inherent difficulty of characterizing rare events. Indeed, the interest of a fault tolerant scheme is that it results in a very low logical fault-rate. Thus, understanding and characterizing such faults requires an extremely large number of simulations. In classical error correction, it is possible to use importance sampling methods which enhance the probability of these rare events, see, e.g., [@RT09; @CL15; @B12]. Here again, quantum error correction poses an new challenge because quantum errors are generically not described by stochastic processes, and hence importance sampling methods do not directly apply. In , we will present our attempts at developing importance sampling methods tailored to quantum processes. While we obtain some improvements over direct simulations, the number of simulations required for practical quantum computing applications remains prohibitively large.

Numerical simulation {#sec:NS}
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The evolution of a single qubit state $\rho$ over some fixed period of time can be described by a completely positive, trace preserving (CPTP) map $\cE(\rho) = \sum_{kk'} \chi_{kk'} \sigma_k \rho \sigma_k$ where $\sigma_k$ are the Pauli matrices and the complex $4\times 4$ matrix $\chi$ has unit trace [@WBC15]. Furthermore, $\cE$ has 12 independent real parameters [@RSW02]. It can be shown that $\cE$ can always be obtained by considering 