Projector quantum Monte Carlo with matrix product states

We marry tensor network states (TNS) and projector quantum Monte Carlo (PMC) to overcome the high computational scaling of TNS and the sign problem of PMC. Using TNS as trial wave functions provides a route to systematically improve the sign structure and to eliminate the bias in fixed-node and constrained-path PMC. As a specific example, we describe phaseless auxiliary-field quantum Monte Carlo with matrix product states (MPS-AFQMC). MPS-AFQMC improves significantly on the density-matrix renormalization group (DMRG) ground-state energy. For the $J_1-J_2$ model on two-dimensional square lattices, we observe with MPS-AFQMC an order of magnitude reduction in the error for all couplings, compared to DMRG. The improvement is independent of walker bond dimension, and we therefore use bond dimension 1 for the walkers. The computational cost of MPS-AFQMC is then quadratic in the bond dimension of the trial wave function, which is lower than the cubic scaling of DMRG. The error due to the constrained-path bias is proportional to the variational error of the trial wave function. We show that for the $J_1-J_2$ model on two-dimensional square lattices, a linear extrapolation of the MPS-AFQMC energy with the discarded weight from the DMRG calculation allows us to remove the constrained-path bias. Extensions to other tensor networks are briefly discussed.

Figure 1

Imaginary time evolution for a $4\times 4J_1\text{−}{J}_2$ model (PBC) with $J_1=1$ and $J_2=0$, using different values for the trial bond dimension $D_T$ and walker bond dimension $D_W$.

Figure 2

Converged MPS-AFQMC and DMRG energies for different $D_T$ for a $4\times 4J_1\text{−}{J}_2$ model (PBC) with $J_1=1$ and $J_2=0.6$. Inset: extrapolation of the MPS-AFQMC energy with respect to the discarded weight of the trial state.

Figure 3

Relative energy errors (from the exact result) for MPS-AFQMC and DMRG, as a function of $D_T$, for a $6\times 6J_1\text{−}{J}_2$ model (PBC) with $J_2$ ranging from 0 to $J_1$.

Figure 4

Relative energy errors (from the exact result) for MPS-AFQMC and DMRG, as a function of $D_T$, for a $8\times 8J_1\text{−}{J}_2$ model (OBC) with $J_2$ ranging from 0 to $J_1$, compared to PEPS results from Ref. [38].

Figure 5

Relative energy errors (from the exact result) for MPS-AFQMC and DMRG, as a function of $D_T$, for a $10\times 10J_1\text{−}{J}_2$ model (OBC) with $J_2$ ranging from 0 to $J_1$.