Classically Prepared, Quantumly Evolved: Hybrid Algorithm for Molecular Spectra

TL;DR

The paper tackles the computation of zero-temperature dynamical correlation functions $G_A(t)$ and excitation spectra for molecular systems, which are difficult for classical methods due to long-time dynamics and entanglement. It introduces a hybrid classical–quantum workflow: a classical approximation of the perturbed ground state $|\psi_A\rangle$ is sampled to generate product states, which are evolved for short times on quantum hardware to identify a small, physically relevant subspace; the Hamiltonian is then projected onto this subspace and evolved classically to long times. Across small molecules benchmarked against exact diagonalization and, for larger systems, tensor-network methods, the projected dynamics reproduce low-energy spectra with high fidelity and demonstrate access to dynamical timescales beyond purely classical reach. The results indicate a practical path toward near-term quantum hardware: reduced circuit depth, variance-controlled sampling, and a framework compatible with SqDRIFT and adaptive subspace refinement for scalable dynamical properties.

Abstract

We introduce a hybrid classical-quantum algorithm to compute dynamical correlation functions and excitation spectra in many-body quantum systems, with a focus on molecular systems. The method combines classical preparation of a perturbed ground state with short-time quantum evolution of product states sampled from it. The resulting quantum samples define an effective subspace of the Hilbert space, onto which the Hamiltonian is projected to enable efficient classical simulation of long-time dynamics. This subspace-based approach achieves high-resolution spectral reconstruction using shallow circuits and few samples. Benchmarks on molecular systems show excellent agreement with exact diagonalization and demonstrate access to dynamical timescales beyond the reach of purely classical methods, highlighting its suitability for near-term and early fault-tolerant quantum hardware.

Figures (6)

• Figure 1: Hybrid classical–quantum workflow. A classical approximation of the perturbed state $\ket{\psi_A\rangle}$ is sampled to generate product states, which are briefly evolved on quantum hardware. The measured outcomes define small subspaces $\mathcal{S}_x$ onto which the Hamiltonian is projected for efficient classical long-time evolution and reconstruction of $G_A(t)$.
• Figure 2: Energy spectra of small molecular systems. We compare excitation energies of various molecular systems computed using the hybrid classical–quantum method (blue diamonds), against exact diagonalization results (black lines). All systems are small enough to allow full diagonalization of the Hamiltonian, which serves as a reference for validating the projected dynamics. For each system, we report the number of spin-orbitals ($n_o$), number of electrons ($n_e$), interatomic distance $R$ (in Å) and basis set. For each molecule, the excitation operator $\hat{A}$ used to define the perturbed state $\ket{\psi_A} = \hat{A} \ket{\psi_0}$ is designed to selectively target low-energy excitations. These include single and double excitations, such as $\hat{A} = c^\dagger_6 c^\dagger_5 c_4 c_3$ for N$_2$ , $\hat{A} = c^\dagger_9 c_7$ and $\hat{A} = c^\dagger_9 c_8$ for HCl and, and $\hat{A} = c^\dagger_6 c_4$ for CO. In the LiH systems, different perturbations are tested across basis sets: with 6-31G uses $\hat{A} = c^\dagger_2 c_1$, with cc-pvdz uses the same operator, while 6-31G (g) and cc-pvdz use $\hat{A} = c^\dagger_3 c_1$. The agreement between projected dynamics and exact results across all panels confirms the accuracy and flexibility of the method.
• Figure 3: $\text{N}_2$ dynamical correlation using MPS. Excitation spectrum for the $\text{N}_2$ molecule in the 6-31G basis set with frozen core approximation. The ground state is represented with a MPS state optimized using the DMRG algorithm, as described in \ref{['sec:results_tn']}. The peaks are at the following energies $\Delta E =0.735, 0.848, 1.131, 1.633.$
• Figure 4: Dynamical correlation functions of small molecules. Absolute value of the dynamical correlation function, comparison of the exact (black line) with the projected one (blue lines). The grey dashed line at $|\tilde{G}(\omega)| = 5 \times 10^{-3}$ indicates the threshold used to filter out peaks. Parameters as in Fig. \ref{['fig:all_molecules_peaks']}.
• Figure 5: Excited states accuracy scaling with number of samples. Accuracy of the extracted excitation energies as a function of the number of quantum samples used to construct the subspace $\mathcal{S}_x$. While the behavior depends on the type of system and its size, we observe a transition to very high accuracy with a relatively small number of samples ($\sim 2^{10}$). The accuracy is limited by the spectral resolution of the Fourier transform, here indicated as a grey dashed line.