Exploring Quantum Control Landscape and Solution Space Complexity through Dimensionality Reduction & Optimization Algorithms

TL;DR

This paper investigates the complexity of quantum control landscapes for qubits using dimensionality reduction and compares various optimization algorithms, revealing insights into effective control strategies and the importance of reward design.

Contribution

It introduces the use of PCA for visualizing high-dimensional quantum control landscapes and compares traditional and machine learning algorithms, highlighting their relative performance and the impact of reward functions.

Findings

PCA effectively visualizes complex quantum control landscapes.

Genetic Algorithms outperform SGD in quantum control optimization.

Immediate rewards improve the performance of DQN and PPO.

Abstract

Understanding the quantum control landscape (QCL) is important for designing effective quantum control strategies. In this study, we analyze the QCL for a single two-level quantum system (qubit) using various control strategies. We employ Principal Component Analysis (PCA), to visualize and analyze the QCL for higher dimensional control parameters. Our results indicate that dimensionality reduction techniques such as PCA, can play an important role in understanding the complex nature of quantum control in higher dimensions. Evaluations of traditional control techniques and machine learning algorithms reveal that Genetic Algorithms (GA) outperform Stochastic Gradient Descent (SGD), while Q-learning (QL) shows great promise compared to Deep Q-Networks (DQN) and Proximal Policy Optimization (PPO). Additionally, our experiments highlight the importance of reward function design in DQN and PPO demonstrating that using immediate reward results in improved performance rather than delayed rewards for systems with short time steps. A study of solution space complexity was conducted by using Cluster Density Index (CDI) as a key metric for analyzing the density of optimal solutions in the landscape. The CDI reflects cluster quality and helps determine whether a given algorithm generates regions of high fidelity or not. Our results provide insights into effective quantum control strategies, emphasizing the significance of parameter selection and algorithm optimization.