587 research outputs found

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

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    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

    Practical parallel self-testing of Bell states via magic rectangles

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    Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. In this work, we use the 3×n3 \times n magic rectangle games (generalizations of the magic square game) to obtain a self-test for nn Bell states where the one side needs only to measure single-qubit Pauli observables. The protocol requires small input sizes (constant for Alice and O(logn)O(\log n) bits for Bob) and is robust with robustness O(n5/2ε)O(n^{5/2} \sqrt{\varepsilon}), where ε\varepsilon is the closeness of the ideal (perfect) correlations to those observed. To achieve the desired self-test we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, generalize this strategy to the family of 3×n3 \times n magic rectangle games, and supplement these nonlocal games with extra check rounds (of single and pairs of observables).Comment: 29 pages, 6 figures; v3 minor corrections and changes in response to comment

    A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

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    International audienceWe consider blind source separation in chemical analysis focussing on the 3D fluorescence spectroscopy framework. We present an alternative method to process the Fluorescence Excitation-Emission Matrices (FEEM): first, a preprocessing is applied to eliminate the Raman and Rayleigh scattering peaks that clutter the FEEM. To improve its robustness versus possible improper settings, we suggest to associate the classical Zepp's method with a morphological image filtering technique. Then, in a second stage, the Canonical Polyadic (CP or Cande-comp/Parafac) decomposition of a nonnegative 3-way array has to be computed. In the fluorescence spectroscopy context, the constituent vectors of the loading matrices should be nonnegative (since standing for spectra and concentrations). Thus, we suggest a new NonNegative third order CP decomposition algorithm (NNCP) based on a non linear conjugate gradient optimisation algorithm with regularization terms and periodic restarts. Computer simulations performed on real experimental data are provided to enlighten the effectiveness and robustness of the whole processing chain and to validate the approach

    Parameter Estimation of Gaussian Stationary Processes using the Generalized Method of Moments

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    We consider the class of all stationary Gaussian process with explicit parametric spectral density. Under some conditions on the autocovariance function, we defined a GMM estimator that satisfies consistency and asymptotic normality, using the Breuer-Major theorem and previous results on ergodicity. This result is applied to the joint estimation of the three parameters of a stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian motion. The asymptotic normality of its GMM estimator applies for any H in (0,1) and under some restrictions on the remaining parameters. A numerical study is performed in the fOU case, to illustrate the estimator's practical performance when the number of datapoints is moderate

    A sparse-grid isogeometric solver

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    Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.Comment: updated version after revie

    Higher order influence functions and minimax estimation of nonlinear functionals

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    We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions (Robins (2004), Section 9; Li et al. (2004), Tchetgen et al. (2006), Robins et al. (2007)). Higher order influence functions are higher order U-statistics. Our theory extends the first order semiparametric theory of Bickel et al. (1993) and van der Vaart (1991) by incorporating the theory of higher order scores considered by Pfanzagl (1990), Small and McLeish (1994) and Lindsay and Waterman (1996). The theory reproduces many previous results, produces new non-n\sqrt{n} results, and opens up the ability to perform optimal non-n\sqrt{n} inference in complex high dimensional models. We present novel rate-optimal point and interval estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected n\sqrt{n} rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions described by Robins and Rotnitzky (2001) and van der Laan and Robins (2003).Comment: Published in at http://dx.doi.org/10.1214/193940307000000527 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org
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