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

机译：限制的Boltzmann机器和矩阵产品状态一维译文不变稳定器代码

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摘要

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.
机译：我们讨论了限制Boltzmann机器（RBM）状态与矩阵产品状态（MPS）的关系，为1D平移不变稳定剂码的地面态。通用平移不变和有限连接的RBM状态可以表示为MPS，并且所得MP的矩阵为秩1.我们将这种MPS作为RBM-MPS配合。这提供了一种必要的条件，用于将量子状态恰好实现为RBM状态，如果量子状态可以写为MPS。对于具有周期性边界条件的具有非值接地状态的通用1D稳定器代码，我们获得其MPS键尺寸的下限的表达，以及其MPS矩阵的秩的上限。在RBM方面，我们提供了一种算法来导出rbm的rbm for cocycle hamiltonians，其MPS矩阵被证明是秩的1.此外，我们的算法产生的RBM-MP具有最小的粘合尺寸。提供了一个例子以解释算法。我们终于猜想这些特征对于所有具有周期性边界条件的非稳定器代码的所有1D稳定器代码保持真实，只要它们的MPS矩阵为秩1即可。

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  • 来源
    《Physical review》 |2019年第15期|155129.1-155129.32|共32页
  • 作者

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

  • 作者单位

    Princeton Univ Phys Dept Princeton NJ 08544 USA;

    Princeton Univ Phys Dept Princeton NJ 08544 USA;

    Univ Paris Diderot PSL Res Univ UPMC Univ Paris 06 Sorbonne Paris Cite Sorbonne Univ Lab Pierre Aigrain Dept Phys CNRS E F-75005 Paris France;

    Princeton Univ Phys Dept Princeton NJ 08544 USA|Free Univ Berlin Phys Dept Arnimallee 14 D-14195 Berlin Germany|Max Planck Inst Microstruct Phys D-06120 Halle Germany;

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