Efficient Machine Learning Representations of Surface Code with Boundaries, Defects, Domain Walls and Twists

TL;DR

This paper develops an efficient RBM-based quantum state representation for surface codes with complex features like boundaries and defects, extending to generalized models, enabling compact descriptions of these many-body states.

Contribution

It provides a general method to construct RBM representations for stabilizer code states, including surface codes with various features, and demonstrates their efficiency.

Findings

RBM representations for stabilizer states are highly efficient.

All studied surface code variants can be represented via RBM states.

The approach extends to generalized models like Kitaev's D(Z_d) model.

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 explore one of the most important representations, restricted Boltzmann machine (RBM) representation, in stabilizer formalism. We give the general method of constructing RBM representation for stabilizer code states and find the exact RBM representation for several types of stabilizer groups with the number of hidden neurons equal or less than the number of visible neurons, which indicates that the representation is extremely efficient. Then we analyze the surface code with boundaries, defects, domain walls and twists in full detail and find that all the models can be efficiently represented via RBM ansatz states. Besides, the case for Kitaev's $D(\Zb_d)$ model, which is a generalized model of surface code, is also investigated.