Multiscale-Contracted Variational Quantum Eigensolver method: part 2

This is the sequel of "Multiscale-Contracted Variational Quantum Eigensolver method".

Here, I introduce the result of numerical simulation of the energy levels of ground, triplet and singlet state on hydrogen molecule.

Hamiltonian is transformed by Bravyi-Kitaev transformation into Pauli words. Cluster is UUCSD and transformed in same way. Basis is also.

CIS state is derived by Subspace-Search VQE before MCVQE. Optimization is performed by BFGS method same as paper [1]. All calculations are performed by blueqat SDK.

Besides, constraint term of magnetic moment is added into evaluation function same as paper [1].


I show the result of calculation on ground, triplet and singlet state on hydrogen molecule in case diatomic bond length r=0.7 by SSVQE and MCVQE based on the result of SSVQE for 5 times in Fig. 1.

Ground and Triplet states have high accuracy for the result of both SSVQE and MCVQE.

Although, singlet state has low accuracy for the result of SSVQE.

The result on singlet state of trial a and b derived by MCVQE have high accuracy.

Fig. 1 result of calculation on ground, triplet and singlet state on hydrogen molecule in case diatomic bond length r=0.7 by SSVQE and MCVQE based on the result of SSVQE for 5 times.


I show the common logarithm of the difference between calculated energies and exact values (log error) on ground, triplet and singlet state of SSVQE and MCVQE based on the result of SSVQE for 5 trials in Fig. 2.

All states of trial a and b have good accuracy. Ground and triplet states of trial c have also good accuracy.

Fig. 2 log errors on ground, triplet and singlet state of SSVQE and MCVQE based on the result of SSVQE for 5 trials.


The accuracy of MCVQE depends on the accuracy of each state of CIS state.

SSVQE has low accuracy compared to QSE-VQE and the result of paper [1] has same tendency.

Therefore, I will perform MCVQE numerically with QSE-VQE in the next part.


[1] Nakanishi, K. M., and et. al., arXiv quant-ph:1810.09434v2(2018)


[2] Parrish, R. M. and et. al., arXiv quant-ph:1901.01234v2(2019)

Hikaru Wakaura
個人研究者の若浦 光です。量子アルゴリズムの実装結果や論文の紹介などを載せていきます。 mail: hikaruwakaura@gmail.com
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Hikaru Wakaura
個人研究者の若浦 光です。量子アルゴリズムの実装結果や論文の紹介などを載せていきます。 mail: hikaruwakaura@gmail.com
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