  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 multi-marginal 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 which can be solved
by semidefinite programming (SDP). In particular, the semidefinite constraint
is only of size $2L\times 2L$. Moreover, the SDP-based method yields an
approximation of the dual potential needed to the perform self-consistent field
iteration in the so-called Kohn-Sham SCE framework, which, once converged,
yields a lower bound for the total energy of the system. We demonstrate the
effectiveness of our methods on spinless and spinful Hubbard-type models.
Numerical results indicate that our relaxation methods yield tight lower bounds
for the optimal cost, in the sense that the error due to the semidefinite
relaxation is much smaller than the intrinsic modeling error of the Kohn-Sham
SCE method. We also describe how our relaxation methods generalize to arbitrary
MMOT problems with pairwise cost functions.
