New concepts in quantum-metrology: From coherent averaging to Hamiltonian extensions

This thesis is dedicated to the understanding of the metrology of quantum systems by using the tools of quantum parameter estimation, in particular the quantum Fisher information (QFI). Our first project deals with a specific protocol of quantum enhanced measurement known as coherent averaging [Braun and Martin, 2011]. This protocol is based on a star topology, with one central object, the so-called quantum bus, connected to N extra subsystems, called probes. For the estimation of a parameter characteristic of the interaction between the quantum bus and the probes, coherent averaging leads to a Heisenberg limited (HL) scaling for the QFI (QFI proportional to N 2 ). Importantly this HL scaling can be obtained while starting with a separable state. This provides an advantage as generally one needs to use entangled states to achieve this scaling. Another important aspect in coherent averaging is the possibility to obtain the HL scaling by performing a measurement on the quantum bus only. These results were obtained using perturbation theory in the regime of weak interactions. In this thesis we go one step further in the study of the coherent averaging protocol. We extend the formalism of perturbation theory to encompass the possibility of estimating any parameter, in the regimes of strong and weak interactions. To illustrate the validity of our results, we introduce two models as examples for a coherent averaging scheme. In these models both the quantum bus and all the probes are qubits. In the ZZXX model, the free Hamiltonians do not commute with the interaction Hamiltonians and we have to rely on numerics to find non-perturbative solutions .In the ZZZZ model the free evolution Hamiltonians commute with the interaction Hamiltonians and we can find the exact solution analytically. Perturbation theory shows that in the strong interaction regime and starting with a separable state, we can estimate the parameter of the free evolution of the probes with a HL scaling if the free Hamiltonians do not commute with the interaction Hamiltonians. This is confirmed by the non-perturbative numerical results for the ZZXX model. In the weak interaction regime we only obtain a standard quantum limit (SQL) scaling for the parameter of the free evolution of the probes (QFI proportional to N ). When one has only access to the quantum bus, we show that the HL scaling found using the perturbation theory does not necessarily survive outside the regime of validity of the perturbation. This is especially the case as N becomes large. It is shown by comparing the exact analytical result to the perturbative result with the ZZZZ model. The same behaviour is observed with the ZZXX model using the non-perturbative numerical results. In our second project we investigate the estimation of the depolarizing channel and the phase-flip channel under non-ideal conditions. It is known that using an ancilla can lead to an improvement of the channel QFI (QFI maximized over input states feeding the channel) even if we act with the identity on the ancilla. This method is known as channel extension. In all generality the maximal channel QFI can be obtained using an ancilla whose Hilbert space has the same dimension as the dimension of the Hilbert space of the original system. In this ideal scenario using multiple ancillas — or one ancilla with a larger Hilbert space dimension — is useless. To go beyond this ideal result we take into account the possibility of loosing either the probe or a finite number of ancillas. The input states considered are GHZ and W states with n + 1 qubits (the probe plus n ancillas). We show that for any channel, when the probe is lost then all the information is lost, and the use of ancillas cannot help. For the phase-flip channel the introduction of ancillas never improves the channel QFI and ancillas are useless. For the depolarizing channel the maximal channel QFI can be reached using one ancilla and feeding the extended channel with a Bell state, but if the ancilla is lost then all the advantage is lost. We show that the GHZ states do not help to fight the loss of ancillas: If one ancilla or more are lost all the advantage provided by the use of ancillas is lost. More interestingly, we show that the W states with more than one ancilla are robust against loss. For a given number of lost ancillas, there always exists an initial number of ancillas for which a W state provides a higher QFI than the one obtained without ancillas. Our last project is about Hamiltonian parameter estimation for arbitrary Hamiltonians. It is known that channel extension does not help for unitary channels. Instead we apply the idea of extension to the Hamiltonian itself and not to the channel. This is done by adding to the Hamiltonian an extra term, which is independent of the parameter and which possibly encompasses interactions with an ancilla. We call this technique Hamiltonian extension. We show that for arbitrary Hamiltonians there exists an upper bound to the channel QFI that is in general not saturated. This result is known in the context of non-linear metrology. Here we show explicitly the conditions to saturate the bound. We provide two methods for Hamiltonian extensions, called signal flooding and Hamiltonian subtraction, that allow one to saturate the upper bound for any Hamiltonian. We also introduce a third method which does not saturate the upper bound but provides the possibility to restore the quadratic time scaling in the channel QFI when the original Hamiltonian leads only to a periodic time scaling of the channel QFI. We finally show how these methods work using two different examples. We study the estimation of the strength of a magnetic field using a NV center, and show how using signal flooding we saturate the channel QFI. We also consider the estimation of a direction of a magnetic field using a spin-1. We show how using signal flooding or Hamiltonian subtraction we saturate the channel QFI. We also show how by adding an arbitrary magnetic field we restore the quadratic time scaling in the channel QFI. Eventually we explain how coherent averaging can be scrutinized in the formalism of Hamiltonian extensions

