125,222 research outputs found
Quantum Property Testing
A language L has a property tester if there exists a probabilistic algorithm
that given an input x only asks a small number of bits of x and distinguishes
the cases as to whether x is in L and x has large Hamming distance from all y
in L. We define a similar notion of quantum property testing and show that
there exist languages with quantum property testers but no good classical
testers. We also show there exist languages which require a large number of
queries even for quantumly testing
New Results on Quantum Property Testing
We present several new examples of speed-ups obtainable by quantum algorithms
in the context of property testing. First, motivated by sampling algorithms, we
consider probability distributions given in the form of an oracle
. Here the probability \PP_f(j) of an outcome is the
fraction of its domain that maps to . We give quantum algorithms for
testing whether two such distributions are identical or -far in
-norm. Recently, Bravyi, Hassidim, and Harrow \cite{BHH10} showed that if
\PP_f and \PP_g are both unknown (i.e., given by oracles and ), then
this testing can be done in roughly quantum queries to the
functions. We consider the case where the second distribution is known, and
show that testing can be done with roughly quantum queries, which we
prove to be essentially optimal. In contrast, it is known that classical
testing algorithms need about queries in the unknown-unknown case and
about queries in the known-unknown case. Based on this result, we
also reduce the query complexity of graph isomorphism testers with quantum
oracle access. While those examples provide polynomial quantum speed-ups, our
third example gives a much larger improvement (constant quantum queries vs
polynomial classical queries) for the problem of testing periodicity, based on
Shor's algorithm and a modification of a classical lower bound by Lachish and
Newman \cite{lachish&newman:periodicity}. This provides an alternative to a
recent constant-vs-polynomial speed-up due to Aaronson \cite{aaronson:bqpph}.Comment: 2nd version: updated some references, in particular to Aaronson's
Fourier checking proble
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
General formulas for capacity of classical-quantum channels
The capacity of a classical-quantum channel (or in other words the classical
capacity of a quantum channel) is considered in the most general setting, where
no structural assumptions such as the stationary memoryless property are made
on a channel. A capacity formula as well as a characterization of the strong
converse property is given just in parallel with the corresponding classical
results of Verd\'{u}-Han which are based on the so-called information-spectrum
method. The general results are applied to the stationary memoryless case with
or without cost constraint on inputs, whereby a deep relation between the
channel coding theory and the hypothesis testing for two quantum states is
elucidated. no structural assumptions such as the stationary memoryless
property are made on a channel. A capacity formula as well as a
characterization of the strong converse property is given just in parallel with
the corresponding classical results of Verdu-Han which are based on the
so-called information-spectrum method. The general results are applied to the
stationary memoryless case with or without cost constraint on inputs, whereby a
deep relation between the channel coding theory and the hypothesis testing for
two quantum states is elucidated
Quantum algorithms for testing properties of distributions
Suppose one has access to oracles generating samples from two unknown
probability distributions P and Q on some N-element set. How many samples does
one need to test whether the two distributions are close or far from each other
in the L_1-norm ? This and related questions have been extensively studied
during the last years in the field of property testing. In the present paper we
study quantum algorithms for testing properties of distributions. It is shown
that the L_1-distance between P and Q can be estimated with a constant
precision using approximately N^{1/2} queries in the quantum settings, whereas
classical computers need \Omega(N) queries. We also describe quantum algorithms
for testing Uniformity and Orthogonality with query complexity O(N^{1/3}). The
classical query complexity of these problems is known to be \Omega(N^{1/2}).Comment: 20 page
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