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