11 research outputs found
Entropic Measure for Localized Energy Configurations: Kinks, Bounces, and Bubbles
We construct a configurational entropy measure in functional space. We apply
it to several nonlinear scalar field models featuring solutions with
spatially-localized energy, including solitons and bounces in one spatial
dimension, and critical bubbles in three spatial dimensions, typical of
first-order phase transitions. Such field models are of widespread interest in
many areas of physics, from high energy and cosmology to condensed matter.
Using a variational approach, we show that the higher the energy of a trial
function that approximates the actual solution, the higher its relative
configurational entropy, defined as the absolute difference between the
configurational entropy of the actual solution and of the trial function.
Furthermore, we show that when different trial functions have degenerate
energies, the configurational entropy can be used to select the best fit to the
actual solution. The configurational entropy relates the dynamical and
informational content of physical models with localized energy configurations.Comment: 5 pages, 2 PDF figures, uses RevTex4. v2: Modified the normalization
condition in the entropy calculation to be more general and the figures were
replaced to reflect that. Additional comments were added for clarity. v3:
Minor rewordings, version to be published in Physics Letters
Derivative Pricing using Quantum Signal Processing
Pricing financial derivatives on quantum computers typically includes quantum
arithmetic components which contribute heavily to the quantum resources
required by the corresponding circuits. In this manuscript, we introduce a
method based on Quantum Signal Processing (QSP) to encode financial derivative
payoffs directly into quantum amplitudes, alleviating the quantum circuits from
the burden of costly quantum arithmetic. Compared to current state-of-the-art
approaches in the literature, we find that for derivative contracts of
practical interest, the application of QSP significantly reduces the required
resources across all metrics considered, most notably the total number of
T-gates by x and the number of logical qubits by x.
Additionally, we estimate that the logical clock rate needed for quantum
advantage is also reduced by a factor of x. Overall, we find that
quantum advantage will require k logical qubits, and quantum devices that
can execute T-gates at a rate of MHz. While in this work we focus
specifically on the payoff component of the derivative pricing process where
the method we present is most readily applicable, similar techniques can be
employed to further reduce the resources in other applications, such as state
preparation
Generation of Coherent Structures After Cosmic Inflation
We investigate the nonlinear dynamics of hybrid inflation models, which are
characterized by two real scalar fields interacting quadratically. We start by
solving numerically the coupled Klein-Gordon equations in static Minkowski
spacetime, searching for possible coherent structures. We find long-lived,
localized configurations, which we identify as a new kind of oscillon. We
demonstrate that these two-field oscillons allow for "excited" states with much
longer lifetimes than those found in previous studies of single-field
oscillons. We then solve the coupled field equations in an expanding
Friedmann-Robertson-Walker spacetime, finding that as the field responsible for
inflating the Universe rolls down to oscillate about its minimum, it triggers
the formation of long-lived two-field oscillons, which can contribute up to 20%
of the total energy density of the Universe. We show that these oscillons
emerge for a wide range of parameters consistent with WMAP 7-year data. These
objects contain total energy of about 25*10^20 GeV, localized in a region of
approximate radius 6*10^-26 cm. We argue that these structures could have
played a key role during the reheating of the Universe.Comment: 12 pages, 10 .pdf figures, uses RevTex4; v2: expanded discussion in
section IV, accepted for publication in Phys.Rev. D. Results remain the sam
Option Pricing using Quantum Computers
We present a methodology to price options and portfolios of options on a
gate-based quantum computer using amplitude estimation, an algorithm which
provides a quadratic speedup compared to classical Monte Carlo methods. The
options that we cover include vanilla options, multi-asset options and
path-dependent options such as barrier options. We put an emphasis on the
implementation of the quantum circuits required to build the input states and
operators needed by amplitude estimation to price the different option types.
Additionally, we show simulation results to highlight how the circuits that we
implement price the different option contracts. Finally, we examine the
performance of option pricing circuits on quantum hardware using the IBM Q
Tokyo quantum device. We employ a simple, yet effective, error mitigation
scheme that allows us to significantly reduce the errors arising from noisy
two-qubit gates.Comment: Fixed a typo. This article has been accepted in Quantu
Long-Lived Time-Dependent Remnants During Cosmological Symmetry Breaking: From Inflation to the Electroweak Scale
Through a detailed numerical investigation in three spatial dimensions, we
demonstrate that long-lived time-dependent field configurations emerge
dynamically during symmetry breaking in an expanding de Sitter spacetime. We
investigate two situations: a single scalar field with a double-well potential
and the bosonic sector of an SU(2) non-Abelian Higgs model. For the single
scalar, we show that large-amplitude oscillon configurations emerge
spontaneously and persist to contribute about 1.2% of the energy density of the
universe. We also show that for a range of parameters, oscillon lifetimes are
enhanced by the expansion and that this effect is a result of parametric
resonance. For the SU(2) case, we see about 4% of the final energy density in
oscillons.Comment: 10 pages, RevTex4, 6 figures; v2: expanded SU(2) model section, added
2 figures, added one section, improved overall presentation and updated
references, accepted for publication in Phys. Rev. D. Results remain the sam
Information Content of Spontaneous Symmetry Breaking
We propose a measure of order in the context of nonequilibrium field theory
and argue that this measure, which we call relative configurational entropy
(RCE), may be used to quantify the emergence of coherent low-entropy
configurations, such as time-dependent or time-independent topological and
nontopological spatially-extended structures. As an illustration, we
investigate the nonequilibrium dynamics of spontaneous symmetry-breaking in
three spatial dimensions. In particular, we focus on a model where a real
scalar field, prepared initially in a symmetric thermal state, is quenched to a
broken-symmetric state. For a certain range of initial temperatures,
spatially-localized, long-lived structures known as oscillons emerge in
synchrony and remain until the field reaches equilibrium again. We show that
the RCE correlates with the number-density of oscillons, thus offering a
quantitative measure of the emergence of nonperturbative spatiotemporal
patterns that can be generalized to a variety of physical systems.Comment: LaTeX, 9 pages, 5 figures, 1 tabl
End-to-end resource analysis for quantum interior point methods and portfolio optimization
We study quantum interior point methods (QIPMs) for second-order cone
programming (SOCP), guided by the example use case of portfolio optimization
(PO). We provide a complete quantum circuit-level description of the algorithm
from problem input to problem output, making several improvements to the
implementation of the QIPM. We report the number of logical qubits and the
quantity/depth of non-Clifford T-gates needed to run the algorithm, including
constant factors. The resource counts we find depend on instance-specific
parameters, such as the condition number of certain linear systems within the
problem. To determine the size of these parameters, we perform numerical
simulations of small PO instances, which lead to concrete resource estimates
for the PO use case. Our numerical results do not probe large enough instance
sizes to make conclusive statements about the asymptotic scaling of the
algorithm. However, already at small instance sizes, our analysis suggests
that, due primarily to large constant pre-factors, poorly conditioned linear
systems, and a fundamental reliance on costly quantum state tomography,
fundamental improvements to the QIPM are required for it to lead to practical
quantum advantage.Comment: 38 pages, 15 figure
Quantum Resources Required to Block-Encode a Matrix of Classical Data
We provide a modular circuit-level implementation and resource estimates for several methods of block-encoding a dense matrix of classical data to precision ; the minimal-depth method achieves a -depth of while the minimal-count method achieves a -count of . We examine resource tradeoffs between the different approaches, and we explore implementations of two separate models of quantum random access memory. As a part of this analysis, we provide a novel state preparation routine with -depth , improving on previous constructions with scaling . Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms