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We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows…
We propose to analyse the statistical properties of a sequence of vectors using the spectrum of the associated Gram matrix. Such sequences arise e.g. by the repeated action of a deterministic kicked quantum dynamics on an initial condition…
Quantum metrology aims to exploit quantum phenomena to overcome classical limitations in the estimation of relevant parameters. We consider a probe undergoing a phase shift $\varphi$ whose generator is randomly sampled according to a…
Quantum state tomography is a key process in most quantum experiments. In this work, we employ quantum machine learning for state tomography. Given an unknown quantum state, it can be learned by maximizing the fidelity between the output of…
Iterative phase estimation has long been used in quantum computing to estimate Hamiltonian eigenvalues. This is done by applying many repetitions of the same fundamental simulation circuit to an initial state, and using statistical…
The numerical generation of random quantum states (RQS) is an important procedure for investigations in quantum information science. Here we review some methods that may be used for performing that task. We start by presenting a simple…
We consider the problem of phase retrieval, i.e. that of solving systems of quadratic equations. A simple variant of the randomized Kaczmarz method was recently proposed for phase retrieval, and it was shown numerically to have a…
The paper considers the phase retrieval problem in N-dimensional complex vector spaces. It provides two sets of deterministic measurement vectors which guarantee signal recovery for all signals, excluding only a specific subspace and a…
This investigation continues a program aiming at obtaining effective quantum models to describe measurement statistics. In [Stark, arXiv:1209.5737 (2012)], we have described how the Gram matrix associated to the prepared states and the…
Using the formal analysis made by Bohm in his book, {\em "Quantum theory"}, Dover Publications Inc. New York (1979), to calculate approximately the phase time for a transmitted and the reflected wave packets through a potential barrier, we…
We review and extend, in a self-contained way, the mathematical foundations of numerical simulation methods that are based on the use of random states. The power and versatility of this simulation technology is illustrated by calculations…
In the last decade, spectral linear statistics on large dimensional random matrices have attracted significant attention. Within the physics community, a privileged role has been played by invariant matrix ensembles for which a two…
In this work we have explored few tools in Quantum State Tomography for Continuous Variable Systems. The concept of quantum states in phase space representation is introduced in a simple manner by using a few statistical concepts. Unlike…
We introduce matrix quantum phase-space distributions. These extend the idea of a quantum phase-space representation via projections onto a density matrix of global symmetry variables. The method is applied to verification of low-loss…
We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $\mathbf{X}^{\star}$ from $m$ (possibly noisy) observations $Y_\mu = | \sum_{i=1}^n \Phi_{\mu i} X^{\star}_i/\sqrt{n}|$, for a large class of…
Several statistics-based detectors, based on unimodal matrix models, for determining the number of sources in a field are designed. A new variance ratio statistic is proposed, and its asymptotic distribution is analyzed. The variance ratio…
This paper researches how the systematic errors in phase inversions affect the success rate and the number of iterations in optimized quantum random-walk search algorithm. Through geometric description of this algorithm, the model of the…
The wave propagation in random medium plays a critical role in optics and quantum physics. Multiple scattering of coherent wave in a random medium determines the transport procedure. Brownian motions of the scatterers perturb each…
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields…
The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We describe quantum tomography as an inverse statistical problem in which…