19 research outputs found
Distributional Property Testing in a Quantum World
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. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling
Distributional property testing in a quantum world
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 quantum access to a process preparing the samples. Our results build on the recent technique of quantum singular value transformation, combined with more standard tricks such as divide-and-conquer. The presented approach is a natural fit for distributional property testing both in the classical and the quantum case, demonstrating the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling; for classical distributions our algorithms significantly improve the precision dependence of some earlier results
Distributional property testing in a quantum world
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. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon/von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling
Quantum Algorithms for Classical Probability Distributions
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships.
Additionally, we prove 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.
The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings
Succinct quantum testers for closeness and -wise uniformity of probability distributions
We explore potential quantum speedups for the fundamental problem of testing
the properties of closeness and -wise uniformity of probability
distributions.
\textit{Closeness testing} is the problem of distinguishing whether two
-dimensional distributions are identical or at least -far in
- or -distance. We show that the quantum query complexities for
- and -closeness testing are O\rbra{\sqrt{n}/\varepsilon} and
O\rbra{1/\varepsilon}, respectively, both of which achieve optimal dependence
on , improving the prior best results of
\hyperlink{cite.gilyen2019distributional}{Gily{\'e}n and Li~(2019)}.
\textit{-wise uniformity testing} is the problem of distinguishing whether
a distribution over \cbra{0, 1}^n is uniform when restricted to any
coordinates or -far from any such distributions. We propose the
first quantum algorithm for this problem with query complexity
O\rbra{\sqrt{n^k}/\varepsilon}, achieving a quadratic speedup over the
state-of-the-art classical algorithm with sample complexity
O\rbra{n^k/\varepsilon^2} by \hyperlink{cite.o2018closeness}{O'Donnell and
Zhao (2018)}. Moreover, when our quantum algorithm outperforms any
classical one because of the classical lower bound
\Omega\rbra{n/\varepsilon^2}.
All our quantum algorithms are fairly simple and time-efficient, using only
basic quantum subroutines such as amplitude estimation.Comment: We have added the proof of lower bounds and have polished the
languag
Sample Efficient Identity Testing and Independence Testing of Quantum States
In this paper, we study the quantum identity testing problem, i.e., testing whether two given quantum states are identical, and quantum independence testing problem, i.e., testing whether a given multipartite quantum state is in tensor product form. For the quantum identity testing problem of D(Cd) system, we provide a deterministic measurement scheme that uses O(dε22) copies via independent measurements with d being the dimension of the state and ε being the additive error. For the independence testing problem D(Cd1 ⊗ Cd2 ⊗ · · · ⊗ Cdm) system, we show that the sample complexity is Θ(~ Πmi=1ε2di) via collective measurements, and O(Πmi=1ε2d2i) via independent measurements. If randomized choice of independent measurements are allowed, the sample complexity is Θ(d3ε2/2) for the quantum identity testing problem, and Θ(~ Πmi=1ε2d3 i/2) for the quantum independence testing problem
Towards quantum advantage via topological data analysis
Even after decades of quantum computing development, examples of generally
useful quantum algorithms with exponential speedups over classical counterparts
are scarce. Recent progress in quantum algorithms for linear-algebra positioned
quantum machine learning (QML) as a potential source of such useful exponential
improvements. Yet, in an unexpected development, a recent series of
"dequantization" results has equally rapidly removed the promise of exponential
speedups for several QML algorithms. This raises the critical question whether
exponential speedups of other linear-algebraic QML algorithms persist. In this
paper, we study the quantum-algorithmic methods behind the algorithm for
topological data analysis of Lloyd, Garnerone and Zanardi through this lens. We
provide evidence that the problem solved by this algorithm is classically
intractable by showing that its natural generalization is as hard as simulating
the one clean qubit model -- which is widely believed to require
superpolynomial time on a classical computer -- and is thus very likely immune
to dequantizations. Based on this result, we provide a number of new quantum
algorithms for problems such as rank estimation and complex network analysis,
along with complexity-theoretic evidence for their classical intractability.
Furthermore, we analyze the suitability of the proposed quantum algorithms for
near-term implementations. Our results provide a number of useful applications
for full-blown, and restricted quantum computers with a guaranteed exponential
speedup over classical methods, recovering some of the potential for
linear-algebraic QML to become one of quantum computing's killer applications.Comment: 29 pages, 3 figures. New results added and improved expositio