Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes
Abstract
We discuss the relations between restricted Boltzmann machine (RBM) states and the matrix product states (MPS) for the ground states of 1D translational invariant stabilizer codes. A generic translational invariant and finitely connected RBM state can be expressed as an MPS, and the matrices of the resulting MPS are of rank 1. We dub such an MPS as an RBM-MPS. This provides a necessary condition for exactly realizing a quantum state as an RBM state, if the quantum state can be written as an MPS. For generic 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, we obtain an expression for the lower bound of their MPS bond dimension, and an upper bound for the rank of their MPS matrices. In terms of RBM, we provide an algorithm to derive the RBM for the cocycle Hamiltonians whose MPS matrices are proved to be of rank 1. Moreover, the RBM-MPS produced by our algorithm has the minimal bond dimension. A family of examples is provided to explain the algorithm. Furthermore, we finally conjecture that these features hold true for all the 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, as long as their MPSmore »
- Authors:
-
- Princeton Univ., NJ (United States)
- Sorbonne Univ., Paris (France)
- Princeton Univ., NJ (United States); Freie Univ. Berlin (Germany); Max Planck Inst. of Microstructure Physics, Halle (Germany)
- Publication Date:
- Research Org.:
- Princeton Univ., NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1612432
- Alternate Identifier(s):
- OSTI ID: 1507188
- Grant/Contract Number:
- SC0016239; de-sc0016239
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review. B
- Additional Journal Information:
- Journal Volume: 99; Journal Issue: 15; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Materials Science; Physics; Entanglement entropy; Artificial neural networks; Strongly correlated systems; Tensor network methods
Citation Formats
Zheng, Yunqin, He, Huan, Regnault, Nicolas, and Bernevig, B. Andrei. Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes. United States: N. p., 2019.
Web. doi:10.1103/physrevb.99.155129.
Zheng, Yunqin, He, Huan, Regnault, Nicolas, & Bernevig, B. Andrei. Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes. United States. https://doi.org/10.1103/physrevb.99.155129
Zheng, Yunqin, He, Huan, Regnault, Nicolas, and Bernevig, B. Andrei. Mon .
"Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes". United States. https://doi.org/10.1103/physrevb.99.155129. https://www.osti.gov/servlets/purl/1612432.
@article{osti_1612432,
title = {Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes},
author = {Zheng, Yunqin and He, Huan and Regnault, Nicolas and Bernevig, B. Andrei},
abstractNote = {We discuss the relations between restricted Boltzmann machine (RBM) states and the matrix product states (MPS) for the ground states of 1D translational invariant stabilizer codes. A generic translational invariant and finitely connected RBM state can be expressed as an MPS, and the matrices of the resulting MPS are of rank 1. We dub such an MPS as an RBM-MPS. This provides a necessary condition for exactly realizing a quantum state as an RBM state, if the quantum state can be written as an MPS. For generic 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, we obtain an expression for the lower bound of their MPS bond dimension, and an upper bound for the rank of their MPS matrices. In terms of RBM, we provide an algorithm to derive the RBM for the cocycle Hamiltonians whose MPS matrices are proved to be of rank 1. Moreover, the RBM-MPS produced by our algorithm has the minimal bond dimension. A family of examples is provided to explain the algorithm. Furthermore, we finally conjecture that these features hold true for all the 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, as long as their MPS matrices are of rank 1.},
doi = {10.1103/physrevb.99.155129},
journal = {Physical Review. B},
number = 15,
volume = 99,
place = {United States},
year = {Mon Apr 15 00:00:00 EDT 2019},
month = {Mon Apr 15 00:00:00 EDT 2019}
}
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