1,108 research outputs found
Classical Optimizers for Noisy Intermediate-Scale Quantum Devices
We present a collection of optimizers tuned for usage on Noisy Intermediate-Scale Quantum (NISQ) devices. Optimizers have a range of applications in quantum computing, including the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization (QAOA) algorithms. They are also used for calibration tasks, hyperparameter tuning, in machine learning, etc. We analyze the efficiency and effectiveness of different optimizers in a VQE case study. VQE is a hybrid algorithm, with a classical minimizer step driving the next evaluation on the quantum processor. While most results to date concentrated on tuning the quantum VQE circuit, we show that, in the presence of quantum noise, the classical minimizer step needs to be carefully chosen to obtain correct results. We explore state-of-the-art gradient-free optimizers capable of handling noisy, black-box, cost functions and stress-test them using a quantum circuit simulation environment with noise injection capabilities on individual gates. Our results indicate that specifically tuned optimizers are crucial to obtaining valid science results on NISQ hardware, and will likely remain necessary even for future fault tolerant circuits
A variational method for quantitative photoacoustic tomography with piecewise constant coefficients
In this article, we consider the inverse problem of determining spatially
heterogeneous absorption and diffusion coefficients from a single measurement
of the absorbed energy (in the steady-state diffusion approximation of light
transfer). This problem, which is central in quantitative photoacoustic
tomography, is in general ill-posed since it admits an infinite number of
solution pairs. We show that when the coefficients are known to be piecewise
constant functions, a unique solution can be obtained. For the numerical
determination of the coefficients, we suggest a variational method based based
on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional,
which we implemented numerically and tested on simulated two-dimensional data
Uniaxial symmetry in nematic liquid crystals
Within the Landau-de Gennes theory of liquid crystals, we study theoretically
the equilibrium configurations with uniaxial symmetry. We show that the
uniaxial symmetry constraint is very restrictive and can in general not be
satisfied, except in very symmetric situations. For one- and two-dimensional
configurations, we characterize completely the uniaxial equilibria: they must
have constant director. In the three dimensional case we focus on the model
problem of a spherical droplet with radial anchoring, and show that any
uniaxial equilibrium must be spherically symmetric. It was known before that
uniaxiality can sometimes be broken by energy minimizers. Our results shed a
new light on this phenomenon: we prove here that in one or two dimensions
uniaxial symmetry is always broken, unless the director is constant. Moreover,
our results concern all equilibrium configurations, and not merely energy
minimizers.Comment: contains a new presentation of results in arXiv:1307.0295, and new
result
Functional Liftings of Vectorial Variational Problems with Laplacian Regularization
We propose a functional lifting-based convex relaxation of variational
problems with Laplacian-based second-order regularization. The approach rests
on ideas from the calibration method as well as from sublabel-accurate
continuous multilabeling approaches, and makes these approaches amenable for
variational problems with vectorial data and higher-order regularization, as is
common in image processing applications. We motivate the approach in the
function space setting and prove that, in the special case of absolute
Laplacian regularization, it encompasses the discretization-first
sublabel-accurate continuous multilabeling approach as a special case. We
present a mathematical connection between the lifted and original functional
and discuss possible interpretations of minimizers in the lifted function
space. Finally, we exemplarily apply the proposed approach to 2D image
registration problems.Comment: 12 pages, 3 figures; accepted at the conference "Scale Space and
Variational Methods" in Hofgeismar, Germany 201
Classical Optimizers for Noisy Intermediate-Scale Quantum Devices
We present a collection of optimizers tuned for usage on Noisy
Intermediate-Scale Quantum (NISQ) devices. Optimizers have a range of
applications in quantum computing, including the Variational Quantum
Eigensolver (VQE) and Quantum Approximate Optimization (QAOA) algorithms. They
are also used for calibration tasks, hyperparameter tuning, in machine
learning, etc. We analyze the efficiency and effectiveness of different
optimizers in a VQE case study. VQE is a hybrid algorithm, with a classical
minimizer step driving the next evaluation on the quantum processor. While most
results to date concentrated on tuning the quantum VQE circuit, we show that,
in the presence of quantum noise, the classical minimizer step needs to be
carefully chosen to obtain correct results. We explore state-of-the-art
gradient-free optimizers capable of handling noisy, black-box, cost functions
and stress-test them using a quantum circuit simulation environment with noise
injection capabilities on individual gates. Our results indicate that
specifically tuned optimizers are crucial to obtaining valid science results on
NISQ hardware, and will likely remain necessary even for future fault tolerant
circuits.Comment: 11 pages, 17 figure
- …