Resumen de Quantum algorithms for condensed-matter physics, number theory and quantum machine learning
Quantum computing is a novel paradigm that seeks to harness quantum mechanical e ects like superposition, entanglement and interference to outperform classical computers based upon bits (i.e. zeroes and ones) on certain tasks. Research into the eld of quantum algorithms has produced some exponential speed-ups over the best classical algorithms currently known, with potential applications far beyond the academic domain. Yet, the quest for quantum algorithms that dramatically outperform their classical counterparts has proved to be hard. New quantum algorithms are needed, together with a better understanding of the type of problems quantum computers excel at. In this thesis, we explore the capabilities that quantum computers may o er to Condensed-Matter Physics, Number Theory and Quantum Machine Learning. We do so by introducing novel quantum algorithms, both exact and variational, in Chapters 2, 3 and 5. Moreover, we provide in Chapter 4 the rst theoretical study of the so-called overparametrization phenomenon in variational quantum circuits, which is likely to play a relevant role in Quantum Machine Learning. In Chapter 6, we analyze the scalability of certain Hamiltonian learning quantum algorithms that rely on Bayesian inference to a large number of qubits. Chapter 2 also contains original contributions to Quantum Number Theory, a novel approach to Number Theory that employs the tools provided by the formalism of Quantum Information Theory to study mathematical objects like the prime numbers. Finally, in Appendix A we introduce qibo, an open-source, high-performance library for the classical simulation of quantum circuits, that has been used in most of the works presented in this thesis
