Reduction of the molecular hamiltonian matrix using quantum community detection

Abstract

Quantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree–Fock methods as future advances in quantum hardware become available.

Variational Quantum Eigensolver with Reduced Circuit Complexity

“…I A). Alternatively, this can be mapped into a quadratic unconstrained binary optimization (QUBO) problem and solved on a quantum annealer by either the MI-based clustering method (details are given Appendix I B) or the quantum community detection method (QCD) [53] (details are given in Appendix I C). Problems with up to 64 variables or nodes can be solved directly on the D-Wave 2000Q quantum annealer.…”

“…While QCD can cluster a graph into 2 or more clusters [53,82,83], in this work we are interested in splitting only into 2 clusters. The modularity matrix in QUBO form is run on the D-Wave 2000Q quantum annealer:…”

The Basics of Quantum Computing for Chemists

“…However, this review should remain a valuable contribution to introduce chemists, or anyone whose interest is to get acquainted to the basics of gate-based quantum computing with a similar background. Other architectural and computational paradigms exist, and a good example would be those grounded on the adiabatic quantum computing, [116][117][118][119] which has also been employed for molecular electronic structure simulations, [120][121][122][123] but without the same momentum or appeal garnered by logic quantum computers. This further advances the claim that the introductory sections of this review are likely to remain relevant in the near future insofar as one seeks this type of computation.…”

QUBO-based density matrix electronic structure method

“…exploring how feasible it is to construct the density matrix (DM) using linear algebra QUBO algorithms. Previous work on QAs used in quantum chemistry has been recently applied to determine the ground and firsts excited states on the Full Configuration Interaction (CI) method by diagonalizing the Full CI (FCI) matrix 5 ; the downfolding of the FCI matrix to reduce the size of the effective matrix 6 ; and Time Dependent Density Functional Theory (DFT) to compute excitations 7 . The density matrix (DM) formalism is ubiquitous to several methods in computational chemistry that compute the electronic structure of a molecular system.…”