Calculating transition amplitudes by variational quantum deflation

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