Machine Learning Methods for Modeling Synthesizable Molecules

The search for new molecules often involves cycles of design-make-test-analyze steps, where new molecules are designed, synthesized in a lab, tested, and then analyzed to inform what is to be designed next. This thesis proposes new machine learning (ML) methods to augment chemists in the design and make steps of this process, focusing on the tasks of (a) how to use ML to predict chemical reaction outcomes, and (b) how to build generative models to search for new molecules. We take a common approach to both tasks, building our ML models around existing powerful tools and abstractions from the field of chemistry, and in doing so, show that the tasks we tackle are intrinsically linked. Reaction prediction is important for validating synthesis plans before carrying them out. Many previous ML approaches to reaction prediction have treated reactions as either a black box translation or a single graph edit operation. Instead, we propose a model (ELECTRO) that predicts the reaction products through modeling a sequence of electron movements. We show how modeling electron movements in this way has the benefit of being easy for chemists to interpret, and also is a natural format in which to incorporate the constraints of chemistry, such as balanced atom counts before and after a reaction. We show that our model achieves excellent performance on an important subset of chemical reactions and recovers a basic knowledge of chemistry without explicit supervision. In designing new models to search for molecules with particular properties, it is important that the models describe not only what molecule to make, but also crucially how to make it. These instructions form a synthesis plan, describing how easy-to-obtain building blocks can be combined together to form more complex molecules of interest through chemical reactions. Inspired by this real-world process, we develop two machine learning approaches that incorporate reactions into the virtual generation of new molecules. We show that aligning our model with the real-world process allows us to better link up the design and make steps involved in molecule search, and permits chemists to examine the practicability of both the final molecules we suggest and their synthetic routes. Molecule search is inherently an extrapolation task, and we show that by building our methods around the inductive biases of modeling reactions, we can generalize to new chemical spaces, suggesting molecules that not only perform well, but are synthesizable too.EPSR

