On the controllability and quantum speed limit of qubit arrays

Abstract

Universal quantum computing is a coveted goal in the current technological landscape. This thesis presents a theoretical study of qubit arrays, the basic system for processing quantum information, tackling three important problems using results from Lie groups and algebras. Since every real system is subject to decoherence, it is crucial to find quantum devices capable of fast dynamics. The quantum speed limit of a system serves as a measure of the minimum time in which a given unitary evolution can be performed, depending on its controls and qubit couplings. A new method for its estimation is introduced in this thesis. Furthermore, the estimator is extended to determine the quantum speed limit of state-to-state transfers and quantum gates acting on a logical subspace of the total Hilbert space. The number and type of controls and qubit couplings in the system also determine whether all operations in the system are feasible, i.e. whether the system is controllable. Controllability is a necessary property for universality. The thesis presents two controllability tests, which are particularly tailored to the case of qubit arrays: A classical test based on notions from graph theory and a hybrid quantum-classical algorithm that employs parametric quantum circuits. These tests greatly expand the number of cases that can be studied. Finally, a method for designing arbitrarily large controllable qubit arrays is presented. This is achieved by juxtaposing smaller controllable arrays and connecting them via tunable couplings. The modular architecture allows the construction of larger devices that are in principle suitable for universal quantum computing, even if the controllability of the overall system cannot be directly determined by previous tests. The concepts shown here introduce an arsenal of tools that can provide valuable information for the study and development of systems with the aim of bringing quantum technologies closer to universal quantum computing.