qml-expressivity-trainability-paradox

name	qml-expressivity-trainability-paradox
description	Dynamical Lie Algebra framework for understanding and navigating the expressivity-trainability paradox in Quantum Machine Learning. Shows unstructured QML suffers quantum underfitting from barren plateaus. Symmetry-preserving structural regularization guarantees scalable gradient-rich landscapes. Trainability-by-Design approach.
version	1.0.0
tags	["quantum","machine-learning","barren-plateau","dynamical-lie-algebra","trainability","expressivity"]
source	arXiv:2606.31536
authors	["Kung-Ming Lan","Edward Huang"]
published	2026-06-30T00:00:00.000Z
trigger_words	["QML expressivity trainability paradox","barren plateau DLA","dynamical lie algebra quantum","trainability by design","quantum underfitting","symmetry-preserving QML"]

QML Expressivity-Trainability Paradox via DLA

Core Insight

QML suffers from a expressivity-trainability paradox: the vast Hilbert space capacity of PQCs is the direct mathematical cause of Barren Plateaus. Unstructured QML architectures suffer from quantum underfitting, not overfitting.

Key Findings

1. The Paradox

• Classical deep learning: increasing capacity risks overfitting
• Quantum ML: increasing capacity causes barren plateaus (quantum underfitting)
• The vast Hilbert space = the problem, not the solution

2. DLA Framework

• Links circuit generator algebraic dimension to optimization dynamics
• Exponential DLA growth = exponentially flat gradients
• Polynomial DLA growth = trainable landscapes

3. Trainability-by-Design

• Embed group-theoretic geometric priors as structural regularizers
• Restrict DLA growth to polynomial regime
• Sacrifice raw memorization capacity for scalable, gradient-rich landscapes

Implementation Pattern

1. Analyze your PQC's DLA dimension
2. If DLA grows exponentially with qubit count, expect barren plateaus
3. Identify symmetries in your problem domain
4. Embed symmetry-preserving constraints into circuit architecture
5. Verify DLA growth is polynomial after constraint
6. Train with guaranteed gradient-rich landscape

Practical Applications

Financial QML

• Stock price prediction QNNs: embed market symmetry constraints
• Portfolio optimization: embed permutation symmetries
• Avoid barren plateaus by design, not by luck

General QML

• Any QML task where circuit depth/qubit count scales
• Design circuits with built-in trainability guarantees
• Replace heuristic ansatz design with principled DLA analysis

Activation

Use when:

• Designing QML architectures and worried about barren plateaus
• Analyzing why a QNN fails to train
• Need principled ansatz design methodology
• Building scalable quantum neural networks