This dissertation unites variational computation with results and techniques appearing in the theory of ground state computation. It should be readable by graduate students.
The topics covered include: Ising model reductions, stochastic versus quantum processes on graphs, quantum gates and circuits as tensor networks, variational quantum algorithms and Hamiltonian gadgets.
This dissertation unites variational computation with results and techniques appearing in the theory of ground state computation. It should be readable by graduate students.
The topics covered include: Ising model reductions, stochastic versus quantum processes on graphs, quantum gates and circuits as tensor networks, variational quantum algorithms and Hamiltonian gadgets.
Determining properties of ground states of spin Hamiltonians remains a topic of central relevance connecting disciplines of mathematical, theoretical and applied physics. In the last few decades, ground state properties of physical systems have been increasingly considered as computational resources. This thesis develops parts of the mathematical apparatus to create (program) ground states relevant for quantum and classical computation. The core findings presented in this thesis (now over a decade old) including that (i) logic operations (gates) can be embedded into the low-energy sector of Ising spins whereas three (and higher) body Ising interaction terms can be mimicked through the minimisation of 2- and 1-body Ising terms yet require the introduction of slack spins; (ii) Perturbation theory gadgets enable the emulation of interactions not present in a given Hamiltonian, e.g.~YY interactions can be realized from ZZ, XX, the thesis contains a result from 2007 showing that physically relevant two-body model Hamiltonian's have a QMA-hard ground state energy decision problem. Merged with other results, this established that these models provide a universal resource for ground state quantum computation. More recent findings include the proof that an idealised version of the contemporary variational approach to quantum algorithms enables a universal model of quantum computation. Other related results are also presented as they relate to ground state quantum computation and the minimisation of Hamiltonians by quantum circuits. The topics covered include: Ising model reductions, stochastic versus quantum processes on graphs, quantum gates and circuits as tensor networks, variational quantum algorithms and Hamiltonian gadgets.
Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing. This book aims to present the best contemporary practices in the use of tensor networks as a reasoning tool, placing quantum states, operators and processes on the same compositional footing. The book has 7 parts and over 40 subsections which took shape in over a decade of teaching. In addition to covering the foundations, the book covers important applications such as matrix product states, open quantum systems and entanglement − all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.
This dissertation is concerned with a variety of problems in quantum computing and the spectral theory of quantum-mechanical channels and Hamiltonians. We begin with the problem of quantum gate synthesis. The challenge is to find a fixed, finite set S of quantum gates such that, given a quantum circuit component U acting on n qubits and target precision ε, there is a short, efficiently computable composition of gates from S that approximates U to precision ε. This problem has already led to a rich literature in the case n=1 with deep connections to modern number theory. In this dissertation we present the state-of-the-art algorithm for a single qubit, achieving length 7 log (1/ε). The techniques used include the lattice structure of certain dense subgroups of PU(2) and Lenstra's algorithm for integer programming in convex domains. The next topic considered is scientific computing algorithms intended to run on a quantum computer. Specifically, we study a framework for computing functions of normal matrices using a quantum computer, with the objective of controlling the ancilla overhead, probability of success, and end precision. This arises as a special case of the multivariate quantum eigenvalue transform, for computing function of multiple commuting Hermitian matrices. The techniques applied include approximation of regular functions by low-degree polynomials and modern quantum primitives like quantum signal processing and linear combinations of unitaries. The third component is a spectral gap comparison result for Davies generators, a class of Lindbladians used for Gibbs state preparation and which are a well-studied example of quantum Markov dynamics, and which contain a canonical classical Markov chain. The main result shows that if the spectrum of the Hamiltonian driving the dynamics is free of long arithmetic progressions, then the spectral gaps of the quantum and classical Markov chains are comparable. Thus, classical techniques may be leveraged to control the behavior of the quantum dynamics. The proof is based on transferring any "off-diagonal" eigenvector of the Davies generator to an observable which commutes with the Hamiltonian and has a comparable Lindbladian Rayleigh quotient. Finally, we study the infimum of the spectrum, or ground state energy, of a discrete Schrödinger operator on the scaled integer grid in n dimensions, parameterized by a nonnegative potential V and a small frequency parameter θ. We relate this ground state energy to that of a corresponding continuous semiclassical Schrödinger operator on the full n-dimensional space with parameter θ, arising from the same choice of potential, and show that these quantities are equal up to vanishing factors depending on the regularity of V. The proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on the scaled grid to low-energy functions for the continuous operator on the full space, and vice versa. Coauthors on these works were J. Basso, T. R. Blackman, Y. Borns-Weil, I. Detherage, S. Ganguly, N. Srivastava, T. Saffat, A. Sinclair, and T.-D. Vuong.
Quantum computing is an emerging area between computer science and physics. Numerous problems in quantum computing involve quantum many-body interactions. This dissertation concerns the problem of simulating arbitrary quantum many-body interactions using realistic two-body interactions. To address this issue, a general class of techniques called perturbative reductions (or perturbative gadgets) is adopted from quantum complexity theory and in this dissertation these techniques are improved for experimental considerations. The idea of perturbative reduction is based on the mathematical machinery of perturbation theory in quantum physics. A central theme of this dissertation is then to analyze the combinatorial structure of the perturbation theory as it is used for perturbative reductions.
Aimed at advanced undergraduate and graduate-level students, this textbook covers the core topics of quantum computing in a format designed for a single-semester course. It will be accessible to learners from a range of disciplines, with an understanding of linear algebra being the primary prerequisite. The textbook introduces central concepts such as quantum mechanics, the quantum circuit model, and quantum algorithms, and covers advanced subjects such as the surface code and topological quantum computation. These topics are essential for understanding the role of symmetries in error correction and the stability of quantum architectures, which situate quantum computation within the wider realm of theoretical physics. Graphical representations and exercises are included throughout the book and optional expanded materials are summarized within boxed 'Remarks'. Lecture notes have been made freely available for download from the textbook's webpage, with instructors having additional online access to selected exercise solutions.
This Ph.D. thesis provides a comprehensive review of the state-of-the-art in the field of Variational Quantum Algorithms and Quantum Machine Learning, including numerous original contributions. The first chapters are devoted to a brief summary of quantum computing and an in-depth analysis of variational quantum algorithms. The discussion then shifts to quantum machine learning, where an introduction to the elements of machine learning and statistical learning theory is followed by a review of the most common quantum counterparts of machine learning models. Next, several novel contributions to the field based on previous work are presented, namely: a newly introduced model for a quantum perceptron with applications to recognition and classification tasks; a variational generalization of such a model to reduce the circuit footprint of the proposed architecture; an industrial use case of a quantum autoencoder followed by a quantum classifier used to analyze classical data from an industrial power plant; a study of the entanglement features of quantum neural network circuits; and finally, a noise deconvolution technique to remove a large class of noise when performing arbitrary measurements on qubit systems.