Principal Component Analysis and t-Distributed Stochastic Neighbor Embedding Analysis in the Study of Quantum...

Area of Science:

• Quantum Computing
• Machine Learning
• Data Analysis

Background:

• The Quantum Approximate Optimization Algorithm (QAOA) is a prominent algorithm for solving optimization problems.
• Understanding the role of mixing operators, particularly entangled ones, is crucial for enhancing QAOA performance.
• Dimensionality reduction techniques like PCA and t-SNE offer powerful tools for analyzing complex quantum system behaviors.

Purpose of the Study:

• To investigate and compare the behavior of entangled and non-entangled mixing operators within QAOA.
• To analyze the impact of QAOA depth on the performance and information preservation capabilities of these operators.
• To leverage PCA and t-SNE for a deeper understanding of parameter landscapes and model distinctions.

Main Methods:

• Utilized Principal Component Analysis (PCA) and t-distributed Stochastic Neighbor Embedding (t-SNE) for data visualization and analysis.
• Employed a dataset of optimized parameters for QAOA applied to max-cut problems (cyclic and complete configurations).
• Analyzed QAOA models at various depths (1L, 2L, 3L) with and without entanglement in the mixing operator, examining RZ, RX, and RY parameters.

Main Results:

• PCA and t-SNE analyses revealed distinct patterns for entangled versus non-entangled QAOA models.
• Entangled QAOA models generally exhibited superior information preservation in their parameter mappings.
• Quantifiable differences were observed in explained variance (PCA) and Kullback-Leibler divergence (t-SNE), with entangled models showing greater information correlation and clustering.

Conclusions:

• Entangled mixing operators offer advantages in information preservation and correlation within QAOA.
• PCA and t-SNE are effective in differentiating the behavior of entangled and non-entangled QAOA models.
• The findings suggest that incorporating entanglement in QAOA mixing operators can lead to more robust and informative parameter spaces.