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The variational quantum eigensolver (VQE) remains one of the most popularnear-term quantum algorithms for solving the electronic structure problem. Yet,for its practicality, the main challenge to overcome is improving the quantummeasurement efficiency. Numerous quantum measurement techniques have beendeveloped recently, but it is unclear how these state-of-the-art measurementtechniques will perform in extensions of VQE for obtaining excited electronicstates. Assessing the measurement techniques' performance in the excited stateVQE is crucial because the measurement requirements in these extensions aretypically much greater than in conventional VQE, as one must measure theexpectation value of multiple observables in addition to that of the electronicHamiltonian. Here, we adapt various measurement techniques to two widely usedexcited state VQE algorithms: multi-state contraction and quantum subspaceexpansion. Then, the measurement requirements of each measurement technique arenumerically compared. We find that the best methods for multi-state contractionare ones utilizing Hamiltonian data and wavefunction information to minimizethe number of measurements. In contrast, randomized measurement techniques aremore appropriate for quantum subspace expansion, with many more observables ofvastly different energy scales to measure. Nevertheless, when the best possiblemeasurement technique for each excited state VQE algorithm is considered,significantly fewer measurements are required in multi-state contraction thanin quantum subspace expansion.

Measurement optimization techniques for excited electronic states in near-term quantum computing algorithms