Measurement Reduction in Orbital-Optimized Variational Quantum Eigensolver via Orbital Compression

Abstract

The variational quantum eigensolver (VQE) has emerged as one of the leading quantum algorithms for solving electronic structure problems on near-term noisy intermediate-scale quantum devices. However, its practical application to quantum chemistry remains challenging due to the limited coherence time, imperfect quantum gate fidelity, and the large number of measurements required, which together confine current electronic structure simulations to relatively small active spaces. In this work, we present an orbital-optimized VQE framework based on orbital compression, designed to improve the accuracy of electronic structure calculations while maintaining relatively small active spaces. Frozen natural orbitals (FNO) and split virtual orbitals (SVO) are first employed to construct compact active spaces for VQE simulations, leading to the FNO/SVO-VQE approach. Orbital optimization is then incorporated to further recover electron correlation effects, resulting in the FNO/SVO-OO-VQE methods. We apply the proposed method to simulate potential energy surfaces for molecular dissociation and the activation energy of formaldehyde decomposition. Numerical results demonstrate that both FNO-OO-VQE and SVO-OO-VQE improve the variational accuracy while substantially reducing measurement cost.

Figures (4)

• Figure 1: Schematic illustration of the FNO-OO-VQE and SVO-OO-VQE workflow.
• Figure 2: (a) Potential energy surfaces of LiH and the corresponding errors in correlation energy ($E_\mathrm{miss}^\mathrm{corr}$) with respect to the exact diagonalization results (labeled as "FCI"). All VQE calculations use the same active space $(4,6)$. (b) Number of iterations in orbital optimization for different OO-VQE methods and the ratio of the total number of measurements to the number of qubit Hamiltonian terms $N_\mathrm{shot}/N_\mathrm{term}$. Dashed lines denote the averages over all LiH geometries for each method.
• Figure 3: Potential energy surfaces of H2O (a) and N2 (b) computed with different VQE methods, together with the corresponding energy errors relative to CASSCF, and the ratio $N_\mathrm{shot}/N_\mathrm{term}$ used to characterize the measurement cost. The active spaces for H2O and N2 are $(10,7)$ and $(10,8)$, respectively. Dashed lines in the measurement cost plots denote the averages of the corresponding methods over all geometries considered for each molecule.
• Figure 4: Energy profiles for formaldehyde decomposition reaction. The left and right panels show the two reaction pathways, respectively. $\mathrm{S3}$ denotes the reactant H2CO, $\mathrm{S1+S2}$ denotes the products H2 and CO, $\mathrm{TS}$ denotes the transition state, and $\mathrm{S4}$ and $\mathrm{S5}$ are intermediates in pathway 2. Unless otherwise specified in the legend, the active space is $(8,8)$.