Dynamical transition in controllable quantum neural networks with large depth
arXiv:2311.18144 · doi:10.1038/s41467-024-53769-2
Abstract
Understanding the training dynamics of quantum neural networks is a fundamental task in quantum information science with wide impact in physics, chemistry and machine learning. In this work, we show that the late-time training dynamics of quantum neural networks with a quadratic loss function can be described by the generalized Lotka-Volterra equations, which lead to a transcritical bifurcation transition in the dynamics. When the targeted value of loss function crosses the minimum achievable value from above to below, the dynamics evolve from a frozen-kernel dynamics to a frozen-error dynamics, showing a duality between the quantum neural tangent kernel and the total error. In both regions, the convergence towards the fixed point is exponential, while at the critical point becomes polynomial. We provide a non-perturbative analytical theory to explain the transition via a restricted Haar ensemble at late time, when the output state approaches the steady state. Via mapping the Hessian to an effective Hamiltonian, we also identify a linearly vanishing gap at the transition point. Compared with the linear loss function, we show that a quadratic loss function within the frozen-error dynamics enables a speedup in the training convergence. The theory findings are verified experimentally on IBM quantum devices.
11+45 pages, comments are welcomed
References in corpus (43)
- Quantum Machine Learning
- A variational eigenvalue solver on a quantum processor
- Variational Quantum Algorithms
- Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
- Supervised learning with quantum enhanced feature spaces
- Barren plateaus in quantum neural network training landscapes
- The theory of variational hybrid quantum-classical algorithms
- A Quantum Approximate Optimization Algorithm
- Quantum machine learning in feature Hilbert spaces
- Quantum computational chemistry
- Quantum Convolutional Neural Networks
- Neural Tangent Kernel: Convergence and Generalization in Neural Networks
- An introduction to quantum machine learning
- Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
- Deep Neural Networks as Gaussian Processes
- Chaos and complexity by design
- Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays
- Theory of variational quantum simulation
- Real-time quantum error correction beyond break-even
- Why does deep and cheap learning work so well?
- On Exact Computation with an Infinitely Wide Neural Net
- Chaos, Complexity, and Random Matrices
- A review of progress in the physics of open quantum systems: theory and experiment
- The Principles of Deep Learning Theory
- Exploring entanglement and optimization within the Hamiltonian Variational Ansatz
- Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps
- Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
- Beating the break-even point with a discrete-variable-encoded logical qubit
- Theory of overparametrization in quantum neural networks
- Adaptive Variational Quantum Dynamics Simulations
- Spectral form factors and late time quantum chaos
- Representation Learning via Quantum Neural Tangent Kernels
- Information Scrambling in Quantum Neural Networks
- Analytic theory for the dynamics of wide quantum neural networks
- Asymptotics of Wide Networks from Feynman Diagrams
- Topological Obstructions to Autoencoding
- Learning Unitaries by Gradient Descent
- Scrambling and decoding the charged quantum information
- A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers
- Critical Points in Quantum Generative Models
- Quantifying scrambling in quantum neural networks
- Symmetric Pruning in Quantum Neural Networks
- Entangling remote microwave quantum computers with hybrid entanglement swap and variational distillation