1,133 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Sequence of penalties method to study excited states using VQE

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    We propose an extension of the Variational Quantum Eigensolver (VQE) that leads to more accurate energy estimations and can be used to study excited states. The method is based on the introduction of a sequence of increasing penalties in the cost function. This approach does not require circuit modifications and thus can be applied with no additional depth cost. Through numerical simulations, we show that we are able to produce variational states with desired physical properties, such as total spin and charge. We assess its performance both on classical simulators and on currently available quantum devices, calculating the potential energy curves of small molecular systems in different physical configurations. Finally, we compare our method to the original VQE and to another extension, obtaining a better agreement with exact simulations for both energy and targeted physical quantities.Comment: 11 pages, 9 figures, accepted in IOP Quantum Science and Technolog

    Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization

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    In this paper, we propose an accelerated quasi-Newton proximal extragradient (A-QPNE) method for solving unconstrained smooth convex optimization problems. With access only to the gradients of the objective, we prove that our method can achieve a convergence rate of O(min{1k2,dlogkk2.5}){O}\bigl(\min\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr), where dd is the problem dimension and kk is the number of iterations. In particular, in the regime where k=O(d)k = {O}(d), our method matches the optimal rate of O(1k2){O}(\frac{1}{k^2}) by Nesterov's accelerated gradient (NAG). Moreover, in the the regime where k=Ω(dlogd)k = \Omega(d \log d), it outperforms NAG and converges at a faster rate of O(dlogkk2.5){O}\bigl(\frac{\sqrt{d\log k}}{k^{2.5}}\bigr). To the best of our knowledge, this result is the first to demonstrate a provable gain of a quasi-Newton-type method over NAG in the convex setting. To achieve such results, we build our method on a recent variant of the Monteiro-Svaiter acceleration framework and adopt an online learning perspective to update the Hessian approximation matrices, in which we relate the convergence rate of our method to the dynamic regret of a specific online convex optimization problem in the space of matrices.Comment: 44 pages, 1 figur

    Algorithmic Shadow Spectroscopy

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    We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by post-processing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals No108N_o\propto 10^8, whose frequencies correspond to Hamiltonian energy differences with Heisenberg-limited precision. We provide strong analytical guarantees that (a) quantum resources scale as O(logNo)O(\log N_o), while the classical computational complexity is linear O(No)O(N_o), (b) the signal-to-noise ratio increases with the number of analysed signals as No\propto \sqrt{N_o}, and (c) peak frequencies are immune to reasonable levels of noise. Moreover, performing shadow spectroscopy to probe model spin systems and the excited state conical intersection of molecular CH2_2 in simulation verifies that the approach is intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses orders of magnitude fewer number of shots than typical near-term quantum algorithms -- as low as 10 shots per timestep is sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily-available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation.Comment: 31 pages, 13 figures, new results with hardware and figure

    Artificial intelligence in histopathology image analysis for cancer precision medicine

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    In recent years, there have been rapid advancements in the field of computational pathology. This has been enabled through the adoption of digital pathology workflows that generate digital images of histopathological slides, the publication of large data sets of these images and improvements in computing infrastructure. Objectives in computational pathology can be subdivided into two categories, first the automation of routine workflows that would otherwise be performed by pathologists and second the addition of novel capabilities. This thesis focuses on the development, application, and evaluation of methods in this second category, specifically the prediction of gene expression from pathology images and the registration of pathology images among each other. In Study I, we developed a computationally efficient cluster-based technique to perform transcriptome-wide predictions of gene expression in prostate cancer from H&E-stained whole-slide-images (WSIs). The suggested method outperforms several baseline methods and is non-inferior to single-gene CNN predictions, while reducing the computational cost with a factor of approximately 300. We included 15,586 transcripts that encode proteins in the analysis and predicted their expression with different modelling approaches from the WSIs. In a cross-validation, 6,618 of these predictions were significantly associated with the RNA-seq expression estimates with FDR-adjusted p-values <0.001. Upon validation of these 6,618 expression predictions in a held-out test set, the association could be confirmed for 5,419 (81.9%). Furthermore, we demonstrated that it is feasible to predict the prognostic cell-cycle progression score with a Spearman correlation to the RNA-seq score of 0.527 [0.357, 0.665]. The objective of Study II is the investigation of attention layers in the context of multiple-instance-learning for regression tasks, exemplified by a simulation study and gene expression prediction. We find that for gene expression prediction, the compared methods are not distinguishable regarding their performance, which indicates that attention mechanisms may not be superior to weakly supervised learning in this context. Study III describes the results of the ACROBAT 2022 WSI registration challenge, which we organised in conjunction with the MICCAI 2022 conference. Participating teams were ranked on the median 90th percentile of distances between registered and annotated target landmarks. Median 90th percentiles for eight teams that were eligible for ranking in the test set consisting of 303 WSI pairs ranged from 60.1 µm to 15,938.0 µm. The best performing method therefore has a score slightly below the median 90th percentile of distances between first and second annotator of 67.0 µm. Study IV describes the data set that we published to facilitate the ACROBAT challenge. The data set is available publicly through the Swedish National Data Service SND and consists of 4,212 WSIs from 1,153 breast cancer patients. Study V is an example of the application of WSI registration for computational pathology. In this study, we investigate the possibility to register invasive cancer annotations from H&E to KI67 WSIs and then subsequently train cancer detection models. To this end, we compare the performance of models optimised with registered annotations to the performance of models that were optimised with annotations generated for the KI67 WSIs. The data set consists of 272 female breast cancer cases, including an internal test set of 54 cases. We find that in this test set, the performance of both models is not distinguishable regarding performance, while there are small differences in model calibration

    Structured Semidefinite Programming for Recovering Structured Preconditioners

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    We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite KRd×d\mathbf{K} \in \mathbb{R}^{d \times d} with nnz(K)\mathrm{nnz}(\mathbf{K}) nonzero entries, computes an ϵ\epsilon-optimal diagonal preconditioner in time O~(nnz(K)poly(κ,ϵ1))\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1})), where κ\kappa^\star is the optimal condition number of the rescaled matrix. We give an algorithm which, given MRd×d\mathbf{M} \in \mathbb{R}^{d \times d} that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in M\mathbf{M} in O~(d2)\widetilde{O}(d^2) time. Our diagonal preconditioning results improve state-of-the-art runtimes of Ω(d3.5)\Omega(d^{3.5}) attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of Ω(dω)\Omega(d^{\omega}) where ω>2.3\omega > 2.3 is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.Comment: Merge of arXiv:1812.06295 and arXiv:2008.0172

    Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

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    We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix AA, defined as tr(f(A))\operatorname{tr}(f(A)) where f(x)=xlogxf(x)=-x\log x. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.Comment: 32 pages, 10 figure

    Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

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    We propose a natural application of Quantum Linear Systems Problem (QLSP) solvers such as the HHL algorithm to efficiently prepare highly excited interior eigenstates of physical Hamiltonians in a variational manner. This is enabled by the efficient computation of inverse expectation values, taking advantage of the QLSP solvers' exponentially better scaling in problem size without concealing exponentially costly pre/post-processing steps that usually accompanies it. We detail implementations of this scheme for both fault-tolerant and near-term quantum computers, analyse their efficiency and implementability, and discuss applications and simulation results in many-body physics and quantum chemistry that demonstrate its superior effectiveness and scalability over existing approaches.Comment: 16 pages, 6 figure

    Are sketch-and-precondition least squares solvers numerically stable?

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    Sketch-and-precondition techniques are popular for solving large least squares (LS) problems of the form Ax=bAx=b with ARm×nA\in\mathbb{R}^{m\times n} and mnm\gg n. This is where AA is ``sketched" to a smaller matrix SASA with SRcn×mS\in\mathbb{R}^{\lceil cn\rceil\times m} for some constant c>1c>1 before an iterative LS solver computes the solution to Ax=bAx=b with a right preconditioner PP, where PP is constructed from SASA. Popular sketch-and-precondition LS solvers are Blendenpik and LSRN. We show that the sketch-and-precondition technique is not numerically stable for ill-conditioned LS problems. Instead, we propose using an unpreconditioned iterative LS solver on (AP)y=b(AP)y=b with x=Pyx=Py when accuracy is a concern. Provided the condition number of AA is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a comparable backward error to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to provide a convincing argument that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems.Comment: 22 page

    Extending Kernel PCA through Dualization: Sparsity, Robustness and Fast Algorithms

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    The goal of this paper is to revisit Kernel Principal Component Analysis (KPCA) through dualization of a difference of convex functions. This allows to naturally extend KPCA to multiple objective functions and leads to efficient gradient-based algorithms avoiding the expensive SVD of the Gram matrix. Particularly, we consider objective functions that can be written as Moreau envelopes, demonstrating how to promote robustness and sparsity within the same framework. The proposed method is evaluated on synthetic and real-world benchmarks, showing significant speedup in KPCA training time as well as highlighting the benefits in terms of robustness and sparsity.Comment: 15 pages, ICML 202
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