  Quantifying the effect of noise on unitary operations is an essential task in
quantum information processing. We propose the quantum Wasserstein distance
between unitary operations, which shows an explanation for quantum circuit
complexity and characterizes local distinguishability of multi-qudit
operations. We show analytical calculation of the distance between identity and
widely-used quantum gates including SWAP, CNOT, and other controlled gates. As
an application, we estimate the closeness between quantum gates in circuit, and
show that the noisy operation simulates the ideal one well when they become
close under the distance. Further we introduce the $W_1$ error rate by the
distance, and establish the relation between the $W_1$ error rate and two
practical cost measures of recovery operation in quantum error-correction under
typical noise scenarios.
