  In order to realize fault-tolerant quantum computation, tight evaluation of
error threshold under practical noise models is essential. While non-Clifford
noise is ubiquitous in experiments, the error threshold under non-Clifford
noise cannot be efficiently treated with known approaches. We construct an
efficient scheme for estimating the error threshold of one-dimensional quantum
repetition code under non-Clifford noise. To this end, we employ non-unitary
free-fermionic formalism for efficient simulation of the one-dimensional
repetition code under coherent noise. This allows us to evaluate the effect of
coherence in noise on the error threshold without any approximation. The result
shows that the error threshold becomes one third when noise is fully coherent.
Our scheme is also applicable to the surface code undergoing a specific
coherent noise model. The dependence of the error threshold on noise coherence
can be explained with a leading-order analysis with respect to coherence terms
in the noise map. We expect that this analysis is also valid for the surface
code since it is a two-dimensional extension of the one-dimensional repetition
code. Moreover, since the obtained threshold is accurate, our results can be
used as a benchmark for approximation or heuristic schemes for non-Clifford
noise.
