Ground and Excited States from Ensemble Variational Principles

Abstract

The extension of the Rayleigh-Ritz variational principle to ensemble states ${\rho }_{\mathbf{w}}\equiv {\sum }_k{w}_k|{\Psi }_k⟩⟨{\Psi }_k|$ 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 $|{\Psi }_k⟩$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_{-}\Delta E_{\mathbf{w}}\le \Delta Q\le d_{+}\Delta E_{\mathbf{w}}$ 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.

23+7 pages, 9 figures, to appear on Quantum

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