  Calculating the energy spectrum of a quantum system is an important task, for
example to analyse reaction rates in drug discovery and catalysis. There has
been significant progress in developing algorithms to calculate the ground
state energy of molecules on near-term quantum computers. However, calculating
excited state energies has attracted comparatively less attention, and it is
currently unclear what the optimal method is. We introduce a low depth,
variational quantum algorithm to sequentially calculate the excited states of
general Hamiltonians. Incorporating a recently proposed technique, we employ
the low depth swap test to energetically penalise the ground state, and
transform excited states into ground states of modified Hamiltonians. We use
variational imaginary time evolution as a subroutine, which deterministically
propagates towards the target eigenstate. We discuss how symmetry measurements
can mitigate errors in the swap test step. We numerically test our algorithm on
Hamiltonians which encode 3SAT optimisation problems of up to 18 qubits, and
the electronic structure of the Lithium Hydride molecule. As our algorithm uses
only low depth circuits and variational algorithms, it is suitable for use on
near-term quantum hardware.
