1,609,290 research outputs found
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
Quantum Theory of Probability and Decisions
The probabilistic predictions of quantum theory are conventionally obtained
from a special probabilistic axiom. But that is unnecessary because all the
practical consequences of such predictions follow from the remaining,
non-probabilistic, axioms of quantum theory, together with the
non-probabilistic part of classical decision theory
Quantumlike Chaos in the Frequency Distributions of the Bases A, C, G, T in Drosophila DNA
Continuous periodogram power spectral analyses of fractal fluctuations of
frequency distributions of bases A, C, G, T in Drosophila DNA show that the
power spectra follow the universal inverse power-law form of the statistical
normal distribution. Inverse power-law form for power spectra of space-time
fluctuations is generic to dynamical systems in nature and is identified as
self-organized criticality. The author has developed a general systems theory,
which provides universal quantification for observed self-organized criticality
in terms of the statistical normal distribution. The long-range correlations
intrinsic to self-organized criticality in macro-scale dynamical systems are a
signature of quantumlike chaos. The fractal fluctuations self-organize to form
an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling
pattern for the internal structure. Power spectral analysis resolves such a
spiral trajectory as an eddy continuum with embedded dominant wavebands. The
dominant peak periodicities are functions of the golden mean. The observed
fractal frequency distributions of the Drosophila DNA base sequences exhibit
quasicrystalline structure with long-range spatial correlations or
self-organized criticality. Modification of the DNA base sequence structure at
any location may have significant noticeable effects on the function of the DNA
molecule as a whole. The presence of non-coding introns may not be redundant,
but serve to organize the effective functioning of the coding exons in the DNA
molecule as a complete unit.Comment: 46 pages, 9 figure
Quantum Weak Coin Flipping
We investigate weak coin flipping, a fundamental cryptographic primitive
where two distrustful parties need to remotely establish a shared random bit. A
cheating player can try to bias the output bit towards a preferred value. For
weak coin flipping the players have known opposite preferred values. A weak
coin-flipping protocol has a bias if neither player can force the
outcome towards their preferred value with probability more than
. While it is known that all classical protocols have
, Mochon showed in 2007 [arXiv:0711.4114] that quantumly
weak coin flipping can be achieved with arbitrarily small bias (near perfect)
but the best known explicit protocol has bias (also due to Mochon, 2005
[Phys. Rev. A 72, 022341]). We propose a framework to construct new explicit
protocols achieving biases below . In particular, we construct explicit
unitaries for protocols with bias approaching . To go below, we introduce
what we call the Elliptic Monotone Align (EMA) algorithm which, together with
the framework, allows us to numerically construct protocols with arbitrarily
small biases.Comment: 98 pages split into 3 parts, 10 figures; For updates and contact
information see https://atulsingharora.github.io/WCF. Version 2 has minor
improvements. arXiv admin note: text overlap with arXiv:1402.7166 by other
author
Thermal Conductivity and Thermal Rectification in Graphene Nanoribbons: a Molecular Dynamics Study
We have used molecular dynamics to calculate the thermal conductivity of
symmetric and asymmetric graphene nanoribbons (GNRs) of several nanometers in
size (up to ~4 nm wide and ~10 nm long). For symmetric nanoribbons, the
calculated thermal conductivity (e.g. ~2000 W/m-K @400K for a 1.5 nm {\times}
5.7 nm zigzag GNR) is on the similar order of magnitude of the experimentally
measured value for graphene. We have investigated the effects of edge chirality
and found that nanoribbons with zigzag edges have appreciably larger thermal
conductivity than nanoribbons with armchair edges. For asymmetric nanoribbons,
we have found significant thermal rectification. Among various
triangularly-shaped GNRs we investigated, the GNR with armchair bottom edge and
a vertex angle of 30{\deg} gives the maximal thermal rectification. We also
studied the effect of defects and found that vacancies and edge roughness in
the nanoribbons can significantly decrease the thermal conductivity. However,
substantial thermal rectification is observed even in the presence of edge
roughness.Comment: 13 pages, 5 figures, slightly expanded from the published version on
Nano Lett. with some additional note
Dilaton Gravity with a Non-minmally Coupled Scalar Field
We discuss the two-dimensional dilaton gravity with a scalar field as the
source matter. The coupling between the gravity and the scalar, massless, field
is presented in an unusual form. We work out two examples of these couplings
and solutions with black-hole behaviour are discussed and compared with those
found in the literature
Quantum parallel dense coding of optical images
We propose quantum dense coding protocol for optical images. This protocol
extends the earlier proposed dense coding scheme for continuous variables
[S.L.Braunstein and H.J.Kimble, Phys.Rev.A 61, 042302 (2000)] to an essentially
multimode in space and time optical quantum communication channel. This new
scheme allows, in particular, for parallel dense coding of non-stationary
optical images. Similar to some other quantum dense coding protocols, our
scheme exploits the possibility of sending a classical message through only one
of the two entangled spatially-multimode beams, using the other one as a
reference system. We evaluate the Shannon mutual information for our protocol
and find that it is superior to the standard quantum limit. Finally, we show
how to optimize the performance of our scheme as a function of the
spatio-temporal parameters of the multimode entangled light and of the input
images.Comment: 15 pages, 4 figures, RevTeX4. Submitted to the Special Issue on
Quantum Imaging in Journal of Modern Optic
Quantum Walks, Quantum Gates and Quantum Computers
The physics of quantum walks on graphs is formulated in Hamiltonian language,
both for simple quantum walks and for composite walks, where extra discrete
degrees of freedom live at each node of the graph. It is shown how to map
between quantum walk Hamiltonians and Hamiltonians for qubit systems and
quantum circuits; this is done for both a single- and multi-excitation coding,
and for more general mappings. Specific examples of spin chains, as well as
static and dynamic systems of qubits, are mapped to quantum walks, and walks on
hyperlattices and hypercubes are mapped to various gate systems. We also show
how to map a quantum circuit performing the quantum Fourier transform, the key
element of Shor's algorithm, to a quantum walk system doing the same. The
results herein are an essential preliminary to a Hamiltonian formulation of
quantum walks in which coupling to a dynamic quantum environment is included.Comment: 17 pages, 10 figure
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