Variational Quantum Algorithms are promising candidates for near-term quantum computing, yet they face scalability challenges due to barren plateaus, where gradients vanish exponentially relative to system size. Recent conjectures suggest that avoiding these plateaus might inherently lead to classical simulability, thereby limiting the opportunities for quantum advantage. In this work, we advance the theoretical understanding of the relationship between gradient scalability at initialization and the computational complexity of variational quantum algorithms. We first present the Taylor surrogate, a classical simulation technique that matches Pauli path runtime guarantees on near-Clifford regions while offering runtime advantages in specific regimes. Leveraging this surrogate, we prove that beyond previously established classically simulable regions, the computational complexity is at least super-polynomial. Next, we introduce the Linear Clifford Encoder, a classically efficient ansatz modifier that ensures constant-scaling gradients within landscape regions close to Clifford circuits. Finally, numerical experiments on these modified landscapes provide preliminary empirical evidence of a transition zone where constant-scaling gradients may decay polynomially in super-polynomially complex regions rather than exponentially. These findings suggest speculative instances where non-vanishing gradients and super-polynomial complexity could potentially coexist, vindicating the need for future formal proofs.

翻译：变分量子算法是近期量子计算领域颇具前景的候选方案，但其面临由于贫瘠高原（barren plateaus）导致的可扩展性挑战——在此类现象中，梯度随系统规模呈指数级衰减。近期猜想指出，规避此类高原可能本质上导致经典可模拟性，从而限制量子优势的机遇。本研究推进了对初始化阶段梯度可扩展性与变分量子算法计算复杂度之间关系的理论认知。首先，我们提出泰勒替代（Taylor surrogate）这一经典模拟技术，其在近克利福德区域匹配Pauli路径运行时间保障，并在特定场景中提供运行时间优势。借助该替代方法，我们证明在先前建立的经典可模拟区域之外，计算复杂度至少为超多项式级。其次，我们引入线性克利福德编码器（Linear Clifford Encoder）这一经典高效变分修饰方案，确保靠近克利福德电路的景观区域内梯度呈恒定标度。最后，针对经过修饰的量子景观开展数值实验，提供了过渡区域的初步经验证据：在该区域内，恒定标度梯度可能于超多项式复杂区域呈现多项式级而非指数级衰减。这些发现指明了梯度非零与超多项式复杂度可能共存的推测性实例，验证了未来开展形式化证明的必要性。