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Ab-initio quantum chemistry through computer algebra -- a challenge beyond Born-Oppenheimer theory

Ichio Kikuchi, Akihito Kikuchi

Published Sep 30, 2022• 3 citations
Publisher

Abstract

In this article, we demonstrate the first-principles computation of quantum chemistry through symbolic computation, using computational algebraic geometry. We have developed a Python program that generates symbolic formulas of one- and two-electron integrals. The approximations of those integrals by multivariate polynomials yield the set of equations required by quantum chemistry. We solve these equations in hybrid ways where numeric and symbolic computations are intertwined. We discuss the possibility of the extension of our scheme to treat the models beyond the Born-Oppenheimer approximation.

DOI
10.31219/osf.io/qd8nz

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In this article, we demonstrate the first-principles computation of quantum chemistry through symbolic computation, using computational algebraic geometry. We have developed a Python program that generates symbolic formulas of one- and two-electron integrals. The approximations of those integrals by multivariate polynomials yield the set of equations required by quantum chemistry. We solve these equations in hybrid ways where numeric and symbolic computations are intertwined. We discuss the possibility of the extension of our scheme to treat the models beyond the Born-Oppenheimer approximation.
Computational quantum chemistry is a theoretical approach that utilizes the principles of quantum mechanics to perform computations on the electronic structure of molecules and atoms. This method enables the prediction of properties such as molecular structure, energy, electron orbitals, and electron density. In the field of the first principles, or ab initio, which relies solely on computation and not on experimental data, mathematical understanding allows even those without a background in chemistry to perform molecular orbital computations. The computation of quantum chemistry is computationally intensive, making them highly applicable in the field of high-performance computing. However, initiating quantum chemistry can be challenging for newcomers due to the numerous specialized terms and complex equations from quantum mechanics and chemistry. Therefore, this paper aims to provide a clear and accessible compilation of the fundamental and essential algorithm of the Hartree- Fock method, from theory to implementation. The goal is to offer the minimum required knowledge, in an approachable manner, serving as a starting point for those interested in computational quantum chemistry.
Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.
In this article, we discuss how a kind of hybrid computation, which employs symbolic, numeric, classic, and quantum algorithms, allows us to conduct Hartree-Fock electronic structure computation of molecules. In the proposed algorithm, we replace the Hartree-Fock equations with a set of equations composed of multivariate polynomials. We transform those polynomials to the corresponding Gr\"obner bases, and then we investigate the corresponding quotient ring, wherein the orbital energies, the LCAO coefficients, or the atomic coordinates are represented by the variables in the ring. In this quotient ring, the variables generate the transformation matrices that represent the multiplication with the monomial bases, and the eigenvalues of those matrices compose the roots of the equation. The quantum phase estimation (QPE) algorithm enables us to record those roots in the quantum states, which would be used in the input data for more advanced and more accurate quantum computations.