Abstract
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.
Highlights
We are in an era where quantum computing practitioners can regularly use noisy quantum computers with tens of qubits [1]
To support the significance of our method, we provide a comparison of three previously proposed methods to find excited states with numerical simulation of three molecules and find that the variational quantum deflation (VQD) method exhibits the best performance among the three methods within the scope of noiseless simulation
We have observed that the VQD has a better performance compared to the subspace-search VQE (SSVQE) and the multistate contracted VQE (MCVQE) as far as noiseless simulations are concerned
Summary
We are in an era where quantum computing practitioners can regularly use noisy quantum computers with tens of qubits [1]. To expand the potential application of the VQE other than for the ground state, a lot of works have extended the method to evaluate properties of excited states of a target Hamiltonian [13–18] Those methods generally inherit the iterative and variational feature of the VQE, i.e., they iteratively optimize a quantum circuit relative to some cost function. To support the significance of the proposed technique, we present a comprehensive comparison of the SSVQE, the MCVQE, and the VQD by conducting noiseless numerical simulations, where we use exact energy expectation values in the optimization routine of the parametrized circuit. In this test, we use molecular Hamiltonians of LiH and two azo compounds: diazene and azobenzene (AB). As a demonstration of the technique, we conduct a proof-of-principle calculation both on a noisy simulator, i.e., expectation values of observables are simulated with the realistic hardware noise, including shot noise and environmental noise such as T1/T2 and readout errors, and on IBM’s real quantum hardware
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