# Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo

## Abstract

Within *ab initio* Quantum Monte Carlo simulations, the leading numerical cost for large systems is the computation of the values of the Slater determinants in the trial wavefunction. Each Monte Carlo step requires finding the determinant of a dense matrix. This is most commonly iteratively evaluated using a rank-1 Sherman-Morrison updating scheme to avoid repeated explicit calculation of the inverse. The overall computational cost is therefore formally cubic in the number of electrons or matrix size. To improve the numerical efficiency of this procedure, we propose a novel multiple rank delayed update scheme. This strategy enables probability evaluation with application of accepted moves to the matrices delayed until after a predetermined number of moves, *K*. The accepted events are then applied to the matrices *en bloc* with enhanced arithmetic intensity and computational efficiency via matrix-matrix operations instead of matrix-vector operations. Here this procedure does not change the underlying Monte Carlo sampling or its statistical efficiency. For calculations on large systems and algorithms such as diffusion Monte Carlo where the acceptance ratio is high, order of magnitude improvements in the update time can be obtained on both multi- core CPUs and GPUs.

- Authors:

- Univ. of Tennessee, Knoxville, TN (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Laboratory, Oak Ridge Leadership Computing Facility (OLCF); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1407773

- Alternate Identifier(s):
- OSTI ID: 1407834

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 147; Journal Issue: 17; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
McDaniel, Tyler, D’Azevedo, Ed F., Li, Ying Wai, Wong, Kwai, and Kent, Paul R. C. Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo. United States: N. p., 2017.
Web. doi:10.1063/1.4998616.
```

```
McDaniel, Tyler, D’Azevedo, Ed F., Li, Ying Wai, Wong, Kwai, & Kent, Paul R. C. Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo. United States. doi:10.1063/1.4998616.
```

```
McDaniel, Tyler, D’Azevedo, Ed F., Li, Ying Wai, Wong, Kwai, and Kent, Paul R. C. Tue .
"Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo". United States. doi:10.1063/1.4998616. https://www.osti.gov/servlets/purl/1407773.
```

```
@article{osti_1407773,
```

title = {Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo},

author = {McDaniel, Tyler and D’Azevedo, Ed F. and Li, Ying Wai and Wong, Kwai and Kent, Paul R. C.},

abstractNote = {Within ab initio Quantum Monte Carlo simulations, the leading numerical cost for large systems is the computation of the values of the Slater determinants in the trial wavefunction. Each Monte Carlo step requires finding the determinant of a dense matrix. This is most commonly iteratively evaluated using a rank-1 Sherman-Morrison updating scheme to avoid repeated explicit calculation of the inverse. The overall computational cost is therefore formally cubic in the number of electrons or matrix size. To improve the numerical efficiency of this procedure, we propose a novel multiple rank delayed update scheme. This strategy enables probability evaluation with application of accepted moves to the matrices delayed until after a predetermined number of moves, K. The accepted events are then applied to the matrices en bloc with enhanced arithmetic intensity and computational efficiency via matrix-matrix operations instead of matrix-vector operations. Here this procedure does not change the underlying Monte Carlo sampling or its statistical efficiency. For calculations on large systems and algorithms such as diffusion Monte Carlo where the acceptance ratio is high, order of magnitude improvements in the update time can be obtained on both multi- core CPUs and GPUs.},

doi = {10.1063/1.4998616},

journal = {Journal of Chemical Physics},

number = 17,

volume = 147,

place = {United States},

year = {2017},

month = {11}

}

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