  Density matrix embedding theory (DMET) is a powerful quantum embedding method
for solving strongly correlated quantum systems. Theoretically, the performance
of a quantum embedding method should be limited by the computational cost of
the impurity solver. However, the practical performance of DMET is often
hindered by the numerical stability and the computational time of the
correlation potential fitting procedure, which is defined on a single-particle
level. Of particular difficulty are cases in which the effective
single-particle system is gapless or nearly gapless. To alleviate these issues,
we develop a semidefinite programming (SDP) based approach that can
significantly enhance the robustness of the correlation potential fitting
procedure compared to the traditional least squares fitting approach. We also
develop a local correlation potential fitting approach, which allows one to
identify the correlation potential from each fragment independently in each
self-consistent field iteration, avoiding any optimization at the global level.
We prove that the self-consistent solutions of DMET using this local
correlation potential fitting procedure are equivalent to those of the original
DMET with global fitting. We find that our combined approach, called L-DMET, in
which we solve local fitting problems via semidefinite programming, can
significantly improve both the robustness and the efficiency of DMET
calculations. We demonstrate the performance of L-DMET on the 2D Hubbard model
and the hydrogen chain. We also demonstrate with theoretical and numerical
evidence that the use of a large fragment size can be a fundamental source of
numerical instability in the DMET procedure.
