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

TL;DR

The paper tackles visualizing and optimizing the quantum control landscape (QCL) for a single qubit by applying principal component analysis (PCA) to high-dimensional control pulses and introducing a cluster density index (CDI) to quantify solution-space density. It systematically benchmarks traditional optimization (SGD) against genetic algorithms (GA) and reinforcement-learning (QL, DQN, PPO) across 2–4 parameter control spaces, highlighting that GA and QL reliably locate high-fidelity solutions, while RL methods require well-tuned reward schemes for short episodes. A thorough analysis of solution-space structure via CDI reveals that GA and QL form dense, high-fidelity clusters, with CDI increasing as parameter count grows for several algorithms. The findings advocate dimensionality reduction as a practical tool for QCL studies and guide algorithm selection and reward design for quantum control tasks, with potential extensions to multi-qubit and more complex systems.

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.

Figures (8)

• Figure 1: High level diagram: The basic steps involved in this work: (a) the state transfer problem, (b) the brute-force solution (by converting the continuous control signal into discrete values ), (c) optimization algorithms based solution
• Figure 2: Quantum landscape using brute-force data: (a) 2 parameter Larocca_2018, (b) 3 parameter, and (c) 3 parameter after applying PCA. Data points were created by generating every possible combination of values in the range [-1, 1] for each axis, with each axis divided into 100 intervals. Axis labels a1, a2 and a3 represent parameters 1, 2 and 3 in each axis, pc1 and pc2 represent the first two principal components.
• Figure 3: Quantum control landscape after applying PCA and extracting regions of high fidelity (fidelity above 0.95): (a) Two parameter, (b) Three parameter, and (c) Four parameter, the dataset which is used to generate these plots is generated using a brute-force combination of inputs in the range [-1, 1]
• Figure 4: The quantum landscape when using SGD and GA for 1000 tests and after applying PCA (a) SGD - 2 parameter, (b) SGD - 3 parameter, (c) SGD - 4 parameter, (d) GA - 2 parameter, (e) GA - 3 parameter, and (f) GA - 4 parameter
• Figure 5: The quantum landscape when using QL for 1000 tests and after applying PCA (a) 2 parameter, (b) 3 parameter, and (c) 4 parameter