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

Area of Science:

• Computational physics
• Quantum chemistry
• Materials science

Background:

• Stochastic methods in electronic-structure theory reduce computational cost.
• These methods introduce random fluctuations and systematic bias.
• Bias can be significant in nonlinear or self-consistent calculations.

Purpose of the Study:

• To control and remove systematic bias in stochastic electronic-structure calculations.
• To improve the accuracy and reliability of these computational methods.
• To introduce and validate the jackknife-2 estimator for bias reduction.

Main Methods:

• Employing the jackknife-2 estimator to reduce leading-order bias terms.
• Examining bias in stochastic Markovian master equation, Kohn-Sham DFT for warm dense hydrogen, and Hubbard model.
• Developing direct and ΣMTP estimators for the Hubbard model partition function.

Main Results:

• Jackknife-2 estimator reduces bias to O(M^-2) with modest cost.
• Bias control was demonstrated across three distinct computational settings.
• For the Hubbard model, jackknife-2 on the ΣMTP estimator yielded lower total error than the direct estimator.

Conclusions:

• Jackknife bias removal significantly enhances accuracy and reliability of stochastic electronic-structure calculations.
• The jackknife-2 estimator provides an effective and computationally efficient bias control strategy.
• This approach is broadly applicable to various stochastic formulations in electronic-structure theory.