18,528 research outputs found
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
Classical aspects of Hawking radiation verified in analogue gravity experiment
There is an analogy between the propagation of fields on a curved spacetime
and shallow water waves in an open channel flow. By placing a streamlined
obstacle into an open channel flow we create a region of high velocity over the
obstacle that can include wave horizons. Long (shallow water) waves propagating
upstream towards this region are blocked and converted into short (deep water)
waves. This is the analogue of the stimulated Hawking emission by a white hole
(the time inverse of a black hole). The measurements of amplitudes of the
converted waves demonstrate that they appear in pairs and are classically
correlated; the spectra of the conversion process is described by a
Boltzmann-distribution; and the Boltzmann-distribution is determined by the
determined by the change in flow across the white hole horizon.Comment: 17 pages, 10 figures; draft of a chapter submitted to the proceedings
of the IX'th SIGRAV graduate school: Analogue Gravity, Lake Como, Italy, May
201
Exponential Quantum Speed-ups are Generic
A central problem in quantum computation is to understand which quantum
circuits are useful for exponential speed-ups over classical computation. We
address this question in the setting of query complexity and show that for
almost any sufficiently long quantum circuit one can construct a black-box
problem which is solved by the circuit with a constant number of quantum
queries, but which requires exponentially many classical queries, even if the
classical machine has the ability to postselect.
We prove the result in two steps. In the first, we show that almost any
element of an approximate unitary 3-design is useful to solve a certain
black-box problem efficiently. The problem is based on a recent oracle
construction of Aaronson and gives an exponential separation between quantum
and classical bounded-error with postselection query complexities.
In the second step, which may be of independent interest, we prove that
linear-sized random quantum circuits give an approximate unitary 3-design. The
key ingredient in the proof is a technique from quantum many-body theory to
lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction
Measurement of stimulated Hawking emission in an analogue system
There is a mathematical analogy between the propagation of fields in a
general relativistic space-time and long (shallow water) surface waves on
moving water. Hawking argued that black holes emit thermal radiation via a
quantum spontaneous emission. Similar arguments predict the same effect near
wave horizons in fluid flow. By placing a streamlined obstacle into an open
channel flow we create a region of high velocity over the obstacle that can
include wave horizons. Long waves propagating upstream towards this region are
blocked and converted into short (deep water) waves. This is the analogue of
the stimulated emission by a white hole (the time inverse of a black hole), and
our measurements of the amplitudes of the converted waves demonstrate the
thermal nature of the conversion process for this system. Given the close
relationship between stimulated and spontaneous emission, our findings attest
to the generality of the Hawking process.Comment: 7 pages, 5 figures. This version corrects a processing error in the
final graph 5b which multiplied the vertical axis by 2. The graph, and the
data used from it, have been corrected. Some minor typos have also been
corrected. This version also uses TeX rather than Wor
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Quantum computing is powerful because unitary operators describing the
time-evolution of a quantum system have exponential size in terms of the number
of qubits present in the system. We develop a new "Singular value
transformation" algorithm capable of harnessing this exponential advantage,
that can apply polynomial transformations to the singular values of a block of
a unitary, generalizing the optimal Hamiltonian simulation results of Low and
Chuang. The proposed quantum circuits have a very simple structure, often give
rise to optimal algorithms and have appealing constant factors, while usually
only use a constant number of ancilla qubits. We show that singular value
transformation leads to novel algorithms. We give an efficient solution to a
certain "non-commutative" measurement problem and propose a new method for
singular value estimation. We also show how to exponentially improve the
complexity of implementing fractional queries to unitaries with a gapped
spectrum. Finally, as a quantum machine learning application we show how to
efficiently implement principal component regression. "Singular value
transformation" is conceptually simple and efficient, and leads to a unified
framework of quantum algorithms incorporating a variety of quantum speed-ups.
We illustrate this by showing how it generalizes a number of prominent quantum
algorithms, including: optimal Hamiltonian simulation, implementing the
Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude
amplification, robust oblivious amplitude amplification, fast QMA
amplification, fast quantum OR lemma, certain quantum walk results and several
quantum machine learning algorithms. In order to exploit the strengths of the
presented method it is useful to know its limitations too, therefore we also
prove a lower bound on the efficiency of singular value transformation, which
often gives optimal bounds.Comment: 67 pages, 1 figur
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
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