Semidefinite programming is an important generalization of linear programming in which linear inequalities are replaced by linear matrix inequalities [1]. In this lecture we will examine the application of semidefinite programming in quantum chemistry to describe weakly and strongly correlated quantum systems. We will focus on the computation of one- and two-electron reduced density matrices (1- and 2-RDMs) for the study of mean-field behavior and strong correlation in molecules and materials, respectively [2]. By expressing mean-field calculations as semidefinite programs , we can compute upper and lower bounds on the Hartree-Fock energy [3]. When these upper and lower bounds are equal, they provide a polynomial-time calculation of the global Hartree-Fock energy. Semidefinite programs using the 2-RDM as the basic variable can treat strongly correlated electrons at a polynomial computational cost with applications from hydrocarbon chemistry to bioluminescence [2]. We will also examine how measurement-driven reconstruction of quantum processes can be performed with semidefinite programming [4]. Information about many-particle systems can be extracted from experimental data in the form of 1- or 2-RDMs. Finally, we will discuss how a generalization of the Pauli exclusion principle defines a convex polytope of pure 1-RDMs that are derivable from a single wave function. By studying only the relation of a system’s 1-RDM to the boundary of this set, we can address the significant question of whether a quantum system is open or closed to interactions with its environment [5].
[1] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011). "Large-scale semidefinite programming for many-electron quantum mechanics"
[2] L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010). "Strong electron correlation in the decomposition reaction of dioxetanone with implications for firefly bioluminescence"
[3] S. Veeraraghavan and D. A. Mazziotti, Phys. Rev. A 89, 010502 (2014). "Global solutions of Hartree- Fock theory and their consequences for strongly correlated quantum systems"; J. Chem. Phys. 140, 124106 (2014). "Global solutions of restricted open-shell Hartree-Fock theory from semidefinite programming with applications to strongly correlated quantum systems"
[4] J. J. Foley IV and D. A. Mazziotti, Phys. Rev. A 86, 012512 (2012). "Measurement-driven reconstruction of many-particle quantum processes by semidefinite programming with application to photosynthetic light harvesting"
[5] R. Chakraborty and D. A. Mazziotti, Phys. Rev. A 89, 042505 (2014). "Generalized Pauli conditions on the spectra of one-electron reduced density matrices of atoms and molecules"