Quantum computation at the edge of chaos

TL;DR

This paper introduces quantum sparsity using topological Entanglement Entropy as a regularizer to improve variational quantum algorithms by guiding them along the edge of chaos, enhancing convergence and robustness.

Contribution

It proposes a novel quantum sparsity principle with a TEE-based regularizer, linking structural complexity to quantum state encoding and providing a quantum sampling theorem.

Findings

TEE regularizer improves convergence in VQAs

Negative TEE indicates untrainable chaos

Numerical results show enhanced precision in complex tasks

Abstract

A key challenge in classical machine learning is to mitigate overparameterization by selecting sparse solutions. We translate this concept to the quantum domain, introducing quantum sparsity as a principle based on minimizing quantum information shared across multiple parties. This allows us to address fundamental issues in quantum data processing and convergence issues such as the barren plateau problem in Variational Quantum Algorithm (VQA). We propose a practical implementation of this principle using the topological Entanglement Entropy (TEE) as a cost function regularizer. A non-negative TEE is associated with states with a sparse structure in a suitable basis, while a negative TEE signals untrainable chaos. The regularizer, therefore, guides the optimization along the critical edge of chaos that separates these regimes. We link the TEE to structural complexity by analyzing quantum states encoding functions of tunable smoothness, deriving a quantum Nyquist-Shannon sampling theorem that bounds the resource requirements and error propagation in VQA. Numerically, our TEE regularizer demonstrates significantly improved convergence and precision for complex data encoding and ground-state search tasks. This work establishes quantum sparsity as a design principle for robust and efficient VQAs.