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Variational quantum eigensolver (VQE) is an appealing candidate for theapplication of near-term quantum computers. A technique introduced in [Higgotet al., Quantum 3, 156 (2019)], which is named variational quantum deflation(VQD), has extended the ability of the VQE framework for finding excited statesof a Hamiltonian. However, no method to evaluate transition amplitudes betweenthe eigenstates found by the VQD without using any costly Hadamard-test-likecircuit has been proposed despite its importance for computing properties ofthe system such as oscillator strengths of molecules. Here we propose a methodto evaluate transition amplitudes between the eigenstates obtained by the VQDavoiding any Hadamard-test-like circuit. Our method relies only on the abilityto estimate overlap between two states, so it does not restrict to the VQDeigenstates and applies for general situations. To support the significance ofour method, we provide a comprehensive comparison of three previously proposedmethods to find excited states with numerical simulation of three molecules(lithium hydride, diazene, and azobenzene) in a noiseless situation and findthat the VQD method exhibits the best performance among the three methods.Finally, we demonstrate the validity of our method by calculating theoscillator strength of lithium hydride, comparing results from numericalsimulations and real-hardware experiments on the cloud enabled quantum computerIBMQ Rome. Our results illustrate the superiority of the VQD to find excitedstates and widen its applicability to various quantum systems.

Calculating transition amplitudes by variational quantum deflation