Performance of quantum error correction with coherent errors

We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the 5-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error $D'_\diamond$ of the logical qubit with the same quantity at the physical level $D_\diamond$, we show that $D'_\diamond \le c D^d_\diamond$ where $d$ is the distance of the code and $c$ is constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance $d$ will exhibit a scaling $D'_\diamond \sim D_\diamond^{(d+1)/2}$.