  The calculation of excited state energies of electronic structure
Hamiltonians has many important applications, such as the calculation of
optical spectra and reaction rates. While low-depth quantum algorithms, such as
the variational quantum eigenvalue solver (VQE), have been used to determine
ground state energies, methods for calculating excited states currently involve
the implementation of high-depth controlled-unitaries or a large number of
additional samples. Here we show how overlap estimation can be used to deflate
eigenstates once they are found, enabling the calculation of excited state
energies and their degeneracies. We propose an implementation that requires the
same number of qubits as VQE and at most twice the circuit depth. Our method is
robust to control errors, is compatible with error-mitigation strategies and
can be implemented on near-term quantum computers.
