33,978 research outputs found

    Efficient inference in the transverse field Ising model

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    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

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    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

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    Let UdU_d be a unitary operator representing an arbitrary dd-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number kk of calls of UdU_d into its complex conjugate Udˉ\bar{U_d}. Our circuits admit a parallel implementation and are proven to be optimal for any kk and dd with an average fidelity of ⟨F⟩=k+1d(d−k)\left\langle{F}\right\rangle =\frac{k+1}{d(d-k)}. 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 (k=1k=1) of the operation UdU_d, and the special case of k=d−1k=d-1 calls. We then show that our results encompass optimal transformations from kk calls of UdU_d to f(Ud)f(U_d) for any arbitrary homomorphism ff from the group of dd-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

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    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

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    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 π\pi 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 NN points, we can use NN single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or π\pi), 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

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    Considering pure quantum states, entanglement concentration is the procedure where from NN copies of a partially entangled state, a single state with higher entanglement can be obtained. Getting a maximally entangled state is possible for N=1N=1. 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 N=1N=1, regarding a reasonably good probability of success at the expense of having a non-maximal entanglement. Firstly, we define an efficiency function Q\mathcal{Q} 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 Q\mathcal{Q}. 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

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    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

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    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

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    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

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    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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