Complexity and Randomness in the Dynamics of Quantum Systems

Abstract

This thesis explores two complementary facets of quantum dynamics: the growth of operator complexity in open quantum systems and the certification of quantum randomness from temporal correlations in single-qubit dynamics. While these problems differ in motivation and methodology, they are unified by a central theme: understanding how quantum information evolves under coherent and dissipative dynamics, and how this evolution can be harnessed both as a diagnostic and as a resource. The first part of the thesis addresses the question: How can we quantify the complexity of information spreading in open quantum systems? To this end, we extend the framework of Krylov complexity, originally developed for closed, unitary systems to open quantum systems governed by non-unitary Lindbladian evolution. By adapting iterative algorithms such as the Arnoldi and bi-Lanczos procedures, we construct Krylov bases in non-Hermitian settings and define a normalized complexity measure that accounts for probability decay. We derive quantum speed limits on operator growth, identify regimes of universal complexity saturation, and demonstrate how Krylov complexity remains sensitive to signatures of integrability and chaos even in the presence of dissipation and decoherence. The second part of the thesis poses the question: How can we certify genuine quantum randomness without relying on spatially separated entangled systems? We develop a semi-device-independent protocol for randomness certification based solely on temporal quantum correlations. This method leverages violations of the Leggett- Garg inequality, along with compliance with the No-Signaling-in-Time condition, to establish rigorous lower bounds on certifiable randomness. We implement this protocol on both a photonic setup and IBM’s superconducting quantum processors, demonstrating that single-qubit circuits with low depth can produce certified randomness, even in the presence of realistic noise, without requiring spatial separation or entanglement. Together, these investigations provide new tools for characterizing the dynamics of quantum systems and highlight how foundational features of quantum evolution, complexity and randomness, can be practically accessed and applied in near-term quantum technologies.