How an Equi-ensemble Description Systematically Outperforms the Weighted-ensemble Variational Quantum Eigensolver

Abstract

Calculating excited states in chemistry is crucial to provide insight into photoinduced molecular behavior beyond the ground state, enabling innovations in spectroscopy, material sciences, and drug design. While several approaches have been developed to compute excited-state properties, finding the best ratio between computational cost and accuracy remains challenging. The advent of quantum computers brings new perspectives, with the development of quantum algorithms that promise an advantage over classical ones. Most of these new algorithms are inspired from previous classical ones, but with different pros and cons. In this Letter, we focus on the generalization of the variational principle for many-body excited-states that led to the ensemble variational quantum eigensolver (VQE). We compare the performance of two ensemble VQE approaches, the equi-ensemble and weighted-ensemble ones, and conclude that the equi-ensemble is the way to go.

Figures (11)

• Figure 1: Ground and first-excited singlet-state potential energy surfaces (PESs) of the formaldimine molecule with respect to $\alpha$ at $\phi = 89^\circ$. Top panel: $w_{A} > w_{B}$ with $w_{A} = 3/4$ and $w_{B} = 1/4$. Middle panel: $w_{A} < w_{B}$ with $w_{A} = 1/4$ and $w_{B} = 3/4$. Bottom panel: absolute values of the overlaps between the numerically exact eigenstates and the initial states of circuits $A$ and $B$.
• Figure 2: Top panel: behavior of $E_A$ and $E_B$ with respect to the number of VQE iterations until convergence (vanishing gradient of the cost function) for $\alpha = 138^\circ$ with $w_A > w_B$. Curves are smoothened using a Gaussian filter. Horizontal lines (shaded colors) show the value of the last iteration of each curve (converged asymptote). Middle-top panel: error in $E_C^{\bf w}$ and $E_T^{\bf w}$ using 1-GUCCSD (dashed lines) and 2-GUCCSD (dotted lines) in the weighted-ensemble (purple and green) and equi-ensemble (black) cases. Middle-bottom and bottom panels: same as top and middle top panels but for $w_A < w_B$.
• Figure 3: Top panel: Mean values of the occupied orbital energies of H$_{16}$ over 10 trials using random initial ${\bm \theta}$-parameters, within the optimally weighted-ensemble (full lines) and the equi-ensemble with classical postprocessing diagonalization (dashed lines). Shaded areas correspond to standard deviations (only visible for $E_H$ within the weighted-ensemble). 'Exact' (dot markers) refers to the energies obtained by diagonalizing the Kohn--Sham Hamiltonian. Middle panel: mean error in each orbital energy obtained from the weighted-ensemble compared with exact diagonalization. Bottom panel: Same as middle panel but using the equi-ensemble followed by a postprocessing classical diagonalization.
• Figure 4: Potential energy surfaces of the formaldimine molecule with respect to $\alpha$ at $\phi = 89^\circ$ from the equi-ensemble VQE ($w_A = w_B$).
• Figure 5: Top panel: $E_A$ and $E_B$ with respect to the number of VQE iterations for $\alpha = 99^\circ$ with $w_A > w_B$ (wrong initial order). Middle-top panel: errors in $E_T^{\bf w}$ and $E_C^{\bf w}$ using 1-GUCCSD (dashed lines) and 2-GUCCSD (dotted lines) in the weighted-ensemble (purple and green) and equi-ensemble (black) case. Middle-bottom and bottom panels: same as top and middle top panels but for $w_A < w_B$ (right initial order).