Implicit differentiation for measurement-efficient bilevel quantum-classical optimization

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, thus 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 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, thus 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: the Variational Quantum Eigensolver (VQE) admits exact reuse gradients, whereas the Quantum Approximate Optimization Algorithm (QAOA) introduces a state-dependent term that creates a cost–bias trade-off.