Hierarchically discriminating Haar-randomness in quantum states from a black-box device

TL;DR

This paper introduces a hierarchical algorithm to efficiently determine if quantum states from a black-box device are Haar-random, using statistical moments and observable spectrum analysis, aiding quantum device benchmarking.

Contribution

The work presents a novel hierarchical discrimination algorithm that reduces complexity in testing Haar-randomness of quantum states from black-box devices, including analytical Haar moments and extensions for improved accuracy.

Findings

The discriminator can identify non-Haar-random states effectively.

Analytical Haar moments are computed using a new connection with the Dirichlet distribution.

Extensions improve the discrimination accuracy with increased computational resources.

Abstract

The concept of randomness in quantum computing has been central to construct benchmarking tools, cryptographic protocols, as well as a proof of beyond classical computation. Discerning whether quantum states (or unitaries) are randomly distributed is a computational task that requires an enormous amount of quantum computational resources. This work addresses such a challenge, a hierarchical discrimination algorithm to efficiently test the if a set of states $S$ generated from a black-box quantum device with an unknown distribution is (in)compatible with a random distribution. To this end, we reduce the complexity of the problem by selecting an observable with known spectrum to study the statistical properties of its expectation values with respect to the quantum states from an unknown (black-box) quantum device. Concurrently, we use our first technical result, a connection between Haar-randomness and the Dirichlet distribution, to analytically compute Haar-random moments of the observable. Our Haar-random discriminator test is then simply to compare those statistical moments such that if $S$ fails the test it is enough to state that the quantum devices does not output randomly distributed states. Else, we can not (yet) confirm that the states follow a Haar-random distribution. We further provide extension to this algorithm by permutation- and unitary-equivalent randomization of observable at increasing computational resources, which allows us to more accurately state whether $S$ is compatible with Haar-randomness. We envision the use of the discriminator test as quantum device benchmark by discriminating if the states generated are Haar-random incompatible.