Dynamical transition in controllable quantum neural networks with large depth

TL;DR

This paper reveals a dynamical transition in the training behavior of deep quantum neural networks, characterized by a bifurcation in the loss landscape, with implications for training speed and convergence.

Contribution

It introduces a non-perturbative analytical framework describing the transition in quantum neural network training dynamics using generalized Lotka-Volterra equations.

Findings

Identifies a transcritical bifurcation in training dynamics.

Shows quadratic loss enables faster convergence.

Verifies theory with experiments on IBM quantum devices.

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.