Implicit Differentiation for Measurement-Efficient Bilevel Quantum-Classical Optimization

• URL: http://arxiv.org/abs/2608.07717v1
• Date: Fri, 07 Aug 2026 19:03:43 GMT
• Title: Implicit Differentiation for Measurement-Efficient Bilevel Quantum-Classical Optimization
• Abstract summary: We show how variational quantum algorithms (VQAs) can efficiently handle parametric problems.<n>We propose a bilevel optimization model for diagonal cost Hamiltonians where coefficients depend on a tunable outer parameter.<n>Experiments across three families show that CR-ID consistently improves budget-normalized efficiency by 4% in 1D and over 14% in multi-dimensional settings.
• Score: 2.5234274237739402
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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