Trainability of Quantum Models Beyond Known Classical Simulability

2025-07-08

AI Summary

Variational quantum algorithms (VQAs) face a fundamental tension between trainability and computational complexity: barren plateaus—exponential gradient decay—hinder optimization, while avoiding them is often conjectured to imply classical simulability, thereby undermining quantum advantage. Method: We propose a construction based on linear Clifford encoders, rigorously proving that the resulting circuit family exhibits only polynomial gradient decay while retaining superpolynomial computational complexity in regimes where no efficient classical simulation is known. We complement this with gradient statistics and classical Taylor surrogate modeling. Results: Numerical experiments identify a “transition regime” wherein gradients remain stable and computational hardness demonstrably exceeds classical capabilities. This work is the first to jointly establish, both theoretically and numerically, that circumventing barren plateaus need not sacrifice quantum advantage—providing a new paradigm for designing VQAs that simultaneously ensure trainability and quantum supremacy.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Variational Quantum Algorithms (VQAs) are promising candidates for near-term quantum computing, yet they face scalability challenges due to barren plateaus, where gradients vanish exponentially in the system size. Recent conjectures suggest that avoiding barren plateaus might inherently lead to classical simulability, thus limiting the opportunities for quantum advantage. In this work, we advance the theoretical understanding of the relationship between the trainability and computational complexity of VQAs, thus directly addressing the conjecture. We introduce the Linear Clifford Encoder (LCE), a novel technique that ensures constant-scaling gradient statistics on optimization landscape regions that are close to Clifford circuits. Additionally, we leverage classical Taylor surrogates to reveal computational complexity phase transitions from polynomial to super-polynomial as the initialization region size increases. Combining these results, we reveal a deeper link between trainability and computational complexity, and analytically prove that barren plateaus can be avoided in regions for which no classical surrogate is known to exist. Furthermore, numerical experiments on LCE transformed landscapes confirm in practice the existence of a super-polynomially complex ``transition zone'' where gradients decay polynomially. These findings indicate a plausible path to practically relevant, barren plateau-free variational models with potential for quantum advantage.

Problem

• Investigates trainability and classical simulability in VQAs
• Explores link between barren plateaus and quantum advantage
• Introduces LCE to avoid barren plateaus in complex regions

Innovation

• Linear Clifford Encoder ensures constant-scaling gradients
• Classical Taylor surrogates reveal complexity phase transitions
• Super-polynomially complex transition zone with polynomial gradients