用随机微分方程方法研究全组态相互作用量子蒙特卡罗中的布居控制偏置
Stochastic differential equation approach to understanding the population control bias in full configuration interaction quantum Monte Carlo
摘要：
我们研究了一个系统的统计偏差发现在全组态量子蒙特卡罗（FCIQMC），起源于控制一个步行人口与波动的移位参数。当符号问题不存在时，例如在玻色子系统中，这种偏差可能成为主要误差。FCIQMC是一种强有力的统计方法，用于获得关于稀疏和抽象矩阵的基态的信息。我们表明，当没有符号问题时，移位估计具有很好的性质，提供了一个上界的确切基态能量和所有预计的能量估计，而变分估计仍然是一个上界的确切能量与大幅减少的偏见。一般FCIQMC种群动力学的标量模型导致一个精确可解的伊藤随机微分方程。它提供了进一步的见解偏差的性质，并给出了准确的分析预测延迟互协方差和自协方差函数的移位能量估计和沃克数。该模型提供了一个立足点，找到一个治愈的人口控制偏差。我们提供的证据，非普遍的幂律缩放的人口控制偏差与步行数的玻色-哈伯德模型的各种估计的基态能量的基础上的移动或投影能量。对于非相互作用的玻色-哈伯德哈密顿量的特定情况下，我们得到一个完整的分析预测的偏移能量估计的偏差。
英文摘要：
We investigate a systematic statistical bias found in full configuration quantum Monte Carlo (FCIQMC) that originates from controlling a walker population with a fluctuating shift parameter. This bias can become the dominant error when the sign problem is absent, e.g. in bosonic systems. FCIQMC is a powerful statistical method for obtaining information about the ground state of a sparse and abstract matrix. We show that, when the sign problem is absent, the shift estimator has the nice property of providing an upper bound for the exact ground state energy and all projected energy estimators, while a variational estimator is still an upper bound to the exact energy with substantially reduced bias. A scalar model of the general FCIQMC population dynamics leads to an exactly solvable Itô stochastic differential equation. It provides further insights into the nature of the bias and gives accurate analytical predictions for delayed cross-covariance and auto-covariance functions of the shift energy estimator and the walker number. The model provides a toe-hold on finding a cure for the population control bias. We provide evidence for non-universal power-law scaling of the population control bias with walker number in the Bose-Hubbard model for various estimators of the ground state energy based on the shift or on projected energies. For the specific case of the non-interacting Bose-Hubbard Hamiltonian we obtain a full analytical prediction for the bias of the shift energy estimator.
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