Machine learning topological states

Artificial neural networks and machine learning have now reached a new era after several decades of improvement where applications are to explode in many fields of science, industry, and technology. Here, we use artificial neural networks to study an intriguing phenomenon in quantum physics—the topological phases of matter. We find that certain topological states, either symmetry-protected or with intrinsic topological order, can be represented with classical artificial neural networks. This is demonstrated by using three concrete spin systems, the one-dimensional (1D) symmetry-protected topological cluster state and the 2D and 3D toric code states with intrinsic topological orders. For all three cases, we show rigorously that the topological ground states can be represented by short-range neural networks in an exact and efficient fashion—the required number of hidden neurons is as small as the number of physical spins and the number of parameters scales only linearly with the system size. For the 2D toric-code model, we find that the proposed short-range neural networks can describe the excited states with Abelian anyons and their nontrivial mutual statistics as well. In addition, by using reinforcement learning we show that neural networks are capable of finding the topological ground states of nonintegrable Hamiltonians with strong interactions and studying their topological phase transitions. Our results demonstrate explicitly the exceptional power of neural networks in describing topological quantum states, and at the same time provide valuable guidance to machine learning of topological phases in generic lattice models.

Figure 1

The 1D symmetry-protected topological cluster state. (a) A pictorial illustration of the 1D cluster Hamiltonian [see Eq. (1)] with the periodic boundary condition. The shaded region represents a prototypic three-body interaction (a stabilizer) at an arbitrary site $k$. (b) A short-range neural network representation of the 1D topological cluster state. The yellow balls (blue cubes) denote the visible (hidden) artificial neurons.

Figure 2

The 2D toric code Hamiltonian and a neural-network representation. (a) An illustration of $H_{\text{2D}}$ [see Eq. (9)]. The shaded green (red) region depicts a prototypic vertex (face) operator at vertex $\mathcal{V}$ (face $\mathcal{F}$). The Hamiltonian is defined as a summation of these four-body operators with an extra minus sign in front. The four noncontractible loops (labeled by $X_1, X_2, Z_1$, and $Z_2$, respectively) correspond to four nontrivial string operators that define the commutation relations of the two logic qubits living in the topologically protected ground subspace [50, 51]. (b) A neural-network representation of one of the ground state of $H_{\text{2D}}$ with intrinsic topological order. The yellow balls stand for the visible artificial neurons corresponding to the physical spins. The green (red) cubes denotes the hidden neurons corresponding to the vertices (faces).

Figure 3

String operators and nontrivial mutual statistics. A pair of x-type quasiparticles (also called “magnetic vortices”) living on the faces can be created by acting the string operator $S_{{\text{P}}_1^{\text{x}}}^x={\prod }_{j\in {\text{P}}_1^{\text{x}}}{\widehat{\sigma }}_j^x$ on the ground state $|G_{\mathrm{toric}