We proposed VQE-based algorithms for calculating nonadiabatic couplings and Berry’s phase.

Tamiya (intern) and Nakagawa from QunaSys Inc. proposed new near-term quantum algorithms for calculating first- and second-order nonadiabatic couplings (NACs) and Berry’s phase, which are important quantities for analyzing properties of molecules and materials. The preprint is available on arXiv:

"Calculating nonadiabatic couplings and Berry's phase by variational quantum eigensolvers",


Quantum computers consisting of tens or hundreds of imperfect qubits without quantum error correction are attracting growing attention. They are called noisy intermediate-scale quantum (NISQ) devices, and intensive research has been carried out for finding practical applications of NISQ devices. Investigating quantum molecular and spin systems with the variational quantum eigensolver (VQE), an algorithm to obtain eigenenergies and eigenstates of quantum systems, is believed to be one of the candidates of such applications.


For analyzing the properties of molecules and materials, information about energies and eigenstates of systems is not sufficient. For example, to simulate the photochemical reaction (e.g., light production in fireflies), in which nonadiabatic transition plays an important role, we have to calculate the NACs between eigenstates. In addition, the geometric phase of an eigenstate, which is also known as Berry’s phase, is also a key quantity in many fields of modern physics and leads to several physically interesting effects such as the quantum spin Hall effect. However, no efficient way of calculating these quantities (the NACs, Berry's phase) on NISQ devices has been proposed.


In this study, we develop VQE-based methods to evaluate NACs and Berry’s phase. Both quantities are related to the derivatives of the eigenstates with respect to the external parameters of the system.  We derive the analytical formulas of NACs and Berry’s phase based on outputs of the VQE. Moreover, to evaluate the formulas we propose explicit quantum circuits including the projective measurements if needed by referencing the circuits used when performing the VQE. By using our methods, the calculation of NACs reduces to the evaluation of expectation values of observables, which is tractable on NISQ devices. As for Berry’s phase,  while one Hadamard test, which is costly for NISQ devices, is needed, we can also evaluate it by expectation-value measurements. Finally, we demonstrate our methods by numerical simulations of the NACs of the hydrogen molecules and Berry’s phase of spin-½ model with high-speed quantum circuit simulator Qulacs, and the results show good agreements with exact results.


Our proposal widens the applicability of NISQ devices for calculating nonadiabatic couplings and geometric phases. We expect that our algorithms contribute to simulating nonadiabatic molecular dynamics and electron transfers of large-size quantum systems, which is intractable on classical computers.


Category: Research
Category: Research
Year: 2020