  Quantum error correction protocols have been developed to offset the high
sensitivity to noise inherent in quantum systems. However, much is still
unknown about the behaviour of a quantum error-correcting code under general
noise, including noisy measurements. This lack of knowledge is largely due to
the computational cost of simulating quantum systems large enough to perform
nontrivial encodings. In this paper, we develop general methods for
incorporating noisy measurement operations into simulations of quantum
error-correcting codes and show that measurement errors on readout qubits
manifest as a renormalization on the effective logical noise. We also derive
general methods for reducing the computational complexity of calculating the
exact effective logical noise by many orders of magnitude. This reduction is
achieved by determining when different recovery operations produce equivalent
logical noise. These methods could also be used to better approximate soft
decoding schemes for concatenated codes or to reduce the size of a lookup table
to speed up the error correction step in implementations of quantum
error-correcting codes. We give examples of such reductions for the
three-qubit, five-qubit, Steane, concatenated, and toric codes.
