Fidelity of the surface code in the presence of a bosonic bath

We study the resilience of the surface code to decoherence caused by the presence of a bosonic bath. This approach allows us to go beyond the standard stochastic error model commonly used to quantify decoherence and error threshold probabilities in this system. The full quantum mechanical system-bath dynamics is computed exactly over one quantum error correction cycle. Since all physical qubits interact with the bath, space-time correlations between errors are taken into account. We compute the fidelity of the surface code as a function of the quantum error correction time. The calculation allows us to map the problem onto an Ising-like statistical spin model with two-body interactions and a fictitious temperature which is related to the inverse bath coupling constant. The model departs from the usual Ising model in the sense that interactions can be long ranged and can involve complex exchange couplings; in addition, the number of allowed configurations is restricted by the syndrome extraction. Using analytical estimates and numerical calculations, we argue that, in the limit of an infinite number of physical qubits, the spin model sustains a phase transition which can be associated to the existence of an error threshold in the surface code. An estimate of the transition point is given for the case of nearest-neighbor interactions.

Figure 1

A $3\times 3$ two-dimensional square lattice structure for a surface code. Physical qubits (circles) are located at the edges of the lattice, which has open (vertical) and closed (horizontal) boundaries. In the general case, the lattice is $n\times m$ in size, with $nm$ vertical and $(n+1)(m+1)$ horizontal edges and $N=$ physical qubits. The light colored strips show possible paths for the logical operators $\overline{Z}$ and $\overline{X}$. $a$ is the lattice constant.

Figure 2

Schematic representation of the spatial dependence of the correlation functions ${\mathcal{G}}_{\mathbf{r}\mathbf{s}}^{\left(R\right)}\left(\Delta \right)$ and ${\mathcal{G}}_{\mathbf{r}\mathbf{s}}^{\left(I\right)}\left(\Delta \right)$ for $s=-1/2$, 0, and $1/2$. We use $d=|\mathbf{r}-\mathbf{s}|$.

Figure 3

Surface code fidelity of code spaces of 25 and 41 physical qubits in contact with a bosonic bath when star operators are restricted to positive values ($A_{◊}=1$).

Figure 4

Fidelity of a code space of 25 physical qubits in contact with a bosonic bath when star operators are restricted to positive values ($A_{◊}=1$) and an imaginary part is added to the coupling constant: $J=J_R+i{J}_I$. The data sets correspond to different values of $J_I$.

Figure 5

Finite-size scaling of the critical fictitious temperature $T_c$ obtained from cluster mean-field calculations for lattice of sizes 13, 25, and 41. A real Ising interaction of strength $J$ involving only nearest neighbors was used. The circles are the numerical data and the dashed line is a linear fit. $L$ denotes the linear size of the surface code.