Mitigating the Sign Problem Through Basis Rotations

Abstract: Quantum Monte Carlo simulations of quantum many body systems are plagued by the Fermion sign problem. The computational complexity of simulating Fermions scales exponentially in the projection time $\beta$ and system size. The sign problem is basis dependent and an improved basis, for fixed errors, lead to exponentially quicker simulations. We show how to use sign-free quantum Monte Carlo simulations to optimize over the choice of basis on large two-dimensional systems. We numerically illustrate these techniques decreasing the `badness' of the sign problem by optimizing over single-particle basis rotations on one and two-dimensional Hubbard systems. We find a generic rotation which improves the average sign of the Hubbard model for a wide range of $U$ and densities for $L \times 4$ systems. In one example improvement, the average sign (and hence simulation cost at fixed accuracy) for the $16\times 4$ Hubbard model at $U/t=4$ and $n=0.75$ increases by $\exp\left[8.64(6)\beta\right]$. For typical projection times of $\beta\gtrapprox 100$, this accelerates such simulation by many orders of magnitude.