Operational Quantum Average-Case Distances

Abstract

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.

6 pages, 2 figures; v2: changed title, corrected typos; v3: improved narration, corrected typos, extended numerical simulations; v4: major changes, improved narration, extended examples and numerical simulations; v5: version accepted in Quantum, improved narration, expanded appendices; comments and suggestions are welcome; accompanying technical paper: arXiv:2112.14284

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Cited by in corpus (4)

• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• Saturation and recurrence of quantum complexity in random local quantum dynamics
• Certification of quantum state functions under partial information
• Benchmarking the quality of multiplexed qubit readout beyond assignment fidelity