  Quantum error correcting (QEC) codes protect quantum information from
decoherence, as long as error rates fall below critical error thresholds. In
general, obtaining thresholds implies simulating the QEC procedure using, in
general, sub-optimal decoding strategies. In a few cases and for sufficiently
simple noise models, optimal decoding of QEC codes can be framed as a phase
transition in disordered classical spin models. In both situations, accurate
estimation of thresholds demands intensive computational resources. Here we use
the coherent information of the mixed state of noisy QEC codes to accurately
estimate the associated optimal QEC thresholds already from small-distance
codes at moderate computational cost. We show the effectiveness and versatility
of our method by applying it first to the topological surface and color code
under bit-flip and depolarizing noise. We then extend the coherent information
based methodology to phenomenological and quantum circuit level noise settings.
For all examples considered we obtain highly accurate estimates of optimal
error thresholds from small, low-distance instances of the codes, in close
accordance with threshold values reported in the literature. Our findings
establish the coherent information as a reliable competitive practical tool for
the calculation of optimal thresholds of state-of-the-art QEC codes under
realistic noise models.
