Simulation of rare events in quantum error correction

We consider the problem of calculating the logical error probability for a stabilizer quantum code subject to random Pauli errors. To access the regime of large code distances where logical errors are extremely unlikely we adopt the splitting method widely used in Monte Carlo simulations of rare events and Bennett's acceptance ratio method for estimating the free energy difference between two canonical ensembles. To illustrate the power of these methods in the context of error correction, we calculate the logical error probability $P_L$ for the two-dimensional surface code on a square lattice with a pair of holes for all code distances $d\le 20$ and all error rates $p$ below the fault-tolerance threshold. Our numerical results confirm the expected exponential decay $P_L\sim \exp [-\alpha (p\left)d\right]$ and provide a simple fitting formula for the decay rate $\alpha \left(p\right)$. Both noiseless and noisy syndrome readout circuits are considered.

Figure 1

Surface code with a pair of smooth defects representing one logical qubit. Physical qubits are indicated by solid circles. The code space is defined as a common eigenspace of site and plaquette stabilizers—products of Pauli $X$ and $Z$ over stars (red) and plaquettes (green). Logical Pauli operators $\overline{Z}$ and $\overline{X}$ correspond to a loop encircling one of the defects and the path connecting the two defects, respectively.

Figure 2

Noiseless syndrome readout. The decay rate $\alpha \left(p\right)$ in the exponential scaling $P_L\sim \exp [-\alpha (p\left)r\right]$ for looplike and pathlike logical errors computed by the upward splitting method. The dashed line shows the estimate of $\alpha \left(p\right)$ based on the asymptotic low-$p$ formula $P_L\sim \left(\genfrac{}{}{0pt}{}{4r}{2r}\right)p^{2r}$, that is, $\alpha \left(p\right)=-4\ln 2-2\ln \left(p\right)$.

Figure 3

Noisy syndrome readout. The decay rate $\alpha \left(p\right)$ for looplike and pathlike logical errors computed by the downward splitting method. The dashed lines show the estimate of $\alpha \left(p\right)$ based on the low-$p$ asymptotic formula $P_L\sim A_r{p}^{2r}$, that is, $\alpha \left(p\right)=-c-2\ln p$ [see the last line of Eq. (2)].

Figure 4

Noisy syndrome readout. Independent depolarizing errors occur with probability $p$ on each qubit preparation, measurement, and cnot gate. Logical error probability per time step $P_L$ has been computed by the downward splitting method for error rates $p\le p^{*}$ and by Monte Carlo method for $p\ge p^{*}$. Here $p^{*}=0.3\%$ for pathlike errors and $p^{*}=0.5\%$ for looplike errors. The splitting sequence starts at $p_1=p^{*}$. A good agreement between the derivatives of $P_L\left(p\right)$ computed by the two methods is observed at the junction point $p=p^{*}$. Solid lines represent fitting formulas, Eqs. (2) and (3). Generating the data presented in Figs. 4 and 5 took roughly 20 000 CPU hours on PowerPC 450 processors.

Figure 5

Noiseless syndrome readout. Independent bit-flip and phase-flip errors occur with probability $p$ on each physical qubit. Logical error probability $P_L\left(p\right)$ has been computed by the upward splitting method for error rates $p\le p^{*}$ and by Monte Carlo method for $p\ge p^{*}$. Here $p^{*}=3\%$ for pathlike errors and $p^{*}=5\%$ for looplike errors. The splitting sequence starts at error rate $p_1=0.1\%$ where one can use asymptotic formulas $P_L\approx 0.5r\left(\genfrac{}{}{0pt}{}{4r}{2r}\right)p^{2r}$ for pathlike errors and $P_L\approx 0.5\left(\genfrac{}{}{0pt}{}{4r}{2r}\right)p^{2r}$ for looplike errors. Here $r=d/4$ is the linear size of the defects. A good agreement between the values of $P_L\left(p\right)$ obtained by the two methods is observed at the junction point $p=p^{*}$. Solid lines represent fitting formulas, Eqs. (2) and (3).

Figure 6

Four methods of estimating the logical error probability $P_L$. For small code distances $d$ one can compute $P_L$ numerically by the Monte Carlo simulation (if $p$ is close to the threshold) or by the splitting method (if $p$ is sufficiently below the threshold). The fault-path counting (FPC) is applicable for very small error rates. Numerical data are used to estimate the decay rate $\alpha$ in the fitting formula $P_L\sim \exp [-\alpha (p\left)d\right]$, which is applicable for larger values of $d$.

Figure 7

Examples of the decoding graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ used in the simulations of looplike errors (top) and pathlike errors (bottom) for $r=2$. Logical chains $\Gamma$ are highlighted in red. In the case of pathlike errors the graph contains “hanging edges” $(u,v)\in \mathcal{E}$ such that $v\in T$ is a degree-1 vertex that carries no syndrome. To avoid clutter, a hanging edge $(u,v)$ is represented by a solid circle centered at $u$ (the degree-1 vertex $v$ is not shown). The geometry depends on three parameters $r,s,b$, where $r$ is the linear size of the defects, $s$ is the separation between the defects, and $b$ is the length of the buffer zone separating the defects and the external boundary of the lattice. The simulations were performed for $s=b=4r$. This guarantees that the code distance is $d=4r$.

Figure 8

Example of a degenerate syndrome. One can easily check that any error chain $E$ satisfying $\partial E=\{1,2,3,4\}$ has length at least 12. There is a unique even length-12 chain composed of paths $(1,2)$ and $(3,4)$. There are ${\left(\genfrac{}{}{0pt}{}{6}{3}\right)}^2=400$ odd length-12 chains composed of paths $(1,4)$ and $(2,3)$. If the actual error chain has minimum length (which is true in the limit of small error rates), a decoding rule that favors the pairing $(1,4),(2,3)$ is 400 times more likely to correct the error.

Figure 9

Syndrome readout circuits for plaquette and site stabilizers.

Figure 10

Noisy syndrome readout. The decoding graph corresponding to the surface code lattice shown in Fig. 1. Each vertex of the graph represents a space-time location of a syndrome measurement. Any elementary error in the syndrome readout circuit (memory, preparation, measurement, or cnot error) is associated with some edge of the graph. The overall probability of elementary errors associated with a given edge $e$ determines the effective error rate of $e$. Red (blue) color stands for the largest (smallest) effective error rate. Some edges located near the boundary of the defects have only one end point. To avoid clutter, such edges are represented by solid circles. The corresponding surface code has two smooth defects of linear size $r=2$, separation $s=5$, and buffer length $b=3$. The number of syndrome readout rounds is $t=3$. The actual simulations were performed for $s=b=t=4r$ for $r=2,3,4$.

Figure 11

A fragment of the decoding graph indicated by the black rectangle in Fig. 10.