Efficient machine-learning representations of a surface code with boundaries, defects, domain walls, and twists

Zhian Jia, Yuan-Hang Zhang, Yu-Chun Wu, Liang Kong, Guang-Can Guo, Guo-Ping Guo

Abstract:

Machine-learning representations of many-body quantum states have recently been introduced as an ansatz
to describe the ground states and unitary evolutions of many-body quantum systems. We investigate one of the
most important representations, the restricted Boltzmann machine (RBM), in the stabilizer formalism. A general
method to construct RBM representations for stabilizer code states is given, and exact RBM representations for
several types of stabilizer groups with the number of hidden neurons equal to or less than the number of visible
neurons are presented. The result indicates that the representation is extremely efficient. Then we analyze a
surface code with boundaries, defects, domain walls, and twists in full detail and find that almost all the models
can be efficiently represented via the RBM ansatz: the RBM parameters of the perfect case, boundary case, and
defect case are constructed analytically using the method we provide in the stabilizer formalism, and the domain
wall and twist case is studied numerically. In addition, the case for Kitaev’s $D(\mathbb{Z}_d)$ model, which is a generalized
model of the surface code, is also investigated.