Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

Abstract

We perform a systematic study of preparing ground states of correlated multi-orbital impurity models using variational quantum eigensolvers (VQEs). We consider both fixed and adaptive wavefunction ans\"atze and analyze the resulting gate depths and the performance with and without noise. For the adaptive procedure, we develop an operator pool consisting of pairwise commutators of Hamiltonian terms that allows for a fair comparison between the adaptive and fixed Hamiltonian variational ansatz. Using noiseless statevector simulations, we find that the most compact ans\"atze are obtained in an atomic orbital representation and using parity encoding. Focusing on the adaptive algorithms, which yield the circuits with the least number of CNOTs, we then show that in the presence of sampling noise, high-fidelity state preparation can still be achieved with the Hamiltonian commutator pool. By utilizing Hamiltonian integral factorization and a noise resilient optimizer, we show that this approach requires only a modest number of about $2^{12}$ shots per measurement circuit. We discover a dichotomy of the operator pool complexity in the presence of sampling noise, where a small pool size reduces the adaptive overhead but a larger pool size accelerates convergence to the ground state. When considering realistic gate noise in addition, we observe that the variable optimization can still be performed as long as the two-qubit gate error lies below $10^{-3}$, which is close but below current hardware levels. Finally, we measure the ground state energy of the $e_g$ model on IBM and Quantinuum quantum hardware using the converged adaptive ansatz. We perform a systematic error mitigation analysis on the IBM results and obtain a relative error of 0.7\% using symmetry-based postselection and zero-noise extrapolation (ZNE).Comment: 19 pages, 9 figure