Towards Heisenberg Scaling: Measurement-Efficient Non-Orthogonal Quantum Eigensolver

The Non-Orthogonal Quantum Eigensolver (NOQE) provides an accurate framework for electronic-structure calculations, but the estimation of its Hamiltonian and overlap matrix elements relies on sampling and requires O\(1/varepsilon²\) circuit repetitions to achieve additive precision varepsilon. Here, we reformulate this matrix-element estimation step as a collection of amplitude-estimation tasks and integrate iterative quantum amplitude estimation into the NOQE workflow. The resulting protocol achieves near-Heisenberg query complexity O\(1/varepsilon\) for these estimation tasks, by replacing incoherent statistical averaging with coherent amplitude amplification. We present explicit circuit constructions and the corresponding implementation procedure. Numerical simulations for the electronic states of the hydrogen molecule show that the proposed method reaches chemical accuracy with substantially fewer total queries than the original sampling-based protocol. Overall, this work provides a measurement-efficient route to high-precision energy estimation and illustrates how sampling-limited quantum algorithms can be systematically reformulated to leverage quantum coherence and achieve lower measurement costs.