A Unified Variational Framework for Quantum Excited States

• URL: http://arxiv.org/abs/2504.21459v1
• Date: Wed, 30 Apr 2025 09:28:04 GMT
• Title: A Unified Variational Framework for Quantum Excited States
• Authors: Shi-Xin Zhang, Lei Wang,
• Abstract summary: We introduce a novel variational principle that overcomes limitations, enabling the textitsimultaneous determination of multiple low-energy excited states.<n>We demonstrate the power and generality of this method across diverse physical systems and variational ansatzes.<n>In all applications, the method accurately and simultaneously obtains multiple lowest-lying energy levels and their corresponding states.
• Score: 2.935517095941649
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 \textit{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, $\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$, constructed from a set of \textit{non-orthogonal} variational states $\{|\psi_i\rangle\}$. Here, $\mathbf{H}_{ij} = \langle\psi_i | H | \psi_j\rangle$ and $\mathbf{S}_{ij} = \langle\psi_i | \psi_j\rangle$ are the elements of the Hamiltonian and overlap matrices, respectively. This approach variationally optimizes the entire low-energy subspace spanned by $\{|\psi_i\rangle\}$ 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.

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