We introduce operational distance measures between quantum states, measurements, and channels based on their average-case distinguishability. To this end, we analyze the average Total Variation Distance (TVD) between statistics of quantum protocols in which quantum objects are intertwined with random circuits and subsequently measured in a computational basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects, which we call average-case distances. The so-defined distances capture average-case distinguishability via moderate-depth random quantum circuits and satisfy many natural properties. We apply them to analyze the effects of noise in quantum advantage experiments and in the context of efficient discrimination of high-dimensional quantum states and channels without quantum memory. Furthermore, based on analytical and numerical examples, we argue that average-case distances are better suited for assessing the quality of NISQ devices than conventional distance measures such as trace distance and the diamond norm.
The talk is based on recent preprints: arXiv:2112.14283 and arXiv:2112.14284.