  Understanding fault-tolerant properties of quantum circuits is important for
the design of large-scale quantum information processors. In particular,
simulating properties of encoded circuits is a crucial tool for investigating
the relationships between the noise model, encoding scheme, and threshold
value. For general circuits and noise models, these simulations quickly become
intractable in the size of the encoded circuit. We introduce methods for
approximating a noise process by one which allows for efficient Monte Carlo
simulation of properties of encoded circuits. The approximations are as close
to the original process as possible without overestimating their ability to
preserve quantum information, a key property for obtaining more honest
estimates of threshold values. We numerically illustrate the method with
various physically relevant noise models.
