Orthogonal-ansatz VQE: Locating excited states without modifying a cost-function

Abstract: Most literature in the Variational Quantum Eigensolver (VQE) algorithm focuses on finding the ground state of a physical system, by minimizing a quantum-computed cost-function. When excited states are required, the cost-function is usually modified to include additional terms ensuring orthogonality with the ground state. This generally requires additional quantum circuit executions and measurements, increasing algorithmic complexity. Here we present a design strategy for the variational ansatz which enforces orthogonality in candidate excited states while still fully exploring the remaining subset of Hilbert space. The result is an excited-state VQE solver which trades increasing measurement complexity for increasing circuit complexity. The latter is anticipated to become preferable as quantum error mitigation and correction become more refined. We demonstrate our approach with three distinct ansatze, beginning with a simple single-body example, before generalizing to accommodate the full Hilbert space spanned by all qubits, and a constrained Hilbert space obeying particle number conservation.