Dissipative quantum algorithms for excited-state quantum chemistry

TL;DR

A general dissipative algorithm for selectively preparing ab initio electronic excited states is introduced, and three complementary strategies, tailored to different types of prior information about the excited state, such as symmetry and approximate energy are developed.

Abstract

Electronic excited states are central to a vast array of physical and chemical phenomena, yet accurate and efficient methods for preparing them on quantum devices remain challenging and comparatively underexplored. We introduce a general dissipative algorithm for selectively preparing ab initio electronic excited states. The key idea is to recast excited-state preparation as an effective ground-state problem by suitably modifying the underlying Lindblad dynamics so that the target excited state becomes the unique steady state of a designed quantum channel. We develop three complementary strategies, tailored to different types of prior information about the excited state, such as symmetry and approximate energy. We demonstrate the effectiveness and versatility of these schemes through numerical simulations of atomic and molecular spectra, including valence excitations in prototypical planar conjugated molecules and transition-metal complexes. Taken together, these results provide a new pathway for advancing quantum simulation methods for realistic strongly correlated electronic systems.

Figures (11)

• Figure 1: Schematic representation of the dissipative dynamics protocols for preparing excited states.a. The Lindblad dynamics is designed to drive the system from any initial state to the desired excited state. The graph represents the isosurface of electron density differences (EDD) between the $1E_1"$ excited state and the ground state of ferrocene. b. Mechanism of dissipative ground-state preparation. Every state has efficient energy-lowering transition pathways that connect it to the target ground state. c. A symmetry-based approach for excited-state preparation. The initial state is restricted to a symmetry sector e.g. $(N_\alpha, N_\beta)$ matching the target excited state, so that lower-energy states outside this sector become "dark" and are avoided throughout the dissipative dynamics. The problem is then reduced to preparing the ground state within this symmetry sector. d. "Folded-spectrum" approach for dissipative excited-state preparation. The transformation $H \mapsto (H - \mu I)^2$ folds the spectrum around a reference energy $\mu$, making the target excited state the effective ground state of the transformed Hamiltonian. e. A spectral projector approach. We construct an operator $P_{\mu}(H)$ which projects out the energy eigenstates with energy below $\mu$, and then apply $P_\mu(H)$ to the initial state as well as the jump operators, effectively removing the possibility of decaying into states with energy below $\mu$. f. Energy levels and the symmetry sectors of the hydrogen molecule $\rm H_2$ in the 6-31G basis set. The $(N_\alpha,N_\beta)$ sectors of the low-lying states are indicated in the figure. The orange, blue, green markers represent that this target state can be prepared by the symmetry-based, folded-spectrum, and spectral projector methods, respectively.
• Figure 1: Further comparison of the three protocols for preparing the $\rm T_0$ and $\rm T_0^{\pm}$ states for $\rm H_2$ in 6-31G and $\rm H_4$ in STO-3G.a. The convergence of the spin multiplicity for preparing the $\rm T_0$ and $\rm T_0^{\pm}$ states of $\rm H_2$ (6-31G) using the three different protocols. b. The convergence of the spin multiplicity for preparing the $\rm T_0$ and $\rm T_0^{\pm}$ states of $\rm H_4$ (STO-3G) using the three different protocols. c. The Lindbladian gap for these two systems
• Figure 2: Performance of the three protocols for preparing the $\rm T_0$ and $\rm T_0^{\pm}$ states for $\rm H_2$ in 6-31G and $\rm H_4$ in STO-3G. Convergence of the infidelity, energy expectation and energy error for a.$\rm H_2$ in 6-31G and b.$\rm H_4$ chain in STO-3G; c. Lindblad simulation time and Hamiltonian simulation costs required to achieve the chemical accuracy in energy for these three different excited state preparation protocols.
• Figure 2: Comparison between RHF and UHF methods for the $\rm H_4$ system along the bond dissociation pathway. The solid lines represent the potential energy curves obtained from the RHF (blue) and UHF (orange) methods. The Coulson--Fischer point is observed at a bond length of approximately 1.31 Å, where the UHF solution diverges from the RHF solution, providing a more accurate description of the energy as it approaches dissociation. We also see that after the Coulson--Fischer point, the expectation value of the total spin operator $\hat{S}^2$ increases for the UHF solution, indicating a higher degree of spin contamination.
• Figure 3: Dissipative preparation of excited states in atomic systems.a. Energy spectrum of the second-period elements. The dashed lines represent the energy estimates obtained from the final states of the dissipative dynamics. The scatter points and the solid lines denote the FCI reference energies and the results from the EOM-CCSD/EOM-UCCSD methods, respectively. b. Energy errors of the final states from the dissipative dynamics, EOM-CCSD, and EOM-UCCSD methods with respect to the FCI reference energies. c. Spin multiplicities of the final states from the dissipative dynamics. The results show that the final states reproduce the correct spin multiplicities of the target excited states, indicating that they are highly accurate approximations to those states. d.-f. Dissipative preparation of the $2s^12p^3 (^5S)$ state of the carbon atom. The blue and orange lines represent the dynamics with initial state given by the HF ground state $\ket{\rm HF}$ and the high-spin initial state $\ket{D}$ (see Supplementary Note \ref{['sec:highspin']}), respectively. The dashed lines denote the dissipative dynamics using only the reduced quadratic coupling operator set $\mathcal{S}_{\rm II}'$, while the solid lines correspond to the dynamics with the augmented coupling operator set $\mathcal{Q}\cup \mathcal{S}_{\rm II}"$. d. Spin multiplicity during the Lindblad dynamics. e. The overlap with the $^5S$ state during the Lindblad dynamics. f. The overlap with $^1S$ state during the Lindblad dynamics.