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

The concept of randomness in quantum computing has been central to constructing 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 by introducing a hierarchical discrimination algorithm to efficiently test 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 device does not output randomly distributed states. Else, we can not (yet) confirm that the states follow a Haar-random distribution. We further provide an extension to this algorithm by permutation- and unitary-equivalent randomization of the 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 a quantum device benchmark, by discriminating whether the states generated are incompatible with Haar-randomness.