Breakdown of surface-code error correction due to coupling to a bosonic bath

We consider a surface code suffering decoherence due to coupling to a bath of bosonic modes at finite temperature and study the time available before the unavoidable breakdown of error correction occurs as a function of coupling and bath parameters. We derive an exact expression for the error rate on each individual qubit of the code, taking spatial and temporal correlations between the errors into account. We investigate numerically how different kinds of spatial correlations between errors in the surface code affect its threshold error rate. This allows us to derive the maximal duration of each quantum error-correction period by studying when the single-qubit error rate reaches the corresponding threshold. At the time when error correction breaks down, the error rate in the code can be dominated by the direct coupling of each qubit to the bath, by mediated subluminal interactions, or by mediated superluminal interactions. For a two-dimensional Ohmic bath, the time available per quantum error-correction period vanishes in the thermodynamic limit of a large code size $L$ due to induced superluminal interactions, although it does so only like $1/\sqrt{lnL}$. For all other bath types considered, this time remains finite as $L\to \infty$.

Figure 1

Different kinds of spatial correlations between errors in the surface code and how they affect its threshold error rate.

Figure 2

Single-qubit error rate ${\tilde{p}}_c$ for which error correction breaks down for two error models that lead to stringlike error patterns: ballistic and diffusive propagation of anyons.

Figure 3

Single-qubit error rate ${\tilde{p}}_c$ for which error correction breaks down in the $m$-$l$ cluster error models.

Figure 4

Each qubit is independently subjected to an error with probability $p_1$. Furthermore, each pair of nearest-neighbor qubits is subjected to a a pair of errors with probability $p_2$. The blue lines correspond to a constant value of $p_x$, calculated according to Eq. (29), while the red line shows the entropic bound Eq. (30). Diamonds represent threshold error rates $(p_1,p_2)$ when error correction is performed with MWPM and correlations are ignored. Squares represent threshold error rates for an algorithm that takes correlations into account. Threshold error rates have been determined to accuracy ${10}^{-3}$, by comparing logical error rates for code sizes between 10 and 50 (periodic boundary conditions). For each combination of error rates and code sizes, the logical error rates were obtained from as many error configurations as were necessary to obtain ${10}^4$ logical errors.

Figure 5

The three summands $A\left(t\right)$, $B\left(t\right)$, and $C\left(t\right)$ and their sum $p_x\left(t\right)$ compared with $p_c$ for code sizes $L={10}^2$, ${10}^3$, and ${10}^4$. We have used paramters $v=1$, $\lambda =0.1$, $T=0.01$, and ${\omega }_c=30$. Note that the assumptions $\beta {\omega }_c\gg 1$ and ${\tau }_d\gg 1/{\omega }_c$ made during the derivation of $A\left(t\right)$ are well satisfied.

Figure 6

The bold lines show the functions $\Lambda \left(t\right)$ given in Eq. (A7). We have used parameters $\alpha =0.01$, ${\omega }_c/{\omega }_0=30$, and $\beta {\omega }_0=10$. The dashed line shows the critical value $\frac{1}{2}ln\frac{1}{1-2p_c}\simeq 0.123$, when error correction breaks down in the uncorrelated case.