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.

中文翻译：

---

来自集成变分原理的基态和激发态


Rayleigh-Ritz 变分原理扩展到具有固定权重 $w_k$ 的集合态 $\rho_{\mathbf{w}}\equiv\sum_k w_k |\Psi_k\rangle \langle\Psi_k|$ 最终是最近通过变分手段靶向激发能量的几种方法发展的核心。突出的例子是密度和密度矩阵泛函论、蒙特卡洛采样、状态平均完全有源空间自洽场方法和变分量子特征求解器。为了给所有这些方法提供一个坚实的基础并改进它们当前的实现，我们证明了基础关键假设的有效性：每当集成能量收敛良好时，集成状态 $\rho_{\mathbf{w}}$ 以及单个特征态 $|\Psi_k\rangle$ 和特征能 $E_k$ 也是如此。更具体地说，我们根据这些抢手量的误差 $\Delta Q $ 推导出线性边界 $d_-\Delta{E}_{\mathbf{w}} \leq \Delta Q \leq d_+ \Delta{E}_{\mathbf{w}}$。随后的分析分析和数值说明证明了我们普遍不等式的紧密性。我们的结果，特别是 $d_{\pm}\equiv d_{\pm}^{（Q）}（\mathbf{w}，\mathbf{E}）$ 的显式形式，为实际应用中辅助权重 $w_k$ 的最佳选择提供了有价值的见解。