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

Xavier Bonet-Monroig, Hao Wang, Adrián Pérez-Salinas

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.

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

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 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 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 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.
Paper Structure (16 sections, 11 theorems, 79 equations, 1 figure)

This paper contains 16 sections, 11 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Haar meets Dirichlet
  4. Hierarchical Haar-random discriminator
  5. Increasing $t$-design
  6. Permutations
  7. Mutually Unbiased Bases
  8. Mixed states and frame potential
  9. Conclusions
  10. Haar-random Discriminator algorithm and subroutines
  11. Proof of \ref{['le.dirichlet']} and \ref{['th.dirichlet']}
  12. Proof of \ref{['le.moments']}
  13. Proof of \ref{['cor.efficient_moments']}
  14. Proof of \ref{['th.Haar_check']}
  15. Proof of \ref{['prop.nonsymmetric']}
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\ket\psi$ be a random state drawn from the Haar-random distribution, and let $\bm x$ be a random variable defined as $x_i = \vert \braket{\psi}{i}\vert^2$, for an arbitrary unitary operation $U$, where $\ket i$ is the $i$-th element of the reference basis. Then with ${\rm Dir}\left(\bm\alpha\right)$ being the Dirichlet, and $\bm 1$ being a vector where all entries are $1$.

Figures (1)

  • Figure 1: Schematic description of our hierarchical test. The goal is to test the randomness of a set of states generated by a black box with two buttons: copy and new state. In this picture, randomness is represented by circles and spheres. Our test is built on expectation values of an observable $\hat{O}$. The hierarchy of our test grows in two directions: $t$ (from $t$-designs) and complexity of the observable. An increasing number of $t$ -- polyhedra with many edges -- imposes a denser distribution of states. The sampling cost of the test increases exponentially in $t$. For the observables, $\hat{O}$ cannot verify $t$-designs due to the high dimension of quantum states. Analogously, the right projection of a cylinder from a sphere are equivalent. To compare states and $t$-designs, we need to construct increasingly more complex families of observables, represented by more sophisticated observation apparatuses. The algorithms and subroutines of each of the corresponding schemes are given in \ref{['app.algos']}: $t$-design check, \ref{['alg:tdesigntest']}, Permutation check, \ref{['alg:permtest']}, Mutually Unbiased Bases check, \ref{['alg:mubtest']}, and the combined hierarchical Haar-random discriminator, \ref{['alg:discriminator']}.

Theorems & Definitions (16)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Corollary 1
  • Example 1: Clifford circuits
  • Definition 1: $\hat{O}$-shadowed $t$-design
  • Theorem 2: Samples to falsify $\hat{O}$-shadowed $t$-designs
  • Example 2: Relevance of permutations
  • Corollary 2
  • Proposition 1: Non-symmetry of Dirichlet distribution
  • ...and 6 more