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

Author(s): Zheng, Yunqin; He, Huan; Regnault, Nicolas; Bernevig, Bogdan A.

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dc.contributor.authorZheng, Yunqin-
dc.contributor.authorHe, Huan-
dc.contributor.authorRegnault, Nicolas-
dc.contributor.authorBernevig, Bogdan A.-
dc.date.accessioned2019-12-12T16:26:42Z-
dc.date.available2019-12-12T16:26:42Z-
dc.date.issued2019-04-15en_US
dc.identifier.citationZheng, Yunqin, He, Huan, Regnault, Nicolas, Bernevig, B Andrei. (2019). Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes. PHYSICAL REVIEW B, 99, doi:10.1103/PhysRevB.99.155129en_US
dc.identifier.issn2469-9950-
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/pr1dj1b-
dc.description.abstractWe 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.en_US
dc.format.extent155129-1 - 155129-32en_US
dc.language.isoen_USen_US
dc.relation.ispartofPHYSICAL REVIEW Ben_US
dc.rightsFinal published version. Article is made available in OAR by the publisher's permission or policy.en_US
dc.titleRestricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codesen_US
dc.typeJournal Articleen_US
dc.identifier.doidoi:10.1103/PhysRevB.99.155129-
dc.date.eissued2019-04-15en_US
dc.identifier.eissn2469-9969-
pu.type.symplectichttp://www.symplectic.co.uk/publications/atom-terms/1.0/journal-articleen_US

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