Operational Quantum Average-Case Distances

Abstract

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 the 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 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.Introduction. In the era of Noisy Intermediate Scale Quantum (NISQ) devices [1], it is instrumental to have figures of merit that quantify how close two quantum protocols are. The distance measures commonly used for this purpose, for example, in the context of quantum error-correction [2], such as trace distance or diamond norm, have an operational interpretation in terms of optimal statistical distinguishability between two quantum states, measurements, or channels [3][4][5][6]. While it is natural to consider the optimal protocols when one wishes to distinguish between two objects, alas, in reality, such protocols might be not practical. For example, in general, they require high-depth, complicated quantum circuits [7]. From a complementary perspective, quantum distances are often used to compare an ideal implementation (of a state, measurement, or channel) with its noisy experimental version. In this context, using the distances based on optimal distinguishability gives information about the worst-case performance of a device in question. This may be impractical as well -it is not expected that the performance of typical experiments on a quantum device will be comparable to the worst-case scenario.

Quantum simulation of dissipative collective effects on noisy quantum computers

“…That is to say, they rely on the precision of the tomographic measurement. A different figure of merit for the distance between quantum channels has been very recently introduced [82].…”

Saturation and recurrence of quantum complexity in random quantum circuits

“…Our results apply to local random quantum circuits with any arrangement of qudits defined on a graph g with a Hamiltonian path and with gates chosen randomly from any universal gate set G (without 1 We note that this definition of complexity necessarily assumes dichotomic measurements. If this condition is relaxed it can be shown [33] that computational basis measurements preceded by local random circuits of depth O(n) suffice to efficiently distinguish any pure state from maximally mixed state. 2 Indeed, general dimension counting arguments show that almost all (with respect to the Haar measure) unitary circuits have maximal exact complexity (scaling as e Θ(n) ).…”