Efficient Energy Measurement of Chemical Systems via One-Particle Reduced Density Matrix: A NOF-VQE Approach for Optimized Sampling

In this work, we explore the use of the one-particle reduced density matrix (1RDM) to streamline energy measurements of chemical systems on quantum computers, particularly within the variational quantum eigensolver (VQE) framework. This approach leverages the existence of an exact energy functional of the 1RDM, enabling a reduction in both the number of expectation values to measure and the number of circuits to execute, thereby optimizing quantum resource usage. Specifically, sampling the 1RDM involves measuring only [Formula: see text] elements, which is significantly fewer than the [Formula: see text] required for the Hamiltonian's expectation value ⟨<i>Ĥ</i>⟩. We demonstrate our approach by harnessing the well-established natural orbital functional (NOF) theory, using the natural orbitals and occupation numbers derived from the diagonalization of the 1RDM measured from the quantum computer. Starting with the H₂ system, we validate the accuracy of our method by comparing the energy derived from NOF approximations applied to the exact wave function with the value obtained from ⟨<i>Ĥ</i>⟩. This is followed by an optimization of the gate parameters by minimizing the energy using the NOF approximations as the objective function. The analysis is extended to larger systems, such as LiH, Li₂, OH^-, FH, NeH⁺, and F₂ using a wave function ansatz with single and double excitation gates. This NOF-based method reduces the scaling cost of circuit executions compared to standard VQE implementations, achieving around 90% savings in the systems used in this work. Overall, by using a well-performing NOF as the objective function, the proposed NOF-VQE demonstrates the viability of NOF approximations for obtaining accurate energies in the noisy intermediate-scale quantum era and underscores the potential for developing new functionals tailored to quantum computing applications.