Using Deep Learning to Explore Ultra-Large Scale Astronomical Datasets

In every field that deep learning has infiltrated we have seen a reduction in the use of specialist knowledge, to be replaced with knowledge automatically derived from data. We have already seen this process play out in many ‘applied deep learning’ fields such as computer Go, protein folding, natural language processing, and computer vision. This thesis argues that astronomy is no different to these applied deep learning fields. To this end, this thesis’ introduction serves as a historical background on astronomy’s ‘three waves’ of increasingly automated connectionism: initial work on multilayerperceptrons within astronomy required manually selected emergent properties as input; the second wave coincided with the dissemination of convolutional neural networks and recurrent neural networks, models where the multilayer perceptron’s manually selected inputs are replaced with raw data ingestion; and in the current third wave we are seeing the removal of human supervision altogether with deep learning methods inferring labels and knowledge directly from the data. §2, §3, and §4 of this thesis explore these waves through application. In §2 I show that a convolutional/recurrent encoder/decoder network is capable of emulating a complicated semi-manual galaxy processing pipeline. I find that this ‘Pix2Prof’ neural network can satisfactorily carry out this task over 100x faster than the method it emulates. §3 and §4 explore the application of deep generative models to astronomical simulation. §3 uses a generative adversarial network to generate mock deep field surveys, and finds it capable of generating mock images that are statistically indistinguishable from the real thing. Likewise, §4 demonstrates that a Diffusion model is capable of generating galaxy images that are both qualitatively and quantitatively indistinguishable from the training set. The main benefit of these deep learning based simulations is that they do not rely on a possibly flawed (or incomplete) physical knowledge of their subjects and observation processes. Also, once trained, they are capable of rapidly generating a very large amount of mock data. §5 looks to the future and predicts that we will soon enter a fourth wave of astronomical connectionism. If astronomy follows in the footsteps of other applied deep learning fields we will see the removal of expertly crafted deep learning models, to be replaced with finetuned versions of an all-encompassing ‘foundation’ model. As part of this fourth wave I argue for a symbiosis between astronomy and connectionism. This symbiosis is predicated on astronomy’s relative data wealth, and contemporary deep learning’s enormous data appetite; many ultra-large datasets in machine learning are proprietary or of poor quality, and so astronomy as a whole could develop and provide a high quality multimodal public dataset. In turn, this dataset could be used to train an astronomical foundation model that can be used for state-of-the-art downstream tasks. Due to the foundation models’ hunger for data and compute, a single astronomical research group could not bring about such a model alone. Therefore, I conclude that astronomy as a whole has slim chance of keeping up with a research pace set by the Big Tech goliaths—that is, unless we follow the examples of EleutherAI and HuggingFace and pool our resources in a grassroots open source fashion

Single molecule studies of branched polymer dynamics

Polymer architecture plays a major role on the emergent physical and chemical properties of materials such as elasticity and wettability. Branched polymers exhibit strikingly different rheological behavior (e.g. enhanced stress dissipation and strain hardening) compared to linear polymers. In recent years, the dynamic properties of branched polymers have been studied using bulk rheological techniques (Chapter 1), but we still lack a full understanding of how molecular-scale interactions give rise to macroscopic properties for topologically complex polymers. Single molecule studies enable the direct observation of polymer chain dynamics at the molecular level; however, the vast majority of single polymer studies have only focused on linear DNA molecules (Chapter 2). In this dissertation, we extend single molecule techniques to study the dynamics of branched polymers, which effectively bridges the gap between bulk-scale rheological properties and molecular scale behavior. In particular, we explore the synthesis, characterization, single molecule dynamics, and Brownian dynamics simulations of DNA-based branched polymers. This approach enables us to interrogate the impact of distributions in molecular size and architecture, thereby holding the potential to fundamentally change our understanding of the rheological response of topologically complex polymers. We first developed a two-step synthesis method to generate branched polymers for single molecule visualization (Chapter 3). Here, we use a graft-onto synthesis method by linking side branches onto DNA backbones, thereby producing star, H-shaped, and comb-shaped polymers. In these experiments, DNA-based branched polymers are designed to contain short branches (1-10 kilobase pairs) and long backbones (10-40 kilobase pairs), where the branches and backbones are monodisperse and the branch distribution can be controlled in an average sense. Following synthesis and purification, we utilize single molecule fluorescence microscopy to observe the dynamics of these molecules, in particular by tracking the side branches and backbones independently (Chapter 4). In this way, this imaging method allows for characterization of these materials at the single molecule level, including quantification of polymer contour length and branch distributions for varying synthetic conditions. Moving beyond characterization, we study the dynamics of single branched polymers in flow using a molecular rheology approach. In one experiment, we study the dynamics of asymmetric star, H-shaped, and comb-shaped DNA polymers tethered to the surface in a microfluidic flow cell (Chapter 4). In this way, we study the impact of branch frequency and position on backbone chain relaxation from high stretch. In a second experiment, we utilize a microfluidic cross-slot device to hydrodynamically ‘trap’ branched DNA molecules in planar extensional flow, thereby studying the impact of branching on relaxation in solution, as well as transient and steady-state dynamics in flow (Chapter 5). We present results for branched polymer dynamics as functions of branch frequency and flow strength. We also conduct Brownian dynamics simulations based on a coarse-grained model for comb polymers (Chapter 6). Results from simulations and experiments agree qualitatively, and branched polymers exhibit a weaker dependence of relaxation on total polymer molecular weight in comparison to linear polymers. Overall, this work presents molecular-scale investigations of branched polymer dynamics. From a broad perspective, this research provides a molecular-based understanding of topologically complex polymers in flow, thereby holding the potential to advance the large-scale production of polymers. Importantly, this platform can be further extended to study branched polymers in alternate flow fields such as simple shear flow or linear mixed flows, semi-dilute solutions, and concentrated solutions. These experiments will provide a molecular basis for phenomena observed in branched polymers, from viscosity modification of blended branched polymer solutions to hierarchical relaxation mechanisms of entangled branched polymers to enhanced strain hardening of comb polymer melts

