33,978 research outputs found
Efficient inference in the transverse field Ising model
In this paper we introduce an approximate method to solve the quantum cavity
equations for transverse field Ising models. The method relies on a projective
approximation of the exact cavity distributions of imaginary time trajectories
(paths). A key feature, novel in the context of similar algorithms, is the
explicit separation of the classical and quantum parts of the distributions.
Numerical simulations show accurate results in comparison with the sampled
solution of the cavity equations, the exact diagonalization of the Hamiltonian
(when possible) and other approximate inference methods in the literature. The
computational complexity of this new algorithm scales linearly with the
connectivity of the underlying lattice, enabling the study of highly connected
networks, as the ones often encountered in quantum machine learning problems
Fractal functions on the real projective plane
Formerly the geometry was based on shapes, but since the last centuries this
founding mathematical science deals with transformations, projections and
mappings. Projective geometry identifies a line with a single point, like the
perspective on the horizon line and, due to this fact, it requires a
restructuring of the real mathematical and numerical analysis. In particular,
the problem of interpolating data must be refocused. In this paper we define a
linear structure along with a metric on a projective space, and prove that the
space thus constructed is complete. Then we consider an iterated function
system giving rise to a fractal interpolation function of a set of data.Comment: 25 pages, 18 figure
Optimal universal quantum circuits for unitary complex conjugation
Let be a unitary operator representing an arbitrary -dimensional
unitary quantum operation. This work presents optimal quantum circuits for
transforming a number of calls of into its complex conjugate
. Our circuits admit a parallel implementation and are proven to be
optimal for any and with an average fidelity of
. Optimality is shown for
average fidelity, robustness to noise, and other standard figures of merit.
This extends previous works which considered the scenario of a single call
() of the operation , and the special case of calls. We then
show that our results encompass optimal transformations from calls of
to for any arbitrary homomorphism from the group of
-dimensional unitary operators to itself, since complex conjugation is the
only non-trivial automorphisms on the group of unitary operators. Finally, we
apply our optimal complex conjugation implementation to design a probabilistic
circuit for reversing arbitrary quantum evolutions.Comment: 19 pages, 5 figures. Improved presentation, typos corrected, and some
proofs are now clearer. Closer to the published versio
Quantum Mechanics Lecture Notes. Selected Chapters
These are extended lecture notes of the quantum mechanics course which I am
teaching in the Weizmann Institute of Science graduate physics program. They
cover the topics listed below. The first four chapter are posted here. Their
content is detailed on the next page. The other chapters are planned to be
added in the coming months.
1. Motion in External Electromagnetic Field. Gauge Fields in Quantum
Mechanics.
2. Quantum Mechanics of Electromagnetic Field
3. Photon-Matter Interactions
4. Quantization of the Schr\"odinger Field (The Second Quantization)
5. Open Systems. Density Matrix
6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation
7. Mean Field Approaches for Many Body Systems -- Fermions and Boson
A hybrid quantum algorithm to detect conical intersections
Conical intersections are topologically protected crossings between the
potential energy surfaces of a molecular Hamiltonian, known to play an
important role in chemical processes such as photoisomerization and
non-radiative relaxation. They are characterized by a non-zero Berry phase,
which is a topological invariant defined on a closed path in atomic coordinate
space, taking the value when the path encircles the intersection
manifold. In this work, we show that for real molecular Hamiltonians, the Berry
phase can be obtained by tracing a local optimum of a variational ansatz along
the chosen path and estimating the overlap between the initial and final state
with a control-free Hadamard test. Moreover, by discretizing the path into
points, we can use single Newton-Raphson steps to update our state
non-variationally. Finally, since the Berry phase can only take two discrete
values (0 or ), our procedure succeeds even for a cumulative error bounded
by a constant; this allows us to bound the total sampling cost and to readily
verify the success of the procedure. We demonstrate numerically the application
of our algorithm on small toy models of the formaldimine molecule
(\ce{H2C=NH}).Comment: 15 + 10 pages, 4 figure
Optimal high-dimensional entanglement concentration in the bipartite scenario
Considering pure quantum states, entanglement concentration is the procedure
where from copies of a partially entangled state, a single state with
higher entanglement can be obtained. Getting a maximally entangled state is
possible for . However, the associated success probability can be
extremely low while increasing the system's dimensionality. In this work, we
study two methods to achieve a probabilistic entanglement concentration for
bipartite quantum systems with a large dimensionality for , regarding a
reasonably good probability of success at the expense of having a non-maximal
entanglement. Firstly, we define an efficiency function
considering a tradeoff between the amount of entanglement (quantified by the
I-Concurrence) of the final state after the concentration procedure and its
success probability, which leads to solving a quadratic optimization problem.
