Efficient representation of topologically ordered states with restricted Boltzmann machines

Representation by neural networks, in particular by restricted Boltzmann machines (RBMs), has provided a powerful computational tool to solve quantum many-body problems. An important open question is how to characterize which class of quantum states can be efficiently represented with RBMs. Here, we show that RBMs can efficiently represent a wide class of many-body entangled states with rich exotic topological orders. This includes (1) ground states of double semion and twisted quantum double models with intrinsic topological orders, (2) states of the AKLT model and two-dimensional CZX model with symmetry protected topological orders, (3) states of stabilizer Fracton models with fracton topological order, and (4) (generalized) stabilizer states and hypergraph states that are important for quantum information protocols. One twisted quantum double model state considered here harbors non-Abelian anyon excitations. Our result shows that it is possible to study a variety of quantum models with exotic topological orders and rich physics using the RBM computational toolbox.

Figure 1

Different generalizations of graph states. (a) Hypergraph states generalize graph states by introducing three-body correlation factors; (b) stabilizer states generalize graph states by additional parity constraints on qubits; (c) XS-stabilizer states combine the parity constraints from Pauli stabilizers and three-body correlation factors from hypergraph states. Many condensed matter topological models fall into these quantum information formalisms and thus can be represented efficiently by RBM. We will encounter them later. Combining with the unary representation, stabilizer states with added unary constraints can describe the AKLT state. Thus the AKLT state can be efficiently represented by the RBM.

Figure 2

Unary RBM representation for higher spin systems. The three left yellow neurons correspond to the qubit 1 and the three right yellow neurons correspond to the qubit 2. Two blue neurons in the unary hidden layers restrict three visible yellow neurons to be one of 100, 010, and 001 that correspond to $|-1\rangle$, $|0\rangle$, and $|+1\rangle$ in spin 1. The whole graph is still a bipartite graph.

Figure 3

RBM representation of hypergraph state. (a) A hypergraph. There are two three hyperedges that connect $q_1,q_2,q_3$ and $q_3,q_4,q_5$. (b) The encoding circuit of this hypergraph state, where $|+\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$ represents the input state for all the qubits and the connected dots denote the controlled-controlled-$Z$ gate. (c) The corresponding RBM representation of (a). Orange neurons are visible neurons, and both green and yellow vertices are hidden neurons. The two different colors (yellow and green) represent different weight functions.

Figure 4

Closeup of a Clifford circuit. Visible neurons are $v_p,v_q,...$ and hidden neurons are $h_i,h_j,...$. Each $H$ gate corresponds to two neurons with the weight ${(-1)}^{x_i{x}_j}$. Each $S$ gate corresponds to one neuron with the weight $i^{x_j}$; each $CZ$ gate corresponds to two neurons with the weight ${(-1)}^{x_1{x}_2}$. The whole wave function is a DBM which can be eliminated to a RBM.

Figure 5

(a) Ground state of the AKLT model. The spin one on each lattice site can be decomposed into two spin 1/2's which form EPR pairs between nearest neighbor pairs. At the two ends of an open chain, there are two isolated spin 1/2's giving rise to a fourfold ground state degeneracy. Every red line represents a $|00\rangle +|11\rangle$ state. Each shaded circle represents a projection operator $P=|100\rangle (\langle 01|+\langle 10|)+|010\rangle i(\langle 01|-\langle 10\left|\right)+|001\rangle (\langle 00|-\langle 11|)$. (b) The Clifford circuit that realizes the projection $P$ under the unary representation (living in the space spanned by $|100\rangle ,|010\rangle$ and $|001\rangle )$. Here $|+\rangle$ means projecting onto the $|+\rangle$ state.