Quantum computing and quantum information processing technology have attracted attention in recently emerging fields. Among many important and fundamental issues in nowadays science, solving Schroedinger Equation (SE) of atoms and molecules is one of the ultimate goals in chemistry, physics and their related fields. SE is "First Principle" of non-relativistic quantum mechanics, whose solutions termed wave functions can afford any information of electrons within atoms and molecules, predicting their physicochemical properties and chemical reactions. Researchers from Osaka City University (OCU) in Japan, Dr. K. Sugisaki, Profs. K. Sato and T. Takui and coworkers have found a novel quantum algorithm enabling us to determine whether quantum chemical calculations performed on quantum computers give correct wave functions as exact solutions of SE in a desired manner. These issues are intractable with any currently available supercomputers. Such a quantum algorithm contributes to the acceleration of implementing practical quantum computers. Nowadays chemistry and physics have sought to predict complex chemical reactions by invoking Full-CI approaches since 1929, but never been successful until now. Now Full-CI calculations are potentially capable of predicting chemical reactions, and a new Full-CI approach suitable for predicting the physicochemical properties has already been implemented on quantum computers. Now, the possible methodological implementation of "observables on quantum computers" such as calculating the spin quantum numbers of arbitrary wave functions, which is a crucial issue in quantum chemistry, has been established by the OCU research group.

The paper has been published at 4:00 PM on July 4, 2019 (JST, Japan Time Zone) in *Physical Chemistry Chemical Physics* (Royal Chemical Society).

They said, "As Dirac claimed in 1929 when quantum mechanics was established, the exact application of mathematical theories to solve SE leads to equations too complicated to be soluble^{1}. In fact, the number of variables to be determined in the Full-CI method grows exponentially against the system size, and it easily runs into astronomical figures such as exponential explosion. For example, the dimension of the Full-CI calculation for benzene molecule C6H6, in which only 42 electrons are involved, amounts to 1044, which are impossible to be dealt with by any supercomputers. What is worse, molecular systems during the dissociation process are characterized by extremely complex electronic structures (multiconfigurational nature), and relevant numerical calculations are impossible on any supercomputers. Besides these intrinsic difficulties, there has been a difficult issue in the emerging fields such as determining physical quantities relevant to quantum chemistry on quantum computers."

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[1] P.A.M. Dirac, Quantum mechanics of many-electron systems. Proc. R. Soc. London, Ser. A 1929, 123, 714-733.

According to the OCU research group, quantum computers can date back to a Feynman's suggestion in 1982 that the quantum mechanics can be simulated by a computer itself built of quantum mechanical elements which obey quantum mechanical laws. After more than 20 years later, Prof. Aspuru-Guzik, Harvard Univ. (Toronto Univ. since 2018) and coworkers proposed a quantum algorithm capable of calculating the energies of atoms and molecules not exponentially but polynomially against the number of the variables of the systems, making a breakthrough in the field of quantum chemistry on quantum computers^{2}.

[2] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, M. Head-Gordon, Science 2005, 309, 1704.

When Aspuru's quantum algorithm is applied to the Full-CI calculations on quantum computers, good approximate wave functions close to the exact wave functions of SE under study are required, otherwise bad wave functions need an extreme number of steps of repeated calculations to reach the exact ones, hampering the advantages of quantum computing. This problem becomes extremely serious for the analyses of chemical reactions, which have many multiconfigurational nature due to electrons not participating in chemical bonding during the bond dissociation. The OCU researchers have tackled this problem, one of the most intractable issues in quantum science and chemistry, and already made a breakthrough in implementing a new quantum algorithm generating particular wave functions termed configuration state functions (CSFs) in polynomial computing time in 2016^{3} and 2018^{4}.

[3] K. Sugisaki, S. Yamamoto, S. Nakazawa, K. Toyota, K. Sato, D. Shiomi, T. Takui, J. Phys. Chem. A 2016, 120, 6459-6466. DOI: 10.1021/acs.jpca.6b04932

[4] K. Sugisaki, S. Yamamoto, S. Nakazawa, K. Toyota, K. Sato, D. Shiomi, T. Takui, Chem. Phys. Letters: X, in press (2018): https:/

All the previously proposed algorithms for quantum computing, however, involve particularly solving energies of atoms and molecules under study on quantum computers. These physical quantities are very important for understanding the dissociation and formation of many chemical bonds. In quantum chemistry, there are many important physical quantities other than the energies to identify convoluted chemical reactions. Now, the OCU researchers propose a quantum circuit to simulate the time evolution of wave functions under an S2 operator, exp(-iS2t)| as an important physical quantity, and integrate it into the QPE (Quantum Phase Estimation) circuit enabling us to determine the spin quantum number S of the arbitrary wave functions. They demonstrate that the spin quantum numbers of up to three spins can be determined by only one qubit measurement in QPE. This is the first quantum algorithm which enables us to determine the spin quantum numbers as a physical quantity on quantum computers. The OCU group said, "This is another example of a practical quantum algorithm, which makes quantum chemical calculations for predicting chemical reactions on quantum computers equipped with a sizable number of qubits. The implementation empowers practical applications of quantum chemical calculations on quantum computers in many important fields of chemistry and materials science."