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

The Non-Orthogonal Quantum Eigensolver (NOQE) [1] was recently proposed for accurate electronic-structure calculations of systems with both strong and weak correlations, with a measurement cost scaling as 1/ϵ². Here, we develop a protocol that reframes the matrix-element calculation as amplitude-estimation tasks to achieve near-Heisenberg 1/ϵ scaling, significantly reducing the overall measurement cost. We design explicit circuits to estimate overlap and Hamiltonian matrices. Numerical simulations on the hydrogen molecule demonstrate that a target precision of ϵ ≈ 10⁻³ can be achieved with roughly two orders of magnitude fewer total shots than the original protocol, at the cost of only a 3- to 5-fold increase in circuit depth. Overall, this approach provides a promising high-precision, measurement-efficient route to energy estimation suited to the early fault-tolerant regime.

[1] Baek et al., PRX Quantum 4, 030307 (2023)