Numerical evaluation of the fidelity error threshold for the surface code

We study how the resilience of the surface code is affected by the coupling to a non-Markovian environment at zero temperature. The qubits in the surface code experience an effective dynamics due to the coupling to the environment that induces correlations among them. The range of the effective induced qubit-qubit interaction depends on parameters related to the environment and the duration of the quantum error correction cycle. We show numerically that different interaction ranges set different intrinsic bounds on the fidelity of the code. These bounds are unrelated to the error thresholds based on stochastic error models. We introduce a definition of stabilizers based on logical operators that allows us to efficiently implement a Metropolis algorithm to determine upper bounds to the fidelity error threshold.

Figure 1

The geometry of the surface code system. Physical qubits are shown with arrows. A plaquette (star) operator ${\widehat{B}}_p$ $\left({\widehat{A}}_s\right)$ is shown with a shaded (unshaded) enclosed area connecting the corresponding qubits. The dashed green (dotted red) line $l_z$ $\left(l_x\right)$ represents the logical operator ${\widehat{Z}}_{l_z}$$\left({\widehat{X}}_{l_x}\right)$ and runs over corresponding qubits.

Figure 2

The effective interaction induced by the bath, Eq. (9), between a qubit and its nearest (solid black lines) and next-to-nearest neighbors (dashed red lines). The range of the interaction (dotted circle) is related to the QEC period $\Delta$.

Figure 3

Applying two logical $\widehat{Z}$ operators along the paths $l_1$ and $l_2$ is equivalent to a ${\widehat{B}}_p$ operator shown with the hatched rectangle.

Figure 4

The numerical evaluation of $\langle X\rangle =|\frac{\mathcal{B}}{\mathcal{A}}|$ for nonerror syndromes based on the Monte Carlo calculation for different system sizes. $L=20$ is a surface code system with 761 qubits (circle), $L=30$ has 1741 qubits (diamond), and $L=40$ has 3121 qubits. The solid lines are guides to the eye. On the horizontal axis, $\beta$ is proportional to the coupling to the environment and $J$ is the exchange coupling of the effective interaction between nearest-neighbor qubits. In this simulation 80 000 iterations are used for each $\beta$ step.

Figure 5

Finite-size scaling analysis of the heat capacity per qubit that yields ${\beta }_c=0.217$ for $L\to \infty$. The solid lines are guides to the eye.

Figure 6

The ratio $|\frac{\mathcal{B}}{\mathcal{A}}|$ for different interaction ranges as a function of $\beta$ on a lattice with $L=40$ and 80 000 Monte Carlo steps for each data point. The data points correspond to $J_1=1$ (circles), as in Fig. 4; $J_1=1$, $J_2=0.2$, and $J_m=0$ for $m>2$ (diamonds); and $J_1=1$, $J_2=1$, and $J_m=0$ for $m>2$ (triangles) in Eq. (38). The solid lines are guides to the eye.

Figure 7

The location of the single error $\left\{B_{p_0}\right\}$ (bottom) and two errors $\{B_{p_1},B_{p_2}\}$ (top) assumed in the numerical calculations.

Figure 8

The ratio $|\frac{\mathcal{B}}{\mathcal{A}}|$ for a lattice of $L=20$. The left box shows the convergence of $\langle S_1^x\rangle =\left|\frac{\mathcal{B}}{\mathcal{A}}\right|$ for one detected error located as shown in Fig. 7. The right box shows the convergence of $\langle S_2^x\rangle =\left|\frac{\mathcal{B}}{\mathcal{A}}\right|$ for two detected errors located as shown in Fig. 7. In both boxes the number of iterations used for each $\beta$ data point is $90000$ (circle), $180000$ (square), and $900000$ (diamonds). The solid line is obtained from the corresponding no-error sector (Fig. 6).