Quantum Wasserstein distance between unitary operations

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 the local distinguishability of multiqudit operations. We show analytical calculations 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 a circuit and show that the noisy operation simulates the ideal one well when they become close under the distance. Further, we introduce the W1 error rate by the distance and establish the relation between the W1 error rate and two practical cost measures of recovery operations in quantum error correction under typical noise scenarios. These applications allow the distance to quantify the effect of noise on unitary operations from the perspective of experiment resources, which cannot be achieved by existing distance.