Ground and Excited States from Ensemble Variational Principles

Quantum 8, 1525 (2024).

https://doi.org/10.22331/q-2024-11-14-1525

The extension of the Rayleigh-Ritz variational principle to ensemble states $rho_{mathbf{w}}equivsum_k w_k |Psi_krangle langlePsi_k|$ with fixed weights $w_k$ lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state $rho_{mathbf{w}}$ as well as the individual eigenstates $|Psi_krangle$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_-Delta{E}_{mathbf{w}} leq Delta Q leq d_+ Delta{E}_{mathbf{w}}$ on the errors $Delta Q $ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of $d_{pm}equiv d_{pm}^{(Q)}(mathbf{w},mathbf{E})$ provide valuable insights into the optimal choice of the auxiliary weights $w_k$ in practical applications.