密度矩阵量子蒙特卡罗中的符号问题
The sign problem in density matrix quantum Monte Carlo
摘要：
密度矩阵量子Monte Carlo（DMQMC）是最近发展起来的一种随机抽样$N$ -粒子热密度矩阵的方法，用于获得模型和\emph{ab initio}系统的精确平均能量。我们报告了一个系统的数值研究的基础上模拟的原子和分子系统的DMQMC的符号问题。在DMQMC中，密度矩阵被写在Slater行列式的外积基中，并且具有为Slater行列式的数目的平方的空间大小。原则上，这意味着DMQMC需要对在系统大小$N$中缩放的空间进行采样，作为$\mathcal{O}[(\exp(N))^2]$。在实践中，有一个系统依赖的临界步行人口（$N_c$），必须超过，以消除符号的问题，这强加了限制的方式存储和计算机的时间。我们建立了DMQMC的$N_c$是FCIQMC的$N_c$的平方。相比之下，DMQMC（IP-DMQMC）的相互作用图修改中的最小值$N_c$仅与FCIQMC的$N_c$成正比。我们发现，这来自非对称传播的IP-DMQMC相比，规范DMQMC的对称传播。非对称传播模式对于DMQMC来说是极其昂贵的，因为它具有大得多的随机误差。最后，我们发现，IP-DMQMC和FCIQMC之间的等价性似乎延伸到发起人近似，这往往需要研究更大的基组和其他系统。这表明IP-DMQMC提供了一种方法来改善斯莱特行列式空间和外积基之间的移动成本。
英文摘要：
Density matrix quantum Monte Carlo (DMQMC) is a recently-developed method for stochastically sampling the $N$-particle thermal density matrix to obtain exact-on-average energies for model and \emph{ab initio} systems. We report a systematic numerical study of the sign problem in DMQMC based on simulations of atomic and molecular systems. In DMQMC, the density matrix is written in an outer product basis of Slater determinants and has a size of space which is the square of the number of Slater determinants. In principle this means DMQMC needs to sample a space which scales in the system size, $N$, as $\mathcal{O}[(\exp(N))^2]$. In practice, there is a system-dependent critical walker population ($N_c$) which must be exceeded in order to remove the sign problem, and this imposes limitations by way of storage and computer time. We establish that $N_c$ for DMQMC is the square of $N_c$ for FCIQMC. By contrast, the minimum $N_c$ in the interaction picture modification of DMQMC (IP-DMQMC) only is directly proportionate to the $N_c$ for FCIQMC. We find that this comes from the asymmetric propagation of IP-DMQMC compared to the symmetric propagation of canonical DMQMC. An asymmetric mode of propagation is prohibitively expensive for DMQMC because it has a much greater stochastic error. Finally, we find that the equivalence between IP-DMQMC and FCIQMC seems to extend to the initiator approximation, which is often required to study larger basis sets and other systems. This suggests IP-DMQMC offers a way to ameliorate the cost of moving between a Slater determinant space and an outer product basis.
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