Variational Monte Carlo Approach to Partial Differential Equations with Neural Networks

The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here, we develop a variational approach for solving partial differential equations governing the evolution of high dimensional probability distributions. Our approach naturally works on the unbounded continuous domain and encodes the full probability density function through its variational parameters, which are adapted dynamically during the evolution to optimally reflect the dynamics of the density. For the considered benchmark cases we observe excellent agreement with numerical solutions as well as analytical solutions in regimes inaccessible to traditional computational approaches.Comment: 6 + 3 pages, 4 figure

Time-dependent variational principle for open quantum systems with artificial neural networks

We develop a variational approach to simulating the dynamics of open quantum many-body systems using deep autoregressive neural networks. The parameters of a compressed representation of a mixed quantum state are adapted dynamically according to the Lindblad master equation by employing a time-dependent variational principle. We illustrate our approach by solving the dissipative quantum Heisenberg model in one and two dimensions for up to 40 spins and by applying it to the simulation of confinement dynamics in the presence of dissipation.Comment: 7 + 5 pages, 3 figure

Sample-efficient estimation of entanglement entropy through supervised learning

We explore a supervised machine learning approach to estimate the entanglement entropy of multi-qubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network's estimate and benchmark against the best known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for non-unitary evolution by training our model on different noise strengths.Comment: 5 + 1 pages, 4 figure

Zero-Field J-spectroscopy of Quadrupolar Nuclei

Zero- to ultralow-field (ZULF) nuclear magnetic resonance (NMR) is a version of NMR that allows studying molecules and their transformations in the regime dominated by intrinsic spin-spin interactions. While spin dynamics at zero magnetic field can be probed indirectly, J-spectra can also be measured at zero field by using non-inductive sensors, for example, optically-pumped magnetometers (OPMs). A J-spectrum can be detected when a molecule contains at least two different types of magnetic nuclei (i.e., nuclei with different gyromagnetic ratios) that are coupled via J-coupling. Up to date, no pure J-spectra of molecules featuring the coupling to quadrupolar nuclei were reported. Here we show that zero-field J-spectra can be collected from molecules containing quadrupolar nuclei with I = 1 and demonstrate this for solutions containing various isotopologues of ammonium cations. Lower ZULF NMR signals are observed for molecules containing larger numbers of deuterons compared to protons; this is attributed to less overall magnetization and not to the scalar relaxation of the second kind. We analyze the energy structure and allowed transitions for the studied molecular cations in detail using perturbation theory and demonstrate that in the studied systems, different lines in J-spectra have different dependencies on the magnetic pulse length allowing for unique on-demand zero-field spectral editing. Precise values for the 15N-1H, 14N-1H, and D-1H coupling constants are extracted from the spectra and the difference in the reduced coupling constants is explained by the secondary isotope effect. Simple symmetric cations such as ammonium do not require expensive isotopic labeling for the observation of J-spectra and, thus, may expand applicability of ZULF NMR spectroscopy in biomedicine and energy storage.Comment: 39 pages, 5 figure

Zero-field nuclear magnetic resonance of chemically exchanging systems.

Zero- to ultralow-field (ZULF) nuclear magnetic resonance (NMR) is an emerging tool for precision chemical analysis. In this work, we study dynamic processes and investigate the influence of chemical exchange on ZULF NMR J-spectra. We develop a computational approach that allows quantitative calculation of J-spectra in the presence of chemical exchange and apply it to study aqueous solutions of [15N]ammonium (15N[Formula: see text]) as a model system. We show that pH-dependent chemical exchange substantially affects the J-spectra and, in some cases, can lead to degradation and complete disappearance of the spectral features. To demonstrate potential applications of ZULF NMR for chemistry and biomedicine, we show a ZULF NMR spectrum of [2-13C]pyruvic acid hyperpolarized via dissolution dynamic nuclear polarization (dDNP). We foresee applications of affordable and scalable ZULF NMR coupled with hyperpolarization to study chemical exchange phenomena in vivo and in situations where high-field NMR detection is not possible to implement

From Spin Systems to Bose-Einstein Condensates: Computational Approaches to Strongly Correlated Quantum Many-Body Systems

The numerical simulation of quantum many-body systems constitutes a long-standing and challenging problem, as the 'curse of dimensionality' restricts the applicability of exact methods to systems consisting of only a few particles. Thus approximative techniques that reduce the computational complexity are of high fundamental interest. Simultaneously, there exists a strong desire to benchmark the ever-growing capabilities of quantum simulators, thus strengthening the motivation to research tools that are capable of matching their increasing system sizes. In this thesis, we, for one, develop and explore such new computational methods by exploiting the rapid developments in machine learning, allowing us to construct highly versatile ansatz functions to model quantum states based on deep artificial neural networks. Building on this, we establish a new numerical technique capable of modeling the dynamics of dissipative many-body quantum systems, relying on an accurate variational description of an informationally complete probability distribution that corresponds to the quantum system of interest. Additionally, we explore the differences in performance in ground state searches between a multitude of different network architectures and thereby shed light on the question of why some networks significantly outperform others. Secondly, we adapt the developed techniques also for classical systems. This is possible as the only requirement is a probabilistic description with a (closed) evolution equation, thereby emphasizing the wide range of applicability. Finally, we rely on existing approximative techniques to devise an experimental proposal aimed at observing an area to a volume law transition following a quench in a spin-1 Bose-Einstein condensate. Notably, we herein do not rely on quantum entropies but rather on differential entropies of the phase-space distribution describing the system. These quasi probability distributions are importantly readily accessible in experiments and we demonstrate that their entropies can be reliably estimated from a feasible number of samples without assuming a particular type of distribution, such as a Gaussian

