# Matrix product operators, matrix product states, and *ab initio* density matrix renormalization group algorithms

## Abstract

Current descriptions of the *ab initio* density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. Here, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the *ab initio* DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the *ab initio* DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in *ab initio* DMRG and related algorithms.

- Authors:

- Princeton Univ., NJ (United States). Dept. of Chemistry
- Weizmann Inst. of Science, Rehovot (Israel). Dept. of Condensed Matter Physics; Univ. of California, Irvine, CA (United States). Dept. of Physics and Astronomy
- Hokkaido Univ., Sapporo (Japan). Inst. for Catalysis
- Univ. of California, Irvine, CA (United States). Dept. of Physics and Astronomy

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

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

- OSTI Identifier:
- 1467831

- Alternate Identifier(s):
- OSTI ID: 1260285

- Grant/Contract Number:
- SC0010530; SC0008624

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 1; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Hilbert space; singular values; ab initio calculations; tensor methods; configuration interaction; ground states; nonperturbative techniques; renormalization; wave functions; quantum entanglement

### Citation Formats

```
Chan, Garnet Kin-Lic, Keselman, Anna, Nakatani, Naoki, Li, Zhendong, and White, Steven R.
```*Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms*. United States: N. p., 2016.
Web. doi:10.1063/1.4955108.

```
Chan, Garnet Kin-Lic, Keselman, Anna, Nakatani, Naoki, Li, Zhendong, & White, Steven R.
```*Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms*. United States. doi:10.1063/1.4955108.

```
Chan, Garnet Kin-Lic, Keselman, Anna, Nakatani, Naoki, Li, Zhendong, and White, Steven R. Tue .
"Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms". United States. doi:10.1063/1.4955108. https://www.osti.gov/servlets/purl/1467831.
```

```
@article{osti_1467831,
```

title = {Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms},

author = {Chan, Garnet Kin-Lic and Keselman, Anna and Nakatani, Naoki and Li, Zhendong and White, Steven R.},

abstractNote = {Current descriptions of the ab initio density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. Here, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.},

doi = {10.1063/1.4955108},

journal = {Journal of Chemical Physics},

issn = {0021-9606},

number = 1,

volume = 145,

place = {United States},

year = {2016},

month = {7}

}

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