We consider the problem of calculating the logical error probability for astabilizer quantum code subject to random Pauli errors. To access the regime oflarge code distances where logical errors are extremely unlikely we adopt thesplitting method widely used in Monte Carlo simulations of rare events andBennett's acceptance ratio method for estimating the free energy differencebetween two canonical ensembles. To illustrate the power of these methods inthe context of error correction, we calculate the logical error probability$P_L$ for the 2D surface code on a square lattice with a pair of holes for allcode distances $d\le 20$ and all error rates $p$ below the fault-tolerancethreshold. Our numerical results confirm the expected exponential decay$P_L\sim \exp{[-\alpha(p)d]}$ and provide a simple fitting formula for thedecay rate $\alpha(p)$. Both noiseless and noisy syndrome readout circuits areconsidered.
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机译：我们考虑了计算受制于随机Pauli误差的稳定剂量子码的逻辑误差概率的问题。为了访问在极不可能发生逻辑错误的大码距离的情况下，我们采用了罕见事件的蒙特卡罗模拟中广泛使用的拆分方法和贝内特的接受率方法来估计两个规范集合之间的自由能差。为了说明这些方法在纠错环境中的强大功能，我们计算了带有一对孔的方格上二维表面代码的逻辑错误概率$ P_L $，其中所有代码距离$ d \ le 20 $和所有错误率$ p $低于容错阈值。我们的数值结果证实了预期的指数衰减$ P_L \ sim \ exp {[-\ alpha（p）d]} $，并为衰减率$ \ alpha（p）$提供了简单的拟合公式。无噪声和噪声综合症读出电路都被考虑。
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