The Hamiltonian of a quantum system often have some symmetry in various fields of physics, e.g., high energy physics and condensed matter physics, and it is demanding to compute the spectrum of the Hamiltonian that resides in each symmetry sector. For this purpose, we can utilize the variational quantum eigensolver (VQE), which is a promising algorithm to compute eigenstates and their energies of a given quantum system that can be performed on a near-term quantum computer, to compute energy eigenstates possessing such symmetry by adding a penalty term in its cost function. 
The systematic analysis of such penalty terms, however, has been lacking and a heuristic way to set the penalty is adopted so far. In this work, we investigate two types of penalty terms for the VQE that have been previously proposed. We discover a convenient formula to set the penalty for one of the two types, which rigorously guarantees the resulting eigenstates by the VQE have desired symmetry. Meanwhile, we prove that the other type of penalty does not work well in a rigorous sense and gives completely wrong results in some cases. We also provide numerical simulations to validate our analysis.
Our results lay the theoretical foundation for the VQE with the penalty that applies to general quantum systems, and contribute to further application of a near-term quantum computer.

Penalty methods for variational quantum eigensolver

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Quantum 2020

The Institute of Physics and IOP Publishing, in partnership with the Chinese Physical Society (CPS) and the University of Science and Technology of China (USTC) are proud to have brought together the international quantum science community online for this inaugural conference.

Early career and leading researchers from universities, industry and government worldwide came together to share, learn and collaborate on the latest research and emerging areas of high interest across quantum science and technology. 

Kohdai Kuroiwa

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Penalty methods for variational quantum eigensolver

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