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We present a unified theory of the variational Monte Carlo (VMC) anddeterminant quantum Monte Carlo (DQMC) methods using a novel density matrixformulation of VMC. We introduce an efficient algorithm for VMC to computecorrelation functions and expectation values based on the auxiliary fieldHirsch-Hubbard-Stratonovic transformation. We show that this new approach toVMC converges significantly faster than its traditional implementations.Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginarytime steps $\tau \sim O(1)$ and develop a variational quantum Monte Carlo(VQMC) method accordingly, which is more accurate than VMC and can incorporatequantum fluctuations more efficiently. The two extreme limits of the VQMCmethod, namely infinitesimal and infinite imaginary time steps, correspond tothe DQMC and VMC techniques, respectively. We demonstrate that our VQMC allowsus to access lower temperatures in comparison with the conventional DQMC beforethe sign problem comes into play. We finally show that our VQMC can alsoenhance the accuracy of the projector Monte Carlo methods by providing betterand less biased candidates for the trial wave functions, requiring shorterprojection times for a given accuracy and alleviating the sign problem further.

A unified theory of variational and quantum Monte Carlo methods and beyond