E Pluribus Duo

The quantum many-body problem can be reformulated in terms of two particles, avoiding the exponential complexity of N-particle wavefunctions. However, the injection from N particles to two particles introduces a new form of complexity known as the N-representability problem. I will discuss the theoretical formulation of reduced-density-matrix mechanics, specifically the variational approach, as well as the computational method for solving the resulting equations. The computational methods range from the primal-dual method for semidefinite programming to a nonlinear optimization approach which is amenable to problems with millions of constraints and unknowns. Preliminary adventures into parallel algorithms may be discussed.