Ground and Excited States from Ensemble Variational Principles
Faculty of Physics, Arnold Sommerfeld Centre for Theoretical Physics (ASC), Ludwig-Maximilians-Universität München, Theresienstr. 37, 80333 München, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany
| Published: | 2024-11-14, volume 8, page 1525 |
| Eprint: | arXiv:2401.12104v2 |
| Doi: | https://doi.org/10.22331/q-2024-11-14-1525 |
| Citation: | Quantum 8, 1525 (2024). |
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Abstract
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.
Featured image: By resorting to concepts from convex analysis we prove a universal and optimal error estimate which in turn rationalizes the ensemble variational principle.
Popular summary
To accomplish this, our work reveals and exploits a fruitful link between ensemble quantum states and geometry: Inspired by the concept of transport polytopes, which plays a crucial role in economy, computer science and mathematics, we elucidate by analytical means the effect of external influences on the quantum system, whose energy contribution is `transported' into the excitation structure. In particular, we conclusively understand how the (weights of the) ensemble needs to be chosen such that the physical properties of the targeted energy levels can be predicted with maximal accuracy.
Accordingly, our work extends the toolbox of quantum scientists by a scheme for effectively targeting the important excited states with potentially higher accuracy with applications not only in classical but also in quantum computing.
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