2,612 research outputs found

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Quantum-inspired classification based on quantum state discrimination

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    We present quantum-inspired algorithms for classification tasks inspired by the problem of quantum state discrimination. By construction, these algorithms can perform multiclass classification, prevent overfitting, and generate probability outputs. While they could be implemented on a quantum computer, we focus here on classical implementations of such algorithms. The training of these classifiers involves Semi-Definite Programming. We also present a relaxation of these classifiers that utilizes Linear Programming (but that can no longer be interpreted as a quantum measurement). Additionally, we consider a classifier based on the Pretty Good Measurement (PGM) and show how to implement it using an analogue of the so-called Kernel Trick, which allows us to study its performance on any number of copies of the input state. We evaluate these classifiers on the MNIST and MNIST-1D datasets and find that the PGM generally outperforms the other quantum-inspired classifiers and performs comparably to standard classifiers.Comment: 19 pages, 4 figure

    Witnessing environment dimension through temporal correlations

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    We introduce a framework to compute upper bounds for temporal correlations achievable in open quantum system dynamics, obtained by repeated measurements on the system. As these correlations arise by virtue of the environment acting as a memory resource, such bounds are witnesses for the minimal dimension of an effective environment compatible with the observed statistics. These witnesses are derived from a hierarchy of semidefinite programs with guaranteed asymptotic convergence. We compute non-trivial bounds for various sequences involving a qubit system and a qubit environment, and compare the results to the best known quantum strategies producing the same outcome sequences. Our results provide a numerically tractable method to determine bounds on multi-time probability distributions in open quantum system dynamics and allow for the witnessing of effective environment dimensions through probing of the system alone.Comment: 24 pages, 7 figure

    Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers

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    We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and quantum circuits of shorter depths. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones. Additionally, using this method, we establish a relationship between discrete-time and continuous-time quantum walks, making inroads into a long-standing open problem.Comment: 72+16 pages, 3 Figure

    Learning and Control of Dynamical Systems

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    Despite the remarkable success of machine learning in various domains in recent years, our understanding of its fundamental limitations remains incomplete. This knowledge gap poses a grand challenge when deploying machine learning methods in critical decision-making tasks, where incorrect decisions can have catastrophic consequences. To effectively utilize these learning-based methods in such contexts, it is crucial to explicitly characterize their performance. Over the years, significant research efforts have been dedicated to learning and control of dynamical systems where the underlying dynamics are unknown or only partially known a priori, and must be inferred from collected data. However, much of these classical results have focused on asymptotic guarantees, providing limited insights into the amount of data required to achieve desired control performance while satisfying operational constraints such as safety and stability, especially in the presence of statistical noise. In this thesis, we study the statistical complexity of learning and control of unknown dynamical systems. By utilizing recent advances in statistical learning theory, high-dimensional statistics, and control theoretic tools, we aim to establish a fundamental understanding of the number of samples required to achieve desired (i) accuracy in learning the unknown dynamics, (ii) performance in the control of the underlying system, and (iii) satisfaction of the operational constraints such as safety and stability. We provide finite-sample guarantees for these objectives and propose efficient learning and control algorithms that achieve the desired performance at these statistical limits in various dynamical systems. Our investigation covers a broad range of dynamical systems, starting from fully observable linear dynamical systems to partially observable linear dynamical systems, and ultimately, nonlinear systems. We deploy our learning and control algorithms in various adaptive control tasks in real-world control systems and demonstrate their strong empirical performance along with their learning, robustness, and stability guarantees. In particular, we implement one of our proposed methods, Fourier Adaptive Learning and Control (FALCON), on an experimental aerodynamic testbed under extreme turbulent flow dynamics in a wind tunnel. The results show that FALCON achieves state-of-the-art stabilization performance and consistently outperforms conventional and other learning-based methods by at least 37%, despite using 8 times less data. The superior performance of FALCON arises from its physically and theoretically accurate modeling of the underlying nonlinear turbulent dynamics, which yields rigorous finite-sample learning and performance guarantees. These findings underscore the importance of characterizing the statistical complexity of learning and control of unknown dynamical systems.</p

