Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

An Optimisation on the Grassmannian with Applications toQuantum Chemistry

Yuri Alexandre Aoto, Márcio Fabiano da Silva

Published Dec 20, 2021• 1 citation
Publisher

Abstract

In this work we propose an algorithm to find critical points of the inner product between an element of the Grassmannian and a fixed point of the projective space of the exterior algebra where the Grassmannian is embedded. This has interesting applications to electronic structure theory, where the wave functions are represented by elements of the exterior algebra. This method is exemplified for a Grassmannian that is a model for the hydrogen molecule, H 2 .

DOI
10.5540/03.2021.008.01.0356

Related papers

Calculating the distance from an electronic wave function to the manifold of Slater determinants through the geometry of Grassmannians

Yuri Alexandre Aoto, Márcio Fabiano da Silva

The set of all electronic states that can be expressed as a single Slater determinant forms a submanifold, isomorphic to the Grassmannian, of the projective Hilbert space of wave functions. We explored this fact by using tools of Riemannian geometry of Grassmannians as described by [Absil et al., Acta Appl. Math. 80, 199 (2004)] to propose an algorithm that converges to a Slater determinant that is a critical point of the overlap function with a correlated wave function. This algorithm can be applied to quantify the entanglement or correlation of a wave function. We show that this algorithm is equivalent to the Newton method using the standard parametrization of Slater determinants by orbital rotations, but it can be more efficiently implemented because the orbital basis used to express the correlated wave function is kept fixed throughout the iterations. We present the equations of this method for a general configuration interaction wave function and for a wave function with up to double excitations over a reference determinant. Applications of this algorithm to selected electronic systems are also presented and discussed.

arXiv:2012.05283•Physical Review A•Nov 6, 2020•13 citations

Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties

Borovik, Viktoriia, Friedman, Hannah, Hoşten, Serkan +1 more

We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. We first study the Rayleigh-Ritz degree and introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to a deficient number of critical points. We then specialize this framework to tensor train varieties: we identify instances when they are Segre products of projective spaces, report what we know about their defining ideals, and present a birational parametrization from products of Grassmannians. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.

arXiv:2512.06939•Dec 7, 2025•0 citations

Many-body eigenstates from quantum manifold optimization

Scott E. Smart, Prineha Narang

Quantum computing offers several new pathways toward finding many-body eigenstates, with variational approaches being some of the most flexible and near-term oriented. These require particular parametrizations of the state and, for solving multiple eigenstates, must incorporate orthogonality. In this work, we use techniques from manifold optimization to arrive at solutions of the many-body eigenstate problem via direct minimization over the Stiefel and Grassmannian manifolds, avoiding parametrizations of the states and allowing for multiple eigenstates to be simultaneously calculated. These Riemannian manifolds naturally encode orthogonality constraints and have efficient quantum representations of the states and tangent vectors. We provide example calculations for quantum many-body molecular systems and discuss different pathways for solving the multiple eigenstate problem.

arXiv:2402.07100•Physical Review A•Nov 21, 2024•3 citations

Algorithm Design in Computational Quantum Chemistry

Ernest R. Davidson

Quantum chemistry is a diverse discipline which uses many different methods to correlate a wide variety of phenomena. In the earliest period of the subject the Schrödinger equation was solved exactly for a few simple model situations. These model solutions were then used to interpret the spectra, kinetics, and thermodynamics of molecules and solids. During this period, accurate solutions for the electronic structure of helium ( 1 ) and the hydrogen molecule ( 2 ) were obtained in order to verify that the Schrödinger equation was useful. Most of the effort, however, was devoted to developing a simple quantum model of electronic structure. Hartree ( 3 ) and others developed the self-consistent-field model for the structure of light atoms. For heavier atoms, the Thomas-Fermi model ( 4 ) based on total charge density rather than individual orbitals was used. Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model

ACS symposium series•Jun 1, 1977•5 citations

Tensor product approximation with optimal rank in quantum chemistry

Sambasiva Rao Chinnamsetty, Mike Espig, Boris N. Khoromskij +2 more

Tensor product decompositions with optimal separation rank provide an interesting alternative to traditional Gaussian-type basis functions in electronic structure calculations. We discuss various applications for a new compression algorithm, based on the Newton method, which provides for a given tensor the optimal tensor product or so-called best separable approximation for fixed Kronecker rank. In combination with a stable quadrature scheme for the Coulomb interaction, tensor product formats enable an efficient evaluation of Coulomb integrals. This is demonstrated by means of best separable approximations for the electron density and Hartree potential of small molecules, where individual components of the tensor product can be efficiently represented in a wavelet basis. We present a fairly detailed numerical analysis, which provides the basis for further improvements of this novel approach. Our results suggest a broad range of applications within density fitting schemes, which have been recently successfully applied in quantum chemistry.