We found an analytical solution, ensuring that an optimal scheme for
entanglement concentration can always be found in terms of .
Finally, a second method was explored, which is based on fixing the success
probability and searching for the maximum amount of entanglement attainable.
Both ways resemble the Procrustean method applied to a subset of the most
significant Schmidt coefficients but obtaining non-maximally entangled states.Comment: 11 pages, 4 figure
Nonstationary Fractionally Integrated Functional Time Series
We study a functional version of nonstationary fractionally integrated time series, covering the functional unit root as a special case. The time series taking values in an infinite-dimensional separable Hilbert space are projected onto a finite number of sub-spaces, the level of nonstationarity allowed to vary over them. Under regularity conditions, we derive a weak convergence result for the projection of the fractionally integrated functional process onto the asymptotically dominant sub-space, which retains most of the sample information carried by the original functional time series. Through the classic functional principal component analysis of the sample variance operator, we obtain the eigenvalues and eigenfunctions which span a sample version of the dominant sub-space. Furthermore, we introduce a simple ratio criterion to consistently estimate the dimension of the dominant sub-space, and use a semiparametric local Whittle method to estimate the memory parameter. Monte-Carlo simulation studies are given to examine the finite-sample performance of the developed techniques
Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots
We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least squares or an interpolatory approach, yielding a function-valued version of the standard rational interpolation method (V-SRI) and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the V-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the V-SRI method seems to be the best performing one
Sofic approximation sequences and sofic mean dimension
We prove that sofic mean dimension of any amenable group action does not
depend on the choice of sofic approximation sequences. Previously, this result
was known only if the acting group is an infinite amenable group; however, in
the case of a finite group action, this knowledge was restricted to
finite-dimensional compact metrizable spaces only. We also prove that sofic
mean dimension of any full shift depends purely on its alphabet. Previously,
this was shown only when the alphabet is a finite-dimensional compact
metrizable space.
Our method is a refinement of the classical technique in relation to the
estimates for mean dimension from above and below, respectively. The key point
of our results is that both of them apply to all compact metrizable spaces
without any restriction (in particular, any of the alphabets and spaces
concerned in our results is not required to be finite-dimensional).Comment: arXiv admin note: text overlap with arXiv:2108.0253
Stability of space-time isogeometric methods for wave propagation problems
This thesis aims at investigating the first steps toward an unconditionally
stable space-time isogeometric method, based on splines of maximal regularity,
for the linear acoustic wave equation. The unconditional stability of
space-time discretizations for wave propagation problems is a topic of
significant interest, by virtue of the advantages of space-time methods
compared with more standard time-stepping techniques. In the case of continuous
finite element methods, several stabilizations have been proposed. Inspired by
one of these works, we address the stability issue by studying the isogeometric
method for an ordinary differential equation closely related to the wave
equation. As a result, we provide a stabilized isogeometric method whose
effectiveness is supported by numerical tests. Motivated by these results, we
conclude by suggesting an extension of this stabilization tool to the
space-time isogeometric formulation of the acoustic wave equation.Comment: Masters thesi
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