Location
online (email amgrad@uwaterloo.ca for link)
Candidate
Einar Gabbassov| Applied Mathematics, University of Waterloo
Title
Adiabatic Dynamics of Entanglement, Constraint-Aware Quantum Optimization, and Stochastic Schr\"odinger Equations for Quantum Reverse Diffusion
Abstract
This thesis studies dynamical information flow in quantum systems, focusing on two distinct topics. The first topic examines how entanglement restructures itself in adiabatic and discretized adiabatic processes, and how this affects computational complexity. The second topic is about quantum information recovery and stochastic control in noisy quantum processes.
First, we show that, during evolution generated by a time-dependent Hamiltonian, any changes in entanglement are attributed to a succession of avoided level crossings in the Hamiltonian’s instantaneous energy spectrum. At these avoided crossings, eigenvalues swap their associated eigenvectors. These swaps mediate the restructuring of entanglement: creation, redistribution and destruction of entanglement in multipartite systems. The efficiency of this restructuring depends on the narrowness of the avoided level crossings and thus constrains the speed of adiabatic evolution. Moreover, we relate the amount of entanglement restructuring to the ruggedness of the energy landscape, which directly affects the hardness of a computational problem. This enables an analysis of computational complexity and quantum advantage from the point of view of entanglement requirements.
Second, we apply this framework to constrained combinatorial optimization. We show how the algebraic and geometric structure of constraints induces invariant subspaces, quantum state transitions, avoided level crossings, and the entanglement restructuring required during computation. We show how constraint-aware quantum algorithms can exploit these features and avoid unnecessary entanglement restructuring, thereby avoiding computational bottlenecks. These results provide guiding principles for designing effective quantum optimization algorithms across continuous-time, circuit-based, and measurement-based paradigms.
Finally, we investigate information recovery in open quantum systems. Ensemble-averaged noisy dynamics are generally irreversible. Noisy quantum channels have no meaningful inverses, and their time-dependent descriptions via master equations have irreversible dynamics. However, this irreversibility need not persist at the level of individual monitored quantum state trajectories. We derive novel physically principled stochastic Schr\"odinger equations, along with corresponding stochastic master equations, that describe exact and approximate reverse processes (dynamics) for time-dependent Pauli and depolarizing noises. The derived reverse processes recover the original state or its distribution while evolving under the same noise conditions. We show that the newly derived reverse processes generalize the well-known forward processes. The corresponding stochastic differential equations admit closed-form solutions and can be implemented in real time without variational techniques or quantum tomography.