非凸函数全局优化中量子优势的机制
Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions
AI总结：
本文通过 Schrödinger 算子谱性质与 Langevin 扩散混合时间的对应，提出非隧穿型量子加速机制，并证明 RsAA 在多类非凸函数上相较经典算法具有多项式时间优势。
摘要：
我们提出了用于非凸函数全局优化中量子加速的新理论机制，将量子优势的范围扩展到传统的基于隧穿的解释之外。作为我们的主要构建模块，我们证明了 Schrödinger 算子的谱性质与经典 Langevin 扩散的混合时间之间存在严格对应关系。这一对应关系引出了一种在具有唯一全局最小值的函数上实现分离的机制：量子算法在原始势上运行，而经典扩散对应于一个 Schrödinger 算子，其 WKB 势具有近乎简并的全局最小值。我们通过证明实空间绝热量子算法（RsAA）能够对广泛的非凸函数族实现可证明的多项式时间优化，将这些思想形式化。首先，对于块可分函数，我们表明 RsAA 保持多项式运行时间，而已知的现成算法需要指数时间，且利用结构的算法表现出任意大的多项式运行时间。这些结果利用了半经典分析中新颖的非渐近结果。其次，我们利用内禀超压缩性理论的最新进展，证明 RsAA 在缺乏全局结构的、经过适当扰动的强凸函数上具有多项式运行时间，而现成算法仍受到指数瓶颈。与先前基于量子隧穿的工作相比，这些分离并不依赖于局部最小值之间势垒的几何形状。我们关于经典算法运行时间的理论主张得到了严格分析和全面数值基准测试的支持。这些发现为连续优化中的量子优势建立了严格的理论基础，并开辟了连接量子算法、随机过程和半经典分析的新研究方向。
英文摘要：
We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.
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