面向测量高效的双层量子-经典优化的隐式微分方法
Implicit Differentiation for Measurement-Efficient Bilevel Quantum-Classical Optimization
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中文总结 AI 辅助
本文针对含可调外部参数的双层量子优化问题，提出关联复用隐式微分（CR-ID）方法，使预算归一化效率在一维场景提升约4%、多维场景超14%，较有限差分法优势显著。
中文摘要 AI 辅助
量子优化在二次无约束二元优化（QUBO）问题上已展现出良好效果。然而，实际应用中往往存在依赖可调外部因素（如需求预测或风险偏好）的多项式系数，由此产生双层优化结构。本文展示变分量子算法（VQA）如何高效处理这类参数问题，作出三项贡献：其一，针对对角成本哈密顿量提出双层优化模型，其中系数依赖可调外部参数——外层循环调整该参数以重塑成本态势，内层VQA则优化电路变量；其二，由于无导数探测方法在每次外层评估需完成完整内层求解时会产生乘法开销，本文开发关联复用隐式微分（CR-ID）方法，通过复用内层能量估计过程中已采集的量子测量值获取外层梯度，基本无需额外电路执行；其三，实验在三类系数族上开展，结果显示CR-ID在一维场景下使预算归一化效率提升约4%，在多维场景下提升超14%，相比有限差分方法具备显著性能优势，且该特性与架构相关：变分量子本征求解器（VQE）可实现梯度的精确复用，而量子近似优化算法（QAOA）会引入依赖状态的项，造成成本与偏差的权衡。
英文摘要
Quantum optimization has shown promising results for quadratic unconstrained binary optimization (QUBO) problems. Real-world applications, however, often involve polynomial coefficients that depend on tunable external factors - such as demand forecasts or risk preferences - giving rise to bilevel optimization structures. We show how variational quantum algorithms (VQAs) can efficiently handle such parametric problems, making three contributions. First, we propose a bilevel optimization model for diagonal cost Hamiltonians where coefficients depend on a tunable outer parameter: an outer loop adjusts this parameter - reshaping the cost landscape - while an inner VQA optimizes circuit variables. Second, since derivative-free probing methods incur a multiplicative overhead when each outer evaluation requires a complete inner solve, we develop correlator-reuse implicit differentiation (CR-ID), which obtains outer gradients by reusing quantum measurements already collected during inner energy estimation, requiring essentially no additional circuit executions. Experiments across three coefficient families show that CR-ID consistently improves budget-normalized efficiency by ~4\% in 1D and over 14\% in multi-dimensional settings, showing a significant performance advantage compared to finite-difference methods. Third, we show that this property is architecture-dependent: VQE admits exact reuse gradients, whereas QAOA introduces a state-dependent term that creates a cost-bias trade-off.
