The quantum approximate optimization algorithm (QAOA) is a leading platform for combinatorial optimization in the noisy intermediate-scale quantum (NISQ) era. A central challenge for QAOA and related variational algorithms is how to obtain efficient, provably reliable gradient estimators. We develop a rigorous spectral-filtering framework for the per-layer two-point estimator, a hardware-efficient heuristic. The framework represents the single-parameter slice as a finite Fourier series and analytically quantifies systematic error via a computable bias kernel, yielding a uniform error bound that depends solely on the Hamiltonian, independent of the specific quantum state. Within this framework, we prove that a two-point rule cannot be exact beyond the two-eigenvalue case of the generator. Incorporating shot noise, we derive a closed-form mean-square error (MSE)–optimal step-size rule. Its constants are rendered computable via trace moments, providing practical envelope bounds that avoid spectral enumeration. The optimal step size scales as $M^{-1/6}$ and the minimal MSE as $M^{-2/3}$ in the total number of shots $M$, matching the classical optimum for noisy finite differences. The framework also unifies randomized shifts and multipoint schemes, elevating the two-point method from a heuristic to a gradient estimator with predictable performance, controllable error, and formal guarantees, thereby providing a rigorous scheme for optimizing QAOA and broader classes of variational quantum algorithms on NISQ hardware.

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• Quantum algorithms & computation
• Quantum computation