Mitigating sign problem by automatic differentiation

Abstract

As an intrinsically-unbiased method, quantum Monte Carlo (QMC) is of unique importance in simulating interacting quantum systems. Unfortunately, QMC often suffers from the notorious sign problem. Although generically curing sign problem is shown to be hard (NP-hard), sign problem of a given quantum model may be mitigated (sometimes even cured) by finding better choices of simulation scheme. A universal framework in identifying optimal QMC schemes has been desired. Here, we propose a general framework using automatic differentiation (AD) to automatically search for the best continuously-parameterized QMC scheme, which we call "automatic differentiable sign mitigation" (ADSM). We further apply the ADSM framework to the honeycomb lattice Hubbard model with Rashba spin-orbit coupling and demonstrate ADSM's effectiveness in mitigating its sign problem. For the model under study, ADSM leads a significant power-law acceleration in computation time (the computation time is reduced from M to the order of M ν with ν ≈ 2/3).

The ALF (Algorithms for Lattice Fermions) project release 2.3. Documentation for the auxiliary-field quantum Monte Carlo code

“…9.1 has a much more severe sign problem than the formulation (presented in the same section) where the HS field couples to the z-component of the magnetization. Optimization schemes minimize ∆ have been put forward in [111,112].…”

Quantum Monte Carlo Simulation of Generalized Kitaev Models

“…The negative sign problem in the QMC approach is formulation dependent and hence can, in principle, be reduced so as to reach relevant energy scales. In fact, this can be seen as an optimization problem over the space of possible path integral formulations [14,15]. Here we adopt a symmetry based strategy, that pins the phase of the action to 0 and π.…”

Simultaneous Stoquasticity

“…A large body of literature has been built up around the problem of finding stoquastic [14,15] or nearly stoquastic [16,17] bases for Hamiltonians. The corresponding unitary basis change is said to "cure" the non-stoquastic Hamiltonian.…”