  Modeling and simulation is essential for predicting and verifying the
behavior of fabricated quantum circuits, but existing simulation methods are
either impractically costly or require an unrealistic simplification of error
processes. We present a method of simulating noisy Clifford circuits that is
both accurate and practical in experimentally relevant regimes. In particular,
the cost is weakly exponential in the size and the degree of non-Cliffordness
of the circuit. Our approach is based on the construction of exact
representations of quantum channels as quasiprobability distributions over
stabilizer operations, which are then sampled, simulated, and weighted to yield
unbiased statistical estimates of circuit outputs and other observables. As a
demonstration of these techniques we simulate a Steane [[7,1,3]]-encoded
logical operation with non-Clifford errors and compute its fault tolerance
error threshold. We expect that the method presented here will enable studies
of much larger and more realistic quantum circuits than was previously
possible.