Characterisation of self-assembled engineered proteins on gold nanoparticles and their application to biosensing

PhD ThesisThe use of gold nanoparticles (AuNP) has a long and varied history, thought to cover several thousand years. More recently the unique properties of nanoscale materials have stimulated extensive work on nanoparticles and other nanomaterials leading to their use in novel technologies. AuNPs have been of particular interest for bioscience applications due to their biocompatibility and the ease with which biological molecules can be conjugated to their surface. In this study the assembly of engineered proteins, specifically the transmembrane domain of Escherichia coli outer membrane protein A (OmpATM), onto the surface of AuNPs was investigated both in solution and with the particles attached to a SiO2 substrate. AuNPs were adhered to SiO2 surfaces using a novel silane treatment developed by the industrial sponsor and were characterised using spectroscopy, electron and atomic force microscopy. The addition of a single cysteine residue to the OmpATM structure was shown, by UV-Vis and fluorescence spectroscopy, to increase protein binding at equilibrium and form higher stability protein-AuNP complexes in solution. Following this, engineered OmpATM proteins containing tandem antibody-binding domains from Streptococcal protein G were assembled on the AuNP surface and their structure interrogated using neutron and light scattering. This revealed an oriented protein layer where the functional domains extend away from the AuNP surface and are available to bind antibodies. OmpATM-AuNP conjugates were used to develop biosensing assays using both well-established methods, such as lateral flow assays, and novel spectroscopic methods, which use the unique optical properties of AuNPs. Detection of influenza A nucleoprotein, an antigen used to clinically diagnose influenza, was achieved using a bespoke anti-nucleoprotein single-chain antibody domain fused to OmpATM and assembled on 20 nm diameter AuNPs. The results demonstrate that engineered OmpATM proteins conjugated to AuNPs can be used to develop novel diagnostics using a range of read out technologies

Handbook of Mathematical Geosciences

This Open Access handbook published at the IAMG's 50th anniversary, presents a compilation of invited path-breaking research contributions by award-winning geoscientists who have been instrumental in shaping the IAMG. It contains 45 chapters that are categorized broadly into five parts (i) theory, (ii) general applications, (iii) exploration and resource estimation, (iv) reviews, and (v) reminiscences covering related topics like mathematical geosciences, mathematical morphology, geostatistics, fractals and multifractals, spatial statistics, multipoint geostatistics, compositional data analysis, informatics, geocomputation, numerical methods, and chaos theory in the geosciences

