  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.
