UC Santa Barbara

Quantum computers, once their fault-tolerant versions are built, are widely believed to revolutionize the world, bringing an exponential quantum advantage to problems in chemistry, physics and material science. In this thesis, we explore various problems in theoretical condensed matter physics that are in one way or another related to quantum computers. The first part of this thesis is devoted to Majorana zero-energy modes and topological quantum computing. We describe how Majorana modes can be synthesized in superconductor-semiconductor heterostructures, and emphasize vulnerabilities of the topological phase in such heterostructures to external magnetic fields. We further demonstrate how and to what degree the addition of a magnetic insulator or spin-orbit scattering to a heterostructure can help alleviating the negative effects of external magnetic fields. Additionally, we analyze quantum dot-based measurements of Majorana qubits. In particular, we identify the optimal regime for such measurements, and estimate to what degree external noise sources and coupling of the quantum dot to quasiparticle modes in superconductors affect the fidelity of the measurements. The other part of this thesis is devoted to the many-body physics in quantum circuits. We describe a Floquet circuit model in which projective measurements can drive a dynamical spectral phase transition in otherwise purely unitary dynamics of the circuit. Using random-matrix theory, we analytically calculate various properties of this transition, and analyze how the transition can be revealed in the long-time dynamics of the system.