  The Gutzwiller variational wavefunction (GVW) is commonly employed to capture
correlation effects in condensed matter systems such as ferromagnets, ultracold
bosonic gases, correlated superconductors, etc. By noticing that the
grand-canonical and number-conserving Gutzwiller Ans\"atze are in fact the
zero-order approximation of an expansion in the truncation parameter of a
Matrix Product State (MPS), we argue that MPSs, and the algorithms used to
operate on them, are not only flexible computational tools but also a unifying
theoretical framework that can be used to generalize and improve on the GVW. In
fact, we show that a number-conserving GVW is less efficient in capturing the
ground state of a quantum system than a more general MPS which can be optimized
with comparable computational resources. Moreover, we suggest a corrected
time-dependent density matrix renormalization group algorithm that ensures the
conservation of the expectation value of the number of particles when a GVW or
a MPS are not explicitly number-conserving. The GVW dynamics obtained with our
algorithm compares very well with the exact one in 1D. Most importantly, the
algorithm works in any dimension for a GVW. We thus expect it to be of great
value in the study of the dynamics of correlated quantum systems.
