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 2D 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 $P_L\sim \exp{[-\alpha(p)d]}$ and provide a simple fitting formula for the decay rate $\alpha(p)$. Both noiseless and noisy syndrome readout circuits are considered.