Article pubs.acs.org/JPCA
Quantum Chemistry on Quantum Computers: A Polynomial-Time Quantum Algorithm for Constructing the Wave Functions of OpenShell Molecules Kenji Sugisaki,* Satoru Yamamoto, Shigeaki Nakazawa, Kazuo Toyota, Kazunobu Sato,* Daisuke Shiomi, and Takeji Takui* Department of Chemistry and Molecular Materials Science, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan S Supporting Information *
ABSTRACT: Quantum computers are capable to efficiently perform full configuration interaction (FCI) calculations of atoms and molecules by using the quantum phase estimation (QPE) algorithm. Because the success probability of the QPE depends on the overlap between approximate and exact wave functions, efficient methods to prepare accurate initial guess wave functions enough to have sufficiently large overlap with the exact ones are highly desired. Here, we propose a quantum algorithm to construct the wave function consisting of one configuration state function, which is suitable for the initial guess wave function in QPE-based FCI calculations of openshell molecules, based on the addition theorem of angular momentum. The proposed quantum algorithm enables us to prepare the wave function consisting of an exponential number of Slater determinants only by a polynomial number of quantum operations.
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probability of the QPE is proportional to |⟨Ψ0|ΨExact⟩|2, where |ΨExact⟩ and |Ψ0⟩ denote the exact and approximate wave function, respectively. With a Hartree−Fock (HF) wave function |ΨHF⟩ as the initial guess wave function, the experimental demonstrations of the FCI/STO-3G calculation of H2 molecule were reported in 2010, using photonic9 and NMR10 QCs with the aid of an adiabatic state preparation (ASP) technique.6,10 The ASP can systematically improve |Ψ0⟩ and increase the overlap based on the adiabatic theorem,11 which allows slowly changing the Hamiltonian from HF to FCI, and it is one of the most powerful approaches for wave function preparation.12,13 In the early attempts of the quantum FCI calculations, the information on the wave function represented by a linear combination of Slater determinants is often mapped onto quantum registers by means of a direct mapping (DM) approach, in which each quantum bit (qubit) represents the one-to-one occupancy of a particular spin−orbital, i.e., |1⟩ for being occupied and |0⟩ for unoccupied. The electronic Hamiltonian of a system is described in the second quantized formula with creation and annihilation operators, as given in eq 1
INTRODUCTION An ultimate goal of chemistry is to exactly solve a Schrödinger equation that governs chemical and physical laws of atoms and molecules. The exact wave functions allow us to thoroughly understand and predict microscopic molecular properties such as geometrical parameters, chemical reaction pathways and kinetics, and spectroscopic properties based on the first principle. The full configuration interaction (FCI) method is known to provide variationally best possible wave function within a given basis set. The number of variables in the FCI wave function, however, strongly depends on the number of electrons in the systems and the size of the basis set being used, and notoriously CPU time runs into astronomical figures. Currently, the FCI calculation is available only for small molecules like N21 and CN and its anion.2 In 1982, Feynman suggested that quantum systems can be efficiently simulated by using quantum devices.3 A quest for the realization of quantum computers (QCs) and quantum simulators (QSs) that utilize quantum mechanical principles such as superposition and entanglement of quantum states4 has increasingly attracted attention after Shor’s quantum factorization algorithm appeared.5 The first application of the quantum computation to the FCI energies of molecules was reported by Aspuru-Guzik and co-workers in 2005,6 in which the calculation is based on the quantum phase estimation (QPE) algorithm of Abrams and Lloyd.7,8 The QPE relies on a projection theorem to the eigenstate of the Hamiltonian of system, and the success © XXXX American Chemical Society
Received: May 16, 2016 Revised: July 11, 2016
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DOI: 10.1021/acs.jpca.6b04932 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A H=
four MnIV (d3) magnetic centers with antiferromagnetic interactions between the adjacent MnIII and MnIV centers, resulting in the ground state of S = 10.46 The number of primitive spin functions belonging to MS = 10 amounts to NCNβ = 44C12 ≈ 2.1 × 1010, where N and Nβ stand for the number of unpaired electrons and those of spin-β, respectively. The overlap between the Hartree−Fock and exact wave functions is approximately proportional to the inverse of the number of the primitive spin functions, and it is clear that the success probability of the QPE becomes exponentially small for the systems with large N and Nβ. This means that exponential number of QPE experiments are required to obtain the correct ground state energy if |ΨHF⟩ is used as the approximate wave function |Ψ0⟩. From the viewpoint of computational costs, development of new state preparation methods more efficient than the ASP is highly desired. Because the multiconfigurational character of the wave function of open-shell systems originates from the spin symmetry requirement, the construction of spin symmetryadapted configuration state functions (CSFs), which are simultaneous eigenfunctions of the S2 and SZ operators (hereafter denoted as spin eigenfunctions), on QCs may give a facile and useful solution to circumvent the overlap decay. We emphasize that adopting the CSFs as the basis of the FCI expansion instead of the Slater determinants corresponds to the block diagonalization of the FCI Hamiltonian matrix in the basis of spin quantum numbers. Therefore, if the |Ψ0⟩ is constructed by a single CSF, the overlap with the states in different spin quantum numbers from the target state becomes zero, although it is not necessary if the |Ψ0⟩ is composed of a single Slater determinant because the Slater determinants carrying unpaired electrons of spin-β are not spin eigenfunctions. We note that the idea to use the multiconfigurational selfconsistent field (MCSCF) wave function as the approximate wave function in the QPE to obtain energy eigenvalues of a multireference configuration interaction (MRCI) wave function has been discussed by Wang and co-workers in 2008.32 In their approach, the MCSCF calculation was carried out with classical computers in advance, and then the MCSCF wave function was mapped onto quantum registers to apply the QPE. The complete active space self-consistent field (CASSCF) wave function has been also frequently used as the trial wave function in the QPE.12,13,35 Preparations of multiconfigurational wave functions on quantum registers have been discussed.33,38 In this paper, we propose a quantum algorithm capable of creating the |ΨCSF⟩ consisting of a single CSF, based on the structure of spin eigenfunctions and the addition theorem of angular momentum. This quantum algorithm is executable in polynomial time against Nβ. Note that unlike the MCSCF wave functions, the |ΨCSF⟩ does not contain electron correlation effects because the |ΨCSF⟩ consists of a single CSF. To take into account static and dynamical electron correlation effects from the |ΨCSF⟩, additional operations such as ASP are required.
∑ hpr ar†ap + ∑ wpqrsar†as†aqap pr
(1)
pqrs
Because in the DM, the information on the orbital occupancy of indistinguishable electrons is mapped onto distinguishable qubits, the Jordan−Wigner transformation (JWT) as given in eq 2 is required to satisfy Fermionic anticommutation relations of the operators.14,15 ar† = σr+(∏ σiz) i