The Journal of Chemical Physics•Aug 28, 2007•61 citations

Grassmann algebra and fermions in the lattice: A simple example

S. M. de Souza, M. T. Thomaz

Using the exact results for integrals of exponentials of polynomials of Grassmann variables and the Feynman construction of path integrals, an approximate method is presented to calculate the thermodynamic quantities of a fermionic system in equilibrium with a reservoir at temperature T and chemical potential μ. To exemplify, the quantum anharmonic fermionic oscillator is considered in a lattice with N points, where N=2, 3, and 4.

Journal of Mathematical Physics•Dec 1, 1991•4 citations
In this work we propose an algorithm to find critical points of the inner product between an element of the Grassmannian and a fixed point of the projective space of the exterior algebra where the Grassmannian is embedded. This has interesting applications to electronic structure theory, where the wave functions are represented by elements of the exterior algebra. This method is exemplified for a Grassmannian that is a model for the hydrogen molecule, H 2 .
The set of all electronic states that can be expressed as a single Slater determinant forms a submanifold, isomorphic to the Grassmannian, of the projective Hilbert space of wave functions. We explored this fact by using tools of Riemannian geometry of Grassmannians as described by [Absil et al., Acta Appl. Math. 80, 199 (2004)] to propose an algorithm that converges to a Slater determinant that is a critical point of the overlap function with a correlated wave function. This algorithm can be applied to quantify the entanglement or correlation of a wave function. We show that this algorithm is equivalent to the Newton method using the standard parametrization of Slater determinants by orbital rotations, but it can be more efficiently implemented because the orbital basis used to express the correlated wave function is kept fixed throughout the iterations. We present the equations of this method for a general configuration interaction wave function and for a wave function with up to double excitations over a reference determinant. Applications of this algorithm to selected electronic systems are also presented and discussed.
We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. We first study the Rayleigh-Ritz degree and introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to a deficient number of critical points. We then specialize this framework to tensor train varieties: we identify instances when they are Segre products of projective spaces, report what we know about their defining ideals, and present a birational parametrization from products of Grassmannians. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.
Quantum computing offers several new pathways toward finding many-body eigenstates, with variational approaches being some of the most flexible and near-term oriented. These require particular parametrizations of the state and, for solving multiple eigenstates, must incorporate orthogonality. In this work, we use techniques from manifold optimization to arrive at solutions of the many-body eigenstate problem via direct minimization over the Stiefel and Grassmannian manifolds, avoiding parametrizations of the states and allowing for multiple eigenstates to be simultaneously calculated. These Riemannian manifolds naturally encode orthogonality constraints and have efficient quantum representations of the states and tangent vectors. We provide example calculations for quantum many-body molecular systems and discuss different pathways for solving the multiple eigenstate problem.
Quantum chemistry is a diverse discipline which uses many different methods to correlate a wide variety of phenomena. In the earliest period of the subject the Schrödinger equation was solved exactly for a few simple model situations. These model solutions were then used to interpret the spectra, kinetics, and thermodynamics of molecules and solids. During this period, accurate solutions for the electronic structure of helium ( 1 ) and the hydrogen molecule ( 2 ) were obtained in order to verify that the Schrödinger equation was useful. Most of the effort, however, was devoted to developing a simple quantum model of electronic structure. Hartree ( 3 ) and others developed the self-consistent-field model for the structure of light atoms. For heavier atoms, the Thomas-Fermi model ( 4 ) based on total charge density rather than individual orbitals was used. Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model
Tensor product decompositions with optimal separation rank provide an interesting alternative to traditional Gaussian-type basis functions in electronic structure calculations. We discuss various applications for a new compression algorithm, based on the Newton method, which provides for a given tensor the optimal tensor product or so-called best separable approximation for fixed Kronecker rank. In combination with a stable quadrature scheme for the Coulomb interaction, tensor product formats enable an efficient evaluation of Coulomb integrals. This is demonstrated by means of best separable approximations for the electron density and Hartree potential of small molecules, where individual components of the tensor product can be efficiently represented in a wavelet basis. We present a fairly detailed numerical analysis, which provides the basis for further improvements of this novel approach. Our results suggest a broad range of applications within density fitting schemes, which have been recently successfully applied in quantum chemistry.