence of the critical coupling on temperature is weak. In all cases, the critical coupling is determined by microscopic parameters of the environment. For the sub-Ohmic case, it also depends strongly on the duration of the QEC cycle.

Figure 1

Surface code lattice and operators. Physical qubits are located at the edges (black circles). Some star $\widehat{A}$ and plaquette $\widehat{B}$ operators are indicated, as well as some realizations of the logical operators $\widehat{\overline{X}}$ and $\widehat{\overline{Z}}$.

Figure 2

Fidelity in the presence of a super-Ohmic environment ($s=1/2$) in the thermal regime for several system sizes. Data obtained by averaging over ${10}^9$ Monte Carlo steps. Here, $\gamma ={\lambda }^{2}F(\mathrm{\Delta };0;\beta )$. The dashed line marks the phase-transition point of an Ising model on a square lattice with nearest-neighbor interactions.

Figure 3

Fidelity of the surface code in the presence of a super-Ohmic environment ($s=1/2$) for several system sizes when a small imaginary nearest-neighbor interaction is present; see Eq. (58). Data points obtained using Binder's method. Here, $\gamma ={\lambda }^{2}F(\mathrm{\Delta };0;\beta )$ and $\eta =\mathrm{\Phi }(\mathrm{\Delta };d)/F(\mathrm{\Delta };0;\beta )$. The dashed line marks the phase-transition point of an Ising model in a square lattice with nearest-neighbor interactions. The circle marks the threshold position.

Figure 4

Fidelity of the surface code in the presence of an Ohmic bath, as given by the statistical model of Eq. (60), for different lattice sizes. Here, $\gamma ={\lambda }^2\mathrm{\Delta }F,\overline{\mathrm{\Phi }}=0$, and $\overline{F}=0.72\mathrm{\Delta }F$. Data obtained by averaging over ${10}^9$ Monte Carlo steps. The location of the transition point, indicated by the crossing point surrounded by a circle, is marked by the dashed line.

Figure 5

Mass field variable composition.

Figure 6

Thick horizontal lines represent two-body nearest-neighbor horizontal interactions. Short thick double vertical lines represent two-body nearest-neighbor vertical interactions. Diagonal thin lines represent two-body next-to-nearest neighbor interactions. L-shaped lines and squares represent three-body interactions.