    Towards Reliable and Accurate Global Structure-from-Motion

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    Reconstruction of objects or scenes from sparse point detections across multiple views is one of the most tackled problems in computer vision. Given the coordinates of 2D points tracked in multiple images, the problem consists of estimating the corresponding 3D points and cameras\u27 calibrations (intrinsic and pose), and can be solved by minimizing reprojection errors using bundle adjustment. However, given bundle adjustment\u27s nonlinear objective function and iterative nature, a good starting guess is required to converge to global minima. Global and Incremental Structure-from-Motion methods appear as ways to provide good initializations to bundle adjustment, each with different properties. While Global Structure-from-Motion has been shown to result in more accurate reconstructions compared to Incremental Structure-from-Motion, the latter has better scalability by starting with a small subset of images and sequentially adding new views, allowing reconstruction of sequences with millions of images. Additionally, both Global and Incremental Structure-from-Motion methods rely on accurate models of the scene or object, and under noisy conditions or high model uncertainty might result in poor initializations for bundle adjustment. Recently pOSE, a class of matrix factorization methods, has been proposed as an alternative to conventional Global SfM methods. These methods use VarPro - a second-order optimization method - to minimize a linear combination of an approximation of reprojection errors and a regularization term based on an affine camera model, and have been shown to converge to global minima with a high rate even when starting from random camera calibration estimations.This thesis aims at improving the reliability and accuracy of global SfM through different approaches. First, by studying conditions for global optimality of point set registration, a point cloud averaging method that can be used when (incomplete) 3D point clouds of the same scene in different coordinate systems are available. Second, by extending pOSE methods to different Structure-from-Motion problem instances, such as Non-Rigid SfM or radial distortion invariant SfM. Third and finally, by replacing the regularization term of pOSE methods with an exponential regularization on the projective depth of the 3D point estimations, resulting in a loss that achieves reconstructions with accuracy close to bundle adjustment

    Quantum Algorithms for Interpolation and Sampling

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    Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the general case, and particularly under geometric conditions we identify for the critical points of the energy function

    Fundamental and Applied Problems of the String Theory Landscape

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    In this thesis we study quantum corrections to string-derived effective actions \textit{per se} as well as their implications for phenomenologically relevant setups like the \textit{Large Volume Scenario} (LVS) and the \textit{anti-D3-brane} uplift. In the first part of this thesis, we improve the understanding of string loop corrections on general Calabi-Yau orientifolds from an effective field theory perspective by proposing a new classification scheme for quantum corrections. Thereby, we discover new features of string loop corrections, like for instance possible logarithmic effects in the Kahler and scalar potential, which are relevant for phenomenological applications like models of inflation. In the next part of the thesis, we derive a simple and explicit formula, the \textit{LVS parametric tadpole constraint} (PTC), that ensures that the anti-D3-brane uplifted LVS dS vacuum is protected against the most dangerous higher order corrections. The main difficulty appears to be the small uplifting contribution which is necessary due to the exponentially large volume obtained via the LVS. This in turn requires a large negative contribution to the tadpole which is quantified in the PTC. As the negative contribution to the tadpole is limited in weakly coupled string theories, the PTC represents a concrete challenge for the LVS. The last part of the thesis investigates the impact of α′\alpha' corrections to the brane-flux annihilation process discovered by Kachru, Pearson, and Verlinde (KPV) on which the anti-D3-brane uplift is based. We find that α′\alpha' corrections drastically alter the KPV analysis with the result that much more flux in the Klebanov-Strassler throat is required than previously assumed in order to control the leading α′\alpha' corrections on the NS5-brane. The implication for the LVS with standard anti-D3-brane uplift can again be quantified by the PTC. Incorporating this new bound significantly increases the required negative contribution to the tadpole. In addition, we uncover a new uplifting mechanism not relying on large fluxes and hence deep warped throats, thereby sidestepping the main difficulties related to the PTC
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