The Effect of Coherent Errors in Error Correction

The first threshold theorems for fault tolerant quantum computation assumed a stochastic error model. More recently there has been interest in coherent errors, because under these errors the error rate can grow quadratically instead of linearly. Even with the same error rate per gate, a coherent error model could have orders of magnitude larger error for the whole computation. Past work on coherent errors has focused on a constant single-qubit coherent error as a simple case. This work treats coherent, Markovian noise with arbitrary correlations in space using a random ensemble approach. The noise at each time step is drawn from some distribution on the space of channels. We work with an arbitrary family of stabilizer codes paired with a family of ensemble noise models and prove that if the correlations for the noise models satisfy a bound, the logical error rate and the residual logical noise approach the incoherent answer. We present a modified threshold theorem that gives conditions on the noise models such that an arbitrary length computation can be successfully performed.