Quantum computation and analysis of Wigner and Husimi functions: toward a quantum image treatment

We study the efficiency of quantum algorithms which aim at obtaining phase space distribution functions of quantum systems. Wigner and Husimi functions are considered. Different quantum algorithms are envisioned to build these functions, and compared with the classical computation. Different procedures to extract more efficiently information from the final wave function of these algorithms are studied, including coarse-grained measurements, amplitude amplification and measure of wavelet-transformed wave function. The algorithms are analyzed and numerically tested on a complex quantum system showing different behavior depending on parameters, namely the kicked rotator. The results for the Wigner function show in particular that the use of the quantum wavelet transform gives a polynomial gain over classical computation. For the Husimi distribution, the gain is much larger than for the Wigner function, and is bigger with the help of amplitude amplification and wavelet transforms. We also apply the same set of techniques to the analysis of real images. The results show that the use of the quantum wavelet transform allows to lower dramatically the number of measurements needed, but at the cost of a large loss of information.Comment: Revtex, 13 pages, 16 figure

Tensor Network States: Optimizations and Applications in Quantum Many-Body Physics and Machine Learning

Tensor network states are ubiquitous in the investigation of quantum many-body (QMB) physics. Their advantage over other state representations is evident from their reduction in the computational complexity required to obtain various quantities of interest, namely observables. Additionally, they provide a natural platform for investigating entanglement properties within a system. In this dissertation, we develop various novel algorithms and optimizations to tensor networks for the investigation of QMB systems, including classical and quantum circuits. Specifically, we study optimizations for the two-dimensional Ising model in a transverse field, we create an algorithm for the $k$-SAT problem, and we study the entanglement properties of random unitary circuits. In addition to these applications, we reinterpret renormalization group principles from QMB physics in the context of machine learning to develop a novel algorithm for the tasks of classification and regression, and then utilize machine learning architectures for the time evolution of operators in QMB systems

Coarse graining equations for flow in porous media: a HaarWavelets and renormalization approach

Coarse graining of equations for flow in porous media is an important aspect in modelling permeable subsurface geological systems. In the study of hydrocarbon reservoirs as well as in hydrology, there is a need for reducing the size of the numerical models to make them computationally efficient, while preserving all the relevant information which is given at different scales. In the first part, a new renormalization method for upscaling permeability in Darcy’s equation based on Haar wavelets is presented, which differs from other wavelet based methods. The pressure field is expressed as a set of averages and differences, using a one level Haar wavelet transform matrix. Applying this transform to the finite difference discretized form of Darcy’s law, one can deduce which permeability values on the coarse scale would give rise to the average pressure field. Numerical simulations were performed to test this technique on homogeneous and heterogeneous systems. A generalization of the above method was developed designing a hierarchical transform matrix inspired by a full Haar wavelet transform, which allows us to describe pressure as an average and a set of progressively smaller scale differences. Using this transform the pressure solution can be performed at the required level of detail, allowing for different resolutions to be kept in different parts of the system. A natural extension of the methods is the application to two-phase flow. Upscaling mobility allows the saturation profile to be calculated on the fine or coarse scale while based on coarse pressure values. To conclude, an alternative approach to upscaling in multi-phase flow is to upscale the saturation equation itself. Taking its Laplace transform, this equation can be reduced to a simple eigenvalue problem. The wavelet upscaling method can now be applied to calculate the upscaled saturation profile, starting with fine scale velocity data

Towards hybrid molecular simulations

In many biology, chemistry and physics applications molecular simulations can be used to study material and process properties. The level of detail needed in such simulations depends on the application. In some cases quantum mechanical simulations are indispensable. However, traditional ab-initio methods, usually employing plane waves or a linear combination of atomic orbitals as a basis, are extremely expensive in terms of computational as well as memory requirements. The well-known fact that electronic wave functions vary much more rapidly near the atomic nuclei than in inter-atomic regions calls for a multi-resolution approach, allowing one to use low resolution and to add extra resolution only in those regions where necessary, so limiting the costs. This is provided by an alternative basis formed of wavelets. Using such a wavelet basis, a method has been developed for solving electronic structure problems that has been applied successfully to 2D quantum dots and 3D molecular systems. In other cases, it suffices to use effective potentials to describe the atomic interaction instead of the use of the electronic structure, enabling the simulation of larger systems. Molecular dynamics simulations with such effective potentials have been used for a systematic study of surface wettability influence on particle and heat flow in nanochannels, showing that the effects at the solid-gas interface are crucial for the behavior of the whole nanochannel. Again in other cases even coarse grained models can be used where the average behavior of several atoms is combined into a single particle. Such a model, refraining from as much detail as possible while maintaining realistic behavior, has been developed for lipids and with this model the dynamics of membranes and vesicle formation have been studied in detail. A disadvantage of molecular dynamics simulations with effective potentials is that no reactions are possible. Therefore a new method has been developed, where molecular dynamics is coupled with stochastic reactions. Using this method, both unilamellar and multilamellar vesicle formation, and vesicle growth, bursting, and healing are shown. Still larger systems can be simulated using other methods, like the direct simulation Monte Carlo method. However, as shown for nanochannels, these methods are not always accurate enough. But, exploiting again that the finest level of detail is often only needed in part of the domain, a hybrid method has been developed coupling molecular dynamics, where needed for accuracy, and direct simulation Monte Carlo, where possible in order to speed up the calculation. Further development of such hybrid simulations will further increase molecular simulation’s scientific role