Improving variational quantum algorithms : Innovative initialization techniques and extensions to qudit systems

This Ph.D. thesis presents a detailed investigation into Variational Quantum Algorithms (VQAs), a promising class of quantum algorithms that are well suited for near-term quantum computation due to their moderate hardware requirements and resilience to noise. Our primary focus lies on two particular types of VQAs: the Quantum Approximate Optimization Algorithm (QAOA), used for solving binary optimization problems, and the Variational Quantum Eigensolver (VQE), utilized for finding ground states of quantum many-body systems. In the first part of the thesis, we examine the issue of effective parameter initialization for the QAOA. The work demonstrates that random initialization of the QAOA often leads to convergence in local minima with sub-optimal performance. To mitigate this issue, we propose an initialization of QAOA parameters based on the Trotterized Quantum Annealing (TQA). We show that TQA initialization leads to the same performance as the best of an exponentially scaling number of random initializations. The second study introduces Transition States (TS), stationary points with a single direction of descent, as a tool for systematically exploring the QAOA optimization landscape. This leads us to propose a novel greedy parameter initialization strategy that guarantees for the energy to decrease with increasing number of circuit layers. In the third section, we extend the QAOA to qudit systems, which are higher-dimensional generalizations of qubits. This chapter provides theoretical insights and practical strategies for leveraging the increased computational power of qudits in the context of quantum optimization algorithms and suggests a quantum circuit for implementing the algorithm on an ion trap quantum computer. Finally, we propose an algorithm to avoid “barren plateaus”, regions in parameter space with vanishing gradients that obstruct efficient parameter optimization. This novel approach relies on defining a notion of weak barren plateaus based on the entropies of local reduced density matrices and showcases how these can be efficiently quantified using shadow tomography. To illustrate the approach we employ the strategy in the VQE and show that it allows to successfully avoid barren plateaus in the initialization and throughout the optimization. Taken together, this thesis greatly enhances our understanding of parameter initialization and optimization in VQAs, expands the scope of QAOA to higher-dimensional quantum systems, and presents a method to address the challenge of barren plateaus using the VQE. These insights are instrumental in advancing the field of near-term quantum computation.