Material Theories

Material theories is a series of workshops concerned with a broad range of topics related to the mechanics and mathematics of materials. As such, this edition brought together researchers from diverse fields converging toward the interaction between mathematics, mechanics, and material science

Quantum Theory from Principles, Quantum Software from Diagrams

This thesis consists of two parts. The first part is about how quantum theory can be recovered from first principles, while the second part is about the application of diagrammatic reasoning, specifically the ZX-calculus, to practical problems in quantum computing. The main results of the first part include a reconstruction of quantum theory from principles related to properties of sequential measurement and a reconstruction based on properties of pure maps and the mathematics of effectus theory. It also includes a detailed study of JBW-algebras, a type of infinite-dimensional Jordan algebra motivated by von Neumann algebras. In the second part we find a new model for measurement-based quantum computing, study how measurement patterns in the one-way model can be simplified and find a new algorithm for extracting a unitary circuit from such patterns. We use these results to develop a circuit optimisation strategy that leads to a new normal form for Clifford circuits and reductions in the T-count of Clifford+T circuits.Comment: PhD Thesis. Part A is 135 pages. Part B is 95 page

Towards a Theory of Quantum Gravity Through Geometrization of Quantum Mechanics

In this thesis, we adapt an approach by assuming quantum mechanics as a fundamental theory of nature and attempt to recover familiar concepts such as space-time geometry and gravity from quantum wavefunctions and their unitary evolutions. More specifically, we explore a number of approaches in "geometrizing" quantum systems using techniques such as tensor networks and manifold learning. We find that consistency conditions in quantum gravity can be used to put constraints on tensor network models that approximate the anti-de Sitter/Conformal Field Theory correspondence. Furthermore, quantum circuits and tensor networks can also be used to describe cosmological models and reproduce important features of space-time configurations such as de Sitter space. We find that a generic framework using quantum circuit to describe cosmology puts an upper bound on the number of e-folds during the inflationary phase of the Universe's expansion. In addition to tensor network models, we also propose a Bulk Entanglement Gravity framework that analyzes the entanglement data of a quantum state in a Hilbert space without any a priori assumptions on geometry, such as the likes of a boundary conformal field theory. We find that from an amorphous configuration, one can directly recover geometry of bulk space-time from a generic class of wavefunctions that is fully characterized in this thesis via quantum entropy cone techniques. We find that under a number of assumptions, it is possible to derive linearized Einstein's equation from a version of Jacobson's entanglement equilibrium conditions for an emergent spacetime geometry in the weak field limit near Minkowski space. We show that non-local entanglement perturbations display features of wormhole-like configurations. We also clarify connections between Bulk Entanglement Gravity and highly generic features in quantum error correction codes that can be used to derive gravity.</p

Towards music perception by redundancy reduction and unsupervised learning in probabilistic models

PhDThe study of music perception lies at the intersection of several disciplines: perceptual psychology and cognitive science, musicology, psychoacoustics, and acoustical signal processing amongst others. Developments in perceptual theory over the last fifty years have emphasised an approach based on Shannon’s information theory and its basis in probabilistic systems, and in particular, the idea that perceptual systems in animals develop through a process of unsupervised learning in response to natural sensory stimulation, whereby the emerging computational structures are well adapted to the statistical structure of natural scenes. In turn, these ideas are being applied to problems in music perception. This thesis is an investigation of the principle of redundancy reduction through unsupervised learning, as applied to representations of sound and music. In the first part, previous work is reviewed, drawing on literature from some of the fields mentioned above, and an argument presented in support of the idea that perception in general and music perception in particular can indeed be accommodated within a framework of unsupervised learning in probabilistic models. In the second part, two related methods are applied to two different low-level representations. Firstly, linear redundancy reduction (Independent Component Analysis) is applied to acoustic waveforms of speech and music. Secondly, the related method of sparse coding is applied to a spectral representation of polyphonic music, which proves to be enough both to recognise that the individual notes are the important structural elements, and to recover a rough transcription of the music. Finally, the concepts of distance and similarity are considered, drawing in ideas about noise, phase invariance, and topological maps. Some ecologically and information theoretically motivated distance measures are suggested, and put in to practice in a novel method, using multidimensional scaling (MDS), for visualising geometrically the dependency structure in a distributed representation.Engineering and Physical Science Research Counci

Theoretical Concepts of Quantum Mechanics

Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers

Quantum Mechanics for Thinkers

This book provides quick access to quantum mechanics without dealing with a true textbook that demands proper specialized studies in physics (and related mathematics) for about a couple of years. It consists of three parts: basic formalism, formal development, and ontological issues. The 70 figures are a crucial instrument for becoming acquainted in a "representative" way with abstract problems, and the 30 in-section boxes assist readers understand for difficult mathematical problems. The book offers a considerable number of clear and analytical treatments of what are considered the most difficult conceptual problems of the theory