Ground and Excited States from Ensemble Variational Principles
The extension of the Rayleigh-Ritz variational principle to ensemble states $\rho_{\mathbf{w}}\equiv\sum_k w_k |\Psi_k\rangle \langle\Psi_k|$ with fixed weights $w_k$ lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Pr...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2024-11-01
|
| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2024-11-14-1525/pdf/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | The extension of the Rayleigh-Ritz variational principle to ensemble states $\rho_{\mathbf{w}}\equiv\sum_k w_k |\Psi_k\rangle \langle\Psi_k|$ 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 $|\Psi_k\rangle$ 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. |
|---|---|
| ISSN: | 2521-327X |