Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes

Yunqin Zheng, Huan He, Nicolas Regnault, B. Andrei Bernevig

Research output: Contribution to journalArticle

3 Scopus citations

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. 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.

Original languageEnglish (US)
Article number155129
JournalPhysical Review B
Volume99
Issue number15
DOIs
StatePublished - Apr 15 2019

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics

Fingerprint Dive into the research topics of 'Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes'. Together they form a unique fingerprint.

  • Cite this