  We study the performance of quantum error correction codes (QECCs) under the
detection-induced coherent error due to the imperfectness of practical
implementations of stabilizer measurements, after running a quantum circuit.
Considering the most promising surface code, we find that the detection-induced
coherent error will result in undetected error terms, which will accumulate and
evolve into logical errors. However, we show that such errors will be
alleviated by increasing the code size, akin to eliminating other types of
errors discussed previously. We also find that with detection-induced coherent
errors, the exact surface code becomes an approximate QECC.