Exploring the Earth’s subsurface with virtual seismic sources and receivers

Traditional methods of imaging the Earth’s subsurface using seismic waves require an identifiable, impulsive source of seismic energy, for example an earthquake or explosive source. Naturally occurring, ambient seismic waves form an ever-present source of energy that is conventionally regarded as unusable since it is not impulsive. As such it is generally removed from seismic data and subsequent analysis. A new method known as seismic interferometry can be used to extract useful information about the Earth’s subsurface from the ambient noise wavefield. Consequently, seismic interferometry is an important new tool for exploring areas which are otherwise seismically quiet, such as the British Isles in which there are relatively few strong earthquakes. One of the possible applications of seismic interferometry is the ambient noise tomography method (ANT). ANT is a way of using interferometry to image subsurface seismic velocity variations using seismic (surface) waves extracted from the background ambient vibrations of the Earth. To date, ANT has been used to successfully image the Earth’s crust and upper-mantle on regional and continental scales in many locations and has the power to resolve major geological features such as sedimentary basins and igneous and metamorphic cores. In this thesis I provide a review of seismic interferometry and ANT and apply these methods to image the subsurface of north-west Scotland and the British Isles. I show that the seismic interferometry method works well within the British Isles and illustrate the usefulness of the method in seismically quiet areas by presenting the first surface wave group velocity maps of the Scottish Highlands and across the British Isles using only ambient seismic noise. In the Scottish Highlands, these maps show low velocity anomalies in sedimentary basins such as the Moray Firth and high velocity anomalies in igneous and metamorphic centres such as the Lewisian complex. They also suggest that the Moho shallows from south to north across Scotland, which agrees with previous geophysical studies in the region. Rayleigh wave velocity maps from ambient seismic noise across the British Isles for the upper and mid-crust show low velocities in sedimentary basins such as the Midland Valley, the Irish Sea and the Wessex Basin. High velocity anomalies occur predominantly in areas of igneous and metamorphic rock such as the Scottish Highlands, the Southern Uplands, North-West Wales and Cornwall. In the lower crust/upper mantle, the Rayleigh wave maps show higher velocities in the west and lower velocities in the east, suggesting that the Moho shallows generally from east to west across Britain. The extent of the region of higher velocity correlates well with the locations of British earthquakes, agreeing with previous studies that suggest British seismicity might be influenced by a mantle upwelling beneath the west of the British Isles. Until the work described in Chapter 6 of this thesis was undertaken in 2009, seismic interferometry was concerned with cross-correlating recordings at two receivers due to a surrounding boundary of sources, then stacking the cross-correlations to construct the inter-receiver Green’s function. A key element of seismic wave propagation is that of source-receiver reciprocity i.e. the same wavefield will be recorded if its source and receiver locations and component orientations are reversed. By taking the reciprocal of its usual form, in this thesis I show that the impulsive-source form of interferometry can also be used in the opposite sense: to turn any energy source into a virtual sensor. This new method is demonstrated by turning earthquakes in Alaska and south-west USA into virtual seismometers located beneath the Earth’s surface

Controlling the Dynamics of Microstructure Formation in Mixed-Matrix Polymeric-Particle Membranes

Polymer membranes are increasingly important in energy generation, water purification, and resource recovery. Control over chemistry, morphology, and mechanical properties gives organic polymers unparalleled advantages for membrane technology—but only if these complementary functions can be married into a cohesive material. Herein I have sought to expand upon the chemical tools for integrating diverse polymers into multifunctional membrane materials, making them easily tunable to various applications. To overcome a fundamental challenge in polymer science—namely, that polymers with different functions often do not mix—the functional polymer is grown in situ in a solution containing a preformed scaffold polymer, a method pioneered by co-advisor Mamadou Diallo. The hierarchical structure of the resulting mixed matrix polymeric-particle (M2P2) membrane is governed by the kinetic competition between polymerization and phase separation of the functional polymer from the scaffold polymer. This competition is quenched by immersion in a nonsolvent, which rapidly solidifies the material to trap the metastable structure formed during synthesis. In my quest to understand how these competing processes interact to inform multifunctional membrane design, I developed a general method for studying transient structure using ultra-small angle neutron scattering (Chapter II), working closely with Kornfield Group alumnus Dr. Joey Kim. I then investigated the synergistic effects of incorporating different functional polymer architectures in M2P2 membranes (Chapter III), working with fellow graduate student Orland Bateman. By combining low-generation dendrimers with randomly hyperbranched oligomers bearing similar chemical functionality, we can systematically tune the characteristic length of domains formed during synthesis. In the final chapter I discuss the main conclusions and describe future directions for understanding structure during processing in M2P2 membranes. My thesis ultimately provides a broadly relevant platform for membrane design and synthesis, one in which the favorable properties of different polymers may be combined to strike a balance between function, stability, and ease of fabrication.</p