A Unified Variational Framework for Quantum Excited States

Determining quantum excited states is crucial across physics and chemistry but presents significant challenges for variational methods, primarily due to the need to enforce orthogonality to lower-energy states, often requiring state-specific optimization, penalty terms, or specialized ansatz constructions. We introduce a novel variational principle that overcomes these limitations, enabling the $\mathit{\text{simultaneous}}$ determination of multiple low-energy excited states. The principle is based on minimizing the trace of the inverse overlap matrix multiplied by the Hamiltonian matrix, $\mathrm{T}\mathrm{r}({\mathbf{S}}^{-1}\mathbf{H})$, constructed from a set of $\mathit{\text{non-orthogonal}}$ variational states $\{\mathrm{\mid }{\psi }_i⟩\}$. Here, ${\mathbf{H}}_{ij}=⟨{\psi }_i\mathrm{\mid }H\mathrm{\mid }{\psi }_j⟩$ and ${\mathbf{S}}_{ij}=⟨{\psi }_i\mathrm{\mid }{\psi }_j⟩$ are the elements of the Hamiltonian and overlap matrices, respectively. This approach variationally optimizes the entire low-energy subspace spanned by $\{\mathrm{\mid }{\psi }_i⟩\}$ without explicit orthogonality constraints or penalty functions. We demonstrate the power and generality of this method across diverse physical systems and variational ansatzes: calculating the low-energy spectrum of 1D Heisenberg spin chains using matrix product states, finding vibrational spectrum of Morse potential using quantics tensor trains for real-space wavefunctions, and determining excited states for 2D fermionic Hubbard model with variational quantum circuits. In all applications, the method accurately and simultaneously obtains multiple lowest-lying energy levels and their corresponding states, showcasing its potential as a unified and flexible framework for calculating excited states on both classical and quantum computational platforms.