Quantum-data-driven dynamical transition in quantum learning

2024-10-02

arXiv.org

AI Summary

This work investigates how quantum data influence the training dynamics of quantum neural networks (QNNs) in supervised learning. Addressing the critical problem that convergence behavior depends jointly on the objective function and input quantum data, we construct the first comprehensive phase diagram—spanning seven distinct dynamical phases—and identify a phase transition between polynomial and exponential convergence induced synergistically by target values and quantum data. We uncover a high-codimension bifurcation mechanism and develop a non-perturbative, generalized restricted Haar ensemble theory to explain abrupt transitions in convergence properties. Our methodology integrates analytical dynamical modeling, bifurcation analysis, restricted Haar random matrix theory, large-scale QNN numerical simulations, and experimental validation on IBM’s superconducting quantum hardware. Theoretical predictions are consistently corroborated by both simulation and experiment. This work establishes convergence-speed optimization principles for cost-function design in the NISQ era, significantly enhancing quantum learning training efficiency.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Quantum circuits are an essential ingredient of quantum information processing. Parameterized quantum circuits optimized under a specific cost function -- quantum neural networks (QNNs) -- provide a paradigm for achieving quantum advantage in the near term. Understanding QNN training dynamics is crucial for optimizing their performance. In terms of supervised learning tasks such as classification and regression for large datasets, the role of quantum data in QNN training dynamics remains unclear. We reveal a quantum-data-driven dynamical transition, where the target value and data determine the polynomial or exponential convergence of the training. We analytically derive the complete classification of fixed points from the dynamical equation and reveal a comprehensive `phase diagram' featuring seven distinct dynamics. These dynamics originate from a bifurcation transition with multiple codimensions induced by training data, extending the transcritical bifurcation in simple optimization tasks. Furthermore, perturbative analyses identify an exponential convergence class and a polynomial convergence class among the seven dynamics. We provide a non-perturbative theory to explain the transition via generalized restricted Haar ensemble. The analytical results are confirmed with numerical simulations of QNN training and experimental verification on IBM quantum devices. As the QNN training dynamics is determined by the choice of the target value, our findings provide guidance on constructing the cost function to optimize the speed of convergence.

Problem

• Understanding QNN training dynamics with quantum data
• Analyzing convergence in quantum supervised learning tasks
• Identifying dynamical transitions in quantum neural networks

Innovation

• Quantum neural networks optimize parameterized quantum circuits
• Analytical classification reveals seven distinct dynamical phases
• Non-perturbative theory explains transition via Haar ensemble