1,133 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Sequence of penalties method to study excited states using VQE
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
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 , where is the problem dimension and
is the number of iterations. In particular, in the regime where , our method matches the optimal rate of by
Nesterov's accelerated gradient (NAG). Moreover, in the the regime where , it outperforms NAG and converges at a faster rate of
. 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
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 ,
whose frequencies correspond to Hamiltonian energy differences with
Heisenberg-limited precision. We provide strong analytical guarantees that (a)
quantum resources scale as , while the classical computational
complexity is linear , (b) the signal-to-noise ratio increases with the
number of analysed signals as , 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 CH 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
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
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 with
nonzero entries, computes an -optimal
diagonal preconditioner in time , where is the
optimal condition number of the rescaled matrix. We give an algorithm which,
given that is either the pseudoinverse
of a graph Laplacian matrix or a constant spectral approximation of one, solves
linear systems in in time. Our diagonal
preconditioning results improve state-of-the-art runtimes of
attained by general-purpose semidefinite programming, and our solvers improve
state-of-the-art runtimes of where 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
We consider the problem of approximating the von Neumann entropy of a large,
sparse, symmetric positive semidefinite matrix , defined as
where . 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
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?
Sketch-and-precondition techniques are popular for solving large least
squares (LS) problems of the form with and
. This is where is ``sketched" to a smaller matrix with
for some constant before an
iterative LS solver computes the solution to with a right preconditioner
, where is constructed from . 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 with
when accuracy is a concern. Provided the condition number of 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
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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