Representing arbitrary ground states of toric code by a restricted Boltzmann machine

TL;DR

This work analyzes how toric code ground states can be represented by Restricted Boltzmann Machines with local connections and proves the FRRBM's ability to capture the full ground-state manifold through nonlocal hidden units. It provides analytic solutions for face and vertex stabilizers under local FRRBM, then shows how three nonlocal hidden neurons can realize arbitrary ground-state amplitudes and degeneracy sectors, with efficient learning verified on small lattices. The authors generalize the approach from Z2 to Zn, implement flux constraints via an N-dimensional invisible qudit, and discuss extensions to Abelian and some CSS codes, outlining potential non-Abelian generalizations. The results offer a tractable, analytically solvable framework for encoding topological order in neural-network states and suggest practical ML-based strategies for learning and exploring topological phases. Overall, the paper advances a rigorous, scalable route to representing and manipulating toric code and related topological states with neural-network ansatzs.

Abstract

We systematically analyze the representability of toric code ground states by Restricted Boltzmann Machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the Restricted Boltzmann Machine to accommodate arbitrary ground states by introducing essential non-local connections efficiently. The new model is not only analytically solvable but also demonstrates efficient and accurate performance when solved using machine learning techniques. Then we generalize our the model from $Z_2$ to $Z_n$ toric code and discuss future directions.

Figures (23)

• Figure 1: The torus on the left is cut along the edges $E_{v}$ and $E_{h}$ to get the square lattice shown on the right, with opposite edges identified. The $3 \times 3$ lattice shows stabilizer operators $A_{v}$ within the blue range and $B_{f}$ within the red range, logical operators $X_{v}$ and $X_{h}$ along the vertical and horizontal dashed loops, respectively, and logical operators $Z_{v}$ and $Z_{h}$ along the edges $E_{v}$ and $E_{h}$.
• Figure 2: This diagram illustrates a RBM with visible neurons colored gray and hidden neurons colored white. The architecture ensures there are no intra-layer connections; instead, each hidden neuron is connected to all visible neurons. Each neuron and each connection is assigned a weight.
• Figure 3: The right diagram results from collapsing the two layers shown in the left diagram. It illustrates a translation-invariant FRRBM with visible neurons colored gray and hidden neurons colored red and blue, corresponding to faces and vertices, respectively. The architecture ensures that each hidden neuron is connected only to the nearest visible neurons. Each neuron and each connection is assigned a weight.
• Figure 4: The left diagram presents a partial view of a configuration on a larger square lattice. The right diagram is obtained by applying a vertex operator to the vertex $v_{0}$. Green nodes indicate qubits that have flipped states, and the red dashed lines encircle nodes considered in the subsequent calculation.
• Figure 5: Three hidden neurons ($h_x$, $h_y$, $h_z$) are introduced into the FRRBM to simulate an arbitrary ground state. $h_x$ connects to visible neurons along a horizontal loop, $h_y$ connects along a vertical loop, and $h_z$ connects to all neurons connected by $h_x$ and $h_y$. Each connection type from a specific hidden neuron is uniformly weighted ($w_x$, $w_y$, $w_z$).