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 <i>M</i>^-1/2 with the number of samples <i>M</i>, whereas the bias generated in nonlinear or self-consistent settings decays as <i>M</i>^-1 and can remain significant for moderate <i>M</i>. To control this bias we employ the jackknife-2 estimator, which reduces its leading term to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> 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.