  We introduce distance measures between quantum states, measurements, and
channels based on their statistical distinguishability in generic experiments.
Specifically, we analyze the average Total Variation Distance (TVD) between
output statistics of protocols in which quantum objects are intertwined with
random circuits and measured in standard basis. We show that for circuits
forming approximate 4-designs, the average TVDs can be approximated by simple
explicit functions of the underlying objects -- the average-case distances
(ACDs). We apply them to analyze the effects of noise in quantum advantage
experiments and for efficient discrimination of high-dimensional states and
channels without quantum memory. We argue that ACDs are better suited for
assessing the quality of NISQ devices than common distance measures such as
trace distance or the diamond norm.
