  The variational quantum eigensolver (VQE) remains one of the most popular
near-term quantum algorithms for solving the electronic structure problem. Yet,
for its practicality, the main challenge to overcome is improving the quantum
measurement efficiency. Numerous quantum measurement techniques have been
developed recently, but it is unclear how these state-of-the-art measurement
techniques will perform in extensions of VQE for obtaining excited electronic
states. Assessing the measurement techniques' performance in the excited state
VQE is crucial because the measurement requirements in these extensions are
typically much greater than in conventional VQE, as one must measure the
expectation value of multiple observables in addition to that of the electronic
Hamiltonian. Here, we adapt various measurement techniques to two widely used
excited state VQE algorithms: multi-state contraction and quantum subspace
expansion. Then, the measurement requirements of each measurement technique are
numerically compared. We find that the best methods for multi-state contraction
are ones utilizing Hamiltonian data and wavefunction information to minimize
the number of measurements. In contrast, randomized measurement techniques are
more appropriate for quantum subspace expansion, with many more observables of
vastly different energy scales to measure. Nevertheless, when the best possible
measurement technique for each excited state VQE algorithm is considered,
significantly fewer measurements are required in multi-state contraction than
in quantum subspace expansion.
