# We proposed a method for calculating transition amplitudes of molecules on a near-term quantum computer.

We released a paper (preprint) for calculating transition amplitudes of molecules, which are important quantities in photochemistry, authored by Ibe, Nakagawa, Yamamoto, and Mitarai from QunaSys Inc. and Dr. Gao and Dr. Kobayashi at Mitsubishi Chemical Corp. Science & Innovation Center.

The ongoing joint research between QunaSys Inc. and Mitsubishi Chemical Corp. has led to this work.

The preprint is available on arXiv:

“Calculating transition amplitudes by variational quantum eigensolvers”, arXiv:2002.11724.

**BACKGROUND**

Recent progress on the quantum supremacy has drawn our attention to the next playground for quantum computing: quantum chemistry. Quantum computers are expected to speed up the simulation of excited states (quantum states away from the stable state, due to light, heat, or electromagnetic fields, etc.) of molecules, which is a difficult task for classical computers. Properties of excited states of molecules are important for analyzing chemical reactions and optical properties. Recently, a number of algorithms for simulating excited states on near-term quantum computers are proposed, and among them, SSVQE (subspace-search variational quantum eigensolver, proposed by QunaSys Inc.) [1], MCVQE (multistate-contracted variational quantum eigensolver) [2], and VQD (variational quantum deflation) [3] are prominent.

**MOTIVATION**

In photochemistry, which is one of the major applications of simulating excited states, one calculates the transition amplitudes (interstate couplings of interaction operators) between the obtained ground and excited states to predict various response quantities, such as the absorption/emission spectrum of light. Among the three algorithms for simulating excited states, the SSVQE and the MCVQE can readily evaluate the transition amplitudes. The VQD, however, lacks such a method for evaluating the quantity in a hardware-friendly manner.

**METHODS & RESULTS**

Our comparison of the above three algorithms (i.e., the SSVQE, the MCVQE, and the VQD) by numerical simulation for molecules (including azobenzene: an important material for industrial application) revealed that the VQD can simulate excited states the most accurately. Moreover, we developed a hardware-friendly method to calculate the transition amplitude between the ground and excited states obtained by the VQD, which would be feasible on near-term quantum devices. We verified the correctness of our method by calculating the oscillator strengths (which corresponds to photon absorption rates and is an important quantity for photoreaction) of molecules on a realistic simulator of quantum computers, which includes noise in expectation values of physical quantities. It should be noted that our method can evaluate the transition amplitude not only between the excited states obtained by the VQD but also between an arbitrary pair of quantum states.

**OUTLOOK**

Our proposed method will significantly widen the application range of the VQD algorithm, which can precisely simulate the excited states of molecules. This enables quantum computers to evaluate a broader set of physical quantities in quantum chemistry, such as the response function to external fields, various types of transition probabilities, and the correction of energies due to electron correlation.

[1] K. Nakanishi *et al*., Phys. Rev. Research **1**, 033062 (2019)

[2] R. Parrish *et al*., Phys. Rev. Lett. **122**, 230401 (2019)

[3] O. Higgott *et al*., Quantum **3**, 159 (2019)