# Distributional property testing in a quantum world

@inproceedings{Gilyn2020DistributionalPT, title={Distributional property testing in a quantum world}, author={Andr{\'a}s Gily{\'e}n and Tongyang Li}, booktitle={ITCS}, year={2020} }

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. In particular, we give fast quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. The distributions can be either classical or quantum, however our quantum algorithms require coherent…

## 13 Citations

Quantum Identity Testing: A Streaming Algorithm and Applications

- Physics
- 2020

Do one or more unknown quantum states exhibit a particular property or are they ǫ-far from having that property in l1 distance? This is an important question in quantum computing, formulated as a…

Sublinear quantum algorithms for estimating von Neumann entropy

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2021

The problem of obtaining estimates to within a multiplicative factor γ > 1 of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states is studied, and it is proved that no polynomial query algorithm can multiply the entropy of distributions with arbitrarily low entropy.

Quantum Closeness Testing: A Streaming Algorithm and Applications

- Computer Science
- 2019

A novel $\ell_2$ norm connection between quantum property testing problems and the corresponding distribution testing problems is established and this connection opens up the potential to derive efficient testing algorithms using techniques developed for classical property testing.

Near-optimal Quantum algorithms for multivariate mean estimation

- Mathematics, Computer ScienceSTOC
- 2022

It is shown that, outside this low-precision regime, there does exist a quantum estimator that outperforms any classical estimator, and it is proved that the approximation error can be decreased by a factor of about the square root of the ratio between the dimension and the sample complexity.

Towards quantum advantage for topological data analysis

- Computer ScienceArXiv
- 2020

Evidence is provided that certain crucial steps in this quantum algorithm for topological data analysis solve problems that are classically intractable by closely relating them to the one clean qubit model, a restricted model of quantum computation whose power is strongly believed to lie beyond that of classical computation.

Quantum algorithm for estimating
α
-Renyi entropies of quantum states

- Computer SciencePhysical Review A
- 2021

We describe a quantum algorithm to estimate the alpha-Renyi entropy of an unknown d-dimensional density matrix, for alpha not equal to 1, by combining the recent technique of quantum singular value…

Sample Efficient Identity Testing and Independence Testing of Quantum States

- MathematicsITCS
- 2021

This paper provides a deterministic measurement scheme that uses O( d 2 2 ) copies via independent measurements with d being the dimension of the state and being the additive error for the quantum identity testing problem of D(C) system.

Quantum Algorithms for Classical Probability Distributions

- Computer Science, MathematicsESA
- 2019

It is proved that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions.

Multivariate trace estimation in constant quantum depth

- Computer ScienceArXiv
- 2022

This work constructs a constant quantum-depth circuit for the task of multivariate trace estimation, inspired by the method of Shor error correction, and instantiates the latter application with a theorem on estimating nonlinear functions of quantum states with “well-behaved” polynomial approximations.

Quantum Gram-Schmidt processes and their application to efficient state readout for quantum algorithms

- PhysicsPhysical Review Research
- 2021

Many quantum algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their ﬁnal classical description. The additional step to decode…

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