  We present a unified theory of the variational Monte Carlo (VMC) and
determinant quantum Monte Carlo (DQMC) methods using a novel density matrix
formulation of VMC. We introduce an efficient algorithm for VMC to compute
correlation functions and expectation values based on the auxiliary field
Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to
VMC converges significantly faster than its traditional implementations.
Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary
time steps $\tau \sim O(1)$ and develop a variational quantum Monte Carlo
(VQMC) method accordingly, which is more accurate than VMC and can incorporate
quantum fluctuations more efficiently. The two extreme limits of the VQMC
method, namely infinitesimal and infinite imaginary time steps, correspond to
the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows
us to access lower temperatures in comparison with the conventional DQMC before
the sign problem comes into play. We finally show that our VQMC can also
enhance the accuracy of the projector Monte Carlo methods by providing better
and less biased candidates for the trial wave functions, requiring shorter
projection times for a given accuracy and alleviating the sign problem further.
