Bias and Its Control in Stochastic Approaches to Electronic-Structure Theory

Stochastic formulations of electronic-structure theory often reduce computational cost by replacing exact contractions with statistical estimates obtained from random samples, a procedure that inherently introduces random fluctuations and systematic bias. The fluctuations decay as M^-1/2 with the number of samples M, whereas the bias generated in nonlinear or self-consistent settings decays as M^-1 and can remain significant for moderate M. To control this bias we employ the jackknife-2 estimator, which reduces its leading term to $O\left(M^{-2}\right)$ with only modest extra cost. We examine bias formation and removal in three settings: (i) stochastic treatments of the Markovian master equation using bundled dissipators, (ii) stochastic Kohn-Sham density functional theory for warm dense hydrogen, and (iii) stochastic evaluation of the Hubbard-model partition function. The first two settings have been presented in earlier works; accordingly, we review them only briefly and focus primarily on the issue of bias control. The Hubbard-model application is entirely new. For this case, we present two approaches: a direct estimator, which has large variance but no bias, and a "midway transition probability" (ΣMTP) estimator, which has smaller variance but introduces bias. Applying the jackknife-2 procedure to the ΣMTP estimator controls this bias and yields a substantially lower total error than the direct estimator. Across all cases, jackknife bias removal markedly improves the accuracy and reliability of stochastic electronic-structure calculations without increasing the computational cost.