Sign Problem in Quantum Monte Carlo Simulation

Abstract

Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.

Thermodynamic characteristic for correlated flat-band system with quantum anomalous Hall ground state

“…4, with decreasing temperature , the average sign does not decay exponentially to zero, but gradually reaches a finite value, where we can perform monte carlo simulation accurately. This is consistent with the conclusion of the sign bound theory [59,60]: in the zero temperature limit, the sign has a lower bound related to the ground state degeneracy. In addition, since the ground state degeneracy of the model we calculated is a polynomial rather than an exponential function of the system size 𝐿, the average sign does not decay exponentially but algebraically as 𝐿 increases.…”

“…[59], it also decays algebraically with respect to the system size 𝐿. number but not necessarily positive [49]. Fortunately, the sign boundary theory [59,60] tells us that for such a model in the chiral limit, the average of sign at zero temperature has a polynomial lower bound, which is related to the ground state degeneracy. This means that even if the QMC simulations do not have all the weight positive, one can still solve the problem with polynomial computation complexity even at very low temperatures.…”

“…To compute the dynamic and thermodynamic properties of the TBG model in Eq. ( 2), we employ the recently developed momentum space QMC method [49,50], which can fully incorporate the flat-band topological wavefunction and the long-range Coulomb interaction and our sign bound theory can prove there is at most polynomial sign problem for QMC at integer fillings [59,60], rendering the computational complexity also polynomial. The QMC solves the finite size systems in a path-integral of partition function in an unbiased manner and we have simulated 𝐿 = 3, 4, 5, 6 and 𝑇 ∈ [0.5, 20] meV systems.…”

How universal is the relation between sign problem and phase transition

“…However, there exists a fundamental deficiency in QMC approaches. Various QMC methods suffer from notorious sign problems [9][10][11][12][13] in many widely interested quantum systems. With a sign problem, some of the sampling weights in the QMC procedure can be negative or even complex, which does not make sense since a distribution probability must be zero or positive.…”

Partition-function estimation: Quantum and quantum-inspired algorithms

“…Our Letter thus provides a new perspective on the hardness of estimating partition functions, one requiring no mention of the sign problem [24]. It is rather based on the hardness of decomposing a Hamiltonian into a linear combination of tensor products of Pauli operators.…”