Semidefinite Relaxation of Multimarginal Optimal Transport for Strictly Correlated Electrons in Second Quantization

We consider the strictly correlated electron (SCE) limit of the fermionic quantum many-body problem in the second-quantized formalism. This limit gives rise to a multimarginal optimal transport (MMOT) problem. Here the marginal state space for our MMOT problem is the binary set 0,1, and the number of marginals is the number L of sites in the model. The costs of storing and computing the exact solution of the MMOT problem both scale exponentially with respect to L. We propose an efficient convex relaxation to the MMOT which can be solved by semidefinite programming (SDP). In particular, the semidefinite constraint is only of size 2L X 2L. We further prove that the SDP has dual attainment, in spite of the lack of Slater's condition (i.e., the primal SDP does not have any strictly feasible point). In the context of determining the lowest energy of electrons via density functional theory, such dual attainment implies the existence of an effective potential needed to solve a nonlinear Schrödinger equation via self-consistent field iteration. We demonstrate the effectiveness of our methods on computing the ground state energy of spinless and spinful Hubbard-type models. Numerical results indicate that our SDP formulation yields comparable results when using the unrelaxed MMOT formulation. We also describe how our relaxation methods generalize to arbitrary MMOT problems with pairwise cost functions.