发表机构
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles; Electrical and Computer Engineering Department, University of California, Los Angeles; Department of Physics, Tsinghua University; Institute of Advanced Study, Tsinghua University（加州大学洛杉矶分校曼尼·L·巴乌米克理论物理研究所; 加州大学洛杉矶分校电气与计算机工程系; 清华大学物理系; 清华大学高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出连续变量量子近似优化算法CV-QAOA，通过变分初始态和绝热演化，在凸二次、旋转双阱等函数上实现低深度高效优化，并引入旋转方阱问题展示超越经典解量化的优势。
AI 中文摘要
我们研究了一种用于高维连续优化的连续变量量子近似优化算法（CV-QAOA）。我们的公式扩展了早期的CV-QAOA提议，加入了变分优化的初始态，并在高深度极限下恢复了量子哈密顿下降（QHD）的收敛保证。我们在几类代价函数上证明了CV-QAOA的严格性能保证。首先，我们展示了d步CV-QAOA通过2d次对代价函数的量子查询即可最小化任何d维严格凸二次函数。然后，我们分析了arXiv:2311.00811引入的一族具有2^d个局部极小值的非凸“旋转双阱”（RDW）函数。虽然先前的工作表明QHD以Õ(d^3)次查询达到其全局最小值，但我们证明1步CV-QAOA仅用两次量子查询即可解决RDW。尽管通用经典求解器需要超多项式时间来解决RDW，而结构感知可以将成本降低到多项式时间，但我们表明1步CV-QAOA协议可以被高效地解量化，并且梯度对齐的线搜索以O(d)次查询成功，几乎匹配信息论下界Ω(d/log d)。为了超越可解量化区域，我们引入了一个“旋转方阱”（RSW）问题，其全局平坦的景观抑制了有用的局部梯度信息。对于这一族，我们展示了由CV-QAOA模拟的绝热演化可以用d^{o(1)}次查询达到全局最小值。另一方面，任何学习RSW中隐藏旋转的经典算法都需Ω(d^2/log d)次查询，我们用一个显式的Θ(d^2 log d)次查询经典算法几乎匹配了这一界限。在偏转波纹弹簧和Easom函数上的数值模拟展示了CV-QAOA在更一般问题上的有前景的性能。
英文摘要
We study a Continuous-Variable Quantum Approximate Optimization Algorithm (CV-QAOA) for high-dimensional continuous optimization. Our formulation extends an earlier CV-QAOA proposal with a variationally optimized initial state and recovers the convergence guarantees of Quantum Hamiltonian Descent (QHD) in the high-depth limit. We prove rigorous performance guarantees of CV-QAOA on several families of cost functions. First, we show $d$-step CV-QAOA minimizes any $d$-dimensional strictly convex quadratic function with $2d$ quantum queries to the cost function. We then analyze a family of nonconvex "Rotated Double Well" (RDW) functions with $2^d$ local minima introduced by arXiv:2311.00811. While prior work showed QHD reaches its global minimum with $\tilde O(d^3)$ queries, we prove that 1-step CV-QAOA solves RDW with just two quantum queries. Although general-purpose classical solvers need superpolynomial time for RDW and structure-awareness can reduce the cost to polynomial time, we show that the 1-step CV-QAOA protocol can be efficiently dequantized, and that a gradient-aligned line search succeeds with $O(d)$ queries, nearly matching the information-theoretic $Ω(d/\log d)$ query lower bound. To move beyond the dequantizable regime, we introduce a ``Rotated Square Well'' (RSW) problem, whose globally flat landscape suppresses useful local gradient information. For this family, we show that an adiabatic evolution simulated by CV-QAOA can reach the global minimum using $d^{o(1)}$ queries. On the other hand, any classical algorithm that learn the hidden rotation in RSW provably requires $Ω(d^2/\log d)$ queries, a bound we nearly match with an explicit $Θ(d^2\log d)$-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.
Comments40+41 pages, 9 figures