Zero-Field J-spectroscopy of Quadrupolar Nuclei

Zero- to ultralow-field (ZULF) nuclear magnetic resonance (NMR) is a version of NMR that allows studying molecules and their transformations in the regime dominated by intrinsic spin-spin interactions. While spin dynamics at zero magnetic field can be probed indirectly, J-spectra can also be measured at zero field by using non-inductive sensors, for example, optically-pumped magnetometers (OPMs). A J-spectrum can be detected when a molecule contains at least two different types of magnetic nuclei (i.e., nuclei with different gyromagnetic ratios) that are coupled via J-coupling. Up to date, no pure J-spectra of molecules featuring the coupling to quadrupolar nuclei were reported. Here we show that zero-field J-spectra can be collected from molecules containing quadrupolar nuclei with I = 1 and demonstrate this for solutions containing various isotopologues of ammonium cations. Lower ZULF NMR signals are observed for molecules containing larger numbers of deuterons compared to protons; this is attributed to less overall magnetization and not to the scalar relaxation of the second kind. We analyze the energy structure and allowed transitions for the studied molecular cations in detail using perturbation theory and demonstrate that in the studied systems, different lines in J-spectra have different dependencies on the magnetic pulse length allowing for unique on-demand zero-field spectral editing. Precise values for the 15N-1H, 14N-1H, and D-1H coupling constants are extracted from the spectra and the difference in the reduced coupling constants is explained by the secondary isotope effect. Simple symmetric cations such as ammonium do not require expensive isotopic labeling for the observation of J-spectra and, thus, may expand applicability of ZULF NMR spectroscopy in biomedicine and energy storage

Zero-Field Nuclear Magnetic Resonance of Chemically Exchanging Systems

Zero- and ultralow-field (ZULF) nuclear magnetic resonance (NMR) is an emerging tool for precision chemical analysis. Unlike conventional (high-field) NMR, which relies on chemical shifts for molecular identification, zero-field analog reports J-spectra that depend on the nuclear spin-spin coupling topology of molecules under investigation. While chemical shifts are usually a small fraction of the resonance frequencies, J-spectra for various spin systems are completely different from each other. In this work, we use zero-field NMR to study dynamic chemical processes and investigate the influence of chemical exchange on ZULF NMR spectra. We develop a computation approach that allows quantitative calculation of ZULF NMR spectra in the presence of chemical exchange and apply it to study aqueous solutions of [15N]ammonium as a model system. In this system, proton exchange rates span more than three orders of magnitude depending on acidity (pH), as monitored by high-field and ZULF NMR. We show that chemical exchange substantially affects the J-coupled NMR spectra and, in some cases, can lead to degradation and complete disappearance of the spectral features. To demonstrate potential applications of ZULF NMR for chemistry and biomedicine, we show a ZULF NMR spectrum of [2-13C]pyruvic acid hyperpolarized via dissolution dynamic nuclear polarization (dDNP). The metabolism of pyruvate provides valuable biochemical information and its monitoring by zero-field NMR could give spectral resolution that is hard to achieve at high magnetic fields. We foresee applications of affordable and scalable ZULF NMR coupled with hyperpolarization modalities to study chemical exchange phenomena in vivo and in situations where high-field NMR detection is not possible to implement.<br /

Zero-field nuclear magnetic resonance of chemically exchanging systems.

Zero- to ultralow-field (ZULF) nuclear magnetic resonance (NMR) is an emerging tool for precision chemical analysis. In this work, we study dynamic processes and investigate the influence of chemical exchange on ZULF NMR J-spectra. We develop a computational approach that allows quantitative calculation of J-spectra in the presence of chemical exchange and apply it to study aqueous solutions of [15N]ammonium (15N[Formula: see text]) as a model system. We show that pH-dependent chemical exchange substantially affects the J-spectra and, in some cases, can lead to degradation and complete disappearance of the spectral features. To demonstrate potential applications of ZULF NMR for chemistry and biomedicine, we show a ZULF NMR spectrum of [2-13C]pyruvic acid hyperpolarized via dissolution dynamic nuclear polarization (dDNP). We foresee applications of affordable and scalable ZULF NMR coupled with hyperpolarization to study chemical exchange phenomena in vivo and in situations where high-field NMR detection is not possible to implement