coherence-law-noisy-equivariant-qnn

name	coherence-law-noisy-equivariant-qnn
description	Coherence law for trainability in noisy equivariant quantum neural networks. Proves that readout-visible sector coherence determines gradient survival under decoherence, not just symmetry structure.
category	ai_collection
tags	["quantum-neural-networks","equivariant-circuits","noisy-qml","trainability","coherence-law","barren-plateaus"]
trigger_words	["coherence law trainability QNN","equivariant quantum neural network noise","noisy QNN gradient decay","sector coherence trainability","light-cone reduction quantum","open-system quantum trainability"]

Coherence Law for Trainability in Noisy Equivariant QNNs

Background

Symmetry provides structure for quantum neural networks, but does not guarantee trainability once noise is present. The expressivity-trainability paradox shows that unstructured QML architectures suffer from quantum underfitting driven by barren plateaus. This skill provides a coherence law that determines whether gradients survive decoherence.

Core Methodology

1. Light-Cone Reduction

• Causality fixes where the gradient can live — confined to the backward light cone of the readout
• Light-cone reduction pins the noiseless gradient to the sector-restricted cone
• Lower bound independent of total qubit number

2. Coherence Contraction Law

• Coherence determines how fast the gradient decays through contraction of off-diagonal sector modes
• Readout-visible aligned coherence rate defined as Rayleigh quotient of noise generator along gradient-carrying mode
• Perturbative open-system analysis converts this rate into a leading-order training law

3. Training Law

• Finite-noise degradation follows a single accumulated variable built from noise depth and coherence contraction
• Coefficient of determination R² = 0.979 between predicted and observed gradient degradation
• Sector coherence outperforms every standard channel diagnostic

4. Key Insight

• For correlated-dephasing channels with large worst-case rate but near-zero aligned rate:
  • The law predicts no gradient loss → none is observed
  • Sector coherence is the quantity linking equivariant architecture, open-system dynamics, and noisy trainability

Implementation Guidelines

# Pseudocode for coherence-aware QNN training
def coherence_aligned_rate(noise_generator, gradient_mode, readout_projector):
    """Calculate readout-visible aligned coherence rate as Rayleigh quotient"""
    # Project noise generator onto readout-visible subspace
    projected_noise = readout_projector @ noise_generator @ readout_projector
    # Rayleigh quotient along gradient-carrying mode
    rate = (gradient_mode.T @ projected_noise @ gradient_mode) / (gradient_mode.T @ gradient_mode)
    return rate

def predict_gradient_decay(coherence_rate, circuit_depth, noise_strength):
    """Leading-order training law for gradient degradation"""
    accumulated = coherence_rate * circuit_depth * noise_strength
    decay_factor = np.exp(-accumulated)
    return decay_factor

Applications

• Noisy quantum neural network training
• Equivariant quantum circuit design
• Open-system quantum dynamics
• Quantum machine learning robustness
• Barren plateau mitigation strategies

Pitfalls

• Symmetry alone does NOT guarantee trainability under noise — coherence is the deciding factor
• The aligned coherence rate can be near-zero even when worst-case channel rate is large
• Light-cone confinement means gradients only exist in specific sectors — design readout accordingly
• This law applies to U(1)-equivariant brickwork circuits — verify applicability for other symmetries

References

• arXiv:2606.30688 — "A Coherence Law for Trainability in Noisy Equivariant Quantum Neural Networks"
• Ugail, Howard (2026)