Random Error Models in Quantum Error Correction

We examine the performance of quantum error correcting codes subjected to random Haar distribution transformations of weight t. Rather than requiring correction of all errors, we require some high probability that a random error is corrected. We find that, for any integer i and arbitrarily high probability p < 1, there are codes which perfectly correct errors up to weight t and can correct errors up to weight t + i with probability at least p. We also find an analog to the quantum Hamming bound for the new error model. Lastly, we prove that codes generated from classical Reed-Muller codes can correct errors of weight up to 3d/4 with a probability approaching 1 as the length of the code increases, whereas they can only correct up to weight d/2 perfectly.