# Dynamical transition in controllable quantum neural networks with large depth

arXiv: https://arxiv.org/abs/2311.18144 · HTML: https://mlsanity.com/paper/2311.18144/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 64 references that are in the map.
- First posted on arXiv 29 Nov 2023.

## Builds on (most cited first, 26 of 26)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Non-Gaussian processes and neural networks at finite widths](https://mlsanity.com/paper/1910.00019/) (2019)
- [Asymptotics of Wide Networks from Feynman Diagrams](https://mlsanity.com/paper/1909.11304/) (2019)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [On the infinite width limit of neural networks with a standard parameterization](https://mlsanity.com/paper/2001.07301/) (2020)
- [Exponentially Many Local Minima in Quantum Neural Networks](https://mlsanity.com/paper/2110.02479/) (2021)
- [Symmetric Pruning in Quantum Neural Networks](https://mlsanity.com/paper/2208.14057/) (2022)
- [Learning Unitaries by Gradient Descent](https://mlsanity.com/paper/2001.11897/) (2020)
- [Topological Obstructions to Autoencoding](https://mlsanity.com/paper/2102.08380/) (2021)
- [A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers](https://mlsanity.com/paper/2205.12481/) (2022)
- [Laziness, Barren Plateau, and Noise in Machine Learning](https://mlsanity.com/paper/2206.09313/) (2022)
- [Catapult Dynamics and Phase Transitions in Quadratic Nets](https://mlsanity.com/paper/2301.07737/) (2023)
- [Information Scrambling in Quantum Neural Networks](https://mlsanity.com/paper/1909.11887/) (2019)
- [Expressibility-induced Concentration of Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2311.04965/) (2023)

## Built on by (most cited first, 2 of 2)

- [The curse of random quantum data](https://mlsanity.com/paper/2408.09937/) (2024)
- [Quantum-data-driven dynamical transition in quantum learning](https://mlsanity.com/paper/2410.01955/) (2024)

## Abstract (arXiv)

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.