Numerical Optimization Methods in the environment with Quantum Noise

Abstract

The accurate calculation of electronic potential energy surfaces, encompassing both ground and excited states, is paramount in quantum chemistry, particularly for understanding photochemical processes mediated by features like conical intersections. While classical methods like Full Configuration Interaction provides benchmark accuracy, their exponential scaling limits applicability. Approximate methods such as Complete Active Space Self-Consistent Field struggle near degeneracies. The advent of noisy intermediate-scale quantum computers offers potential alternatives, but algorithms like the Variational Quantum Eigensolver are constrained by hardware limitations, often requiring small active spaces.

This thesis focuses on the State-Averaged Orbital-Optimized Variational Quantum Eigensolver algorithm, a hybrid quantum-classical approach designed to provide a balanced description of multiple electronic states by combining quantum state preparation with classical state-averaged orbital optimization. The work involves contributing to the State-Averaged Orbital-Optimized Variational Quantum Eigensolver software package, specifically its optimization module.

A key contribution is the implementation and evaluation of the Differential Evolution, a global optimization algorithm within the State-Averaged Orbital-Optimized Variational Quantum Eigensolver framework, alongside a comparative study against other classical optimizers (Gradient Descent, Broyden-Fletcher-Goldfarb-Shanno algorithm, Constrained Optimization by Linear Approximation, Sequential Least Squares Programming). The performance of these optimizers is assessed for calculating the ground and first excited state energies of H$_2$, H$_4$, and LiH molecules.

Furthermore, the thesis demonstrates the capability of State-Averaged Orbital-Optimized Variational Quantum Eigensolver to accurately model potential energy surfaces near conical intersections, using the formaldimine molecule as a case study. The results highlight the importance of orbital optimization for capturing the correct potential energy surfaces topology in regions of strong non-adiabatic coupling, a task where standard State-Averaged Variational Quantum Eigensolver with fixed orbitals fails. The findings indicate that while Differential Evolution presents challenges in efficiency and robustness for these specific State-Averaged Orbital-Optimized Variational Quantum Eigensolver tasks, gradient-based methods like Broyden-Fletcher-Goldfarb-Shanno algorithm and Sequential Least Squares Programming offer superior performance, and the State-Averaged Orbital-Optimized Variational Quantum Eigensolver approach itself is crucial for treating complex electronic structures like conical intersections.