Exploring Quantum Average-Case Distances: proofs, properties, and examples

Abstract

In this work, we perform an in-depth study of recently introduced average-case quantum distances . The average-case distances approximate, up to relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify average-case statistical distinguishability via random quantum circuits.We prove that once a family of random circuits forms an δ-approximate 4-design, with δ = o(d −8 ), then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed -no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all of the distances utilize the Hilbert-Schmidt (HS) norm in some way. This gives the HS norm an operational interpretation that it did not possess before. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them we provide an example that saturates the bound. Specifically, we show that for each dimension d this ratio is at most d 1 2 , d, d 3 2 for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically. Furthermore, we perform an extensive numerical study of systems up to 10 qubits where circuits have a form of variational ansätze with randomly chosen parameters. Despite the fact that such circuits are not known to form unitary designs, our results suggest that average-case distances are more suitable than the conventional distances for studying the performance of NISQ devices in such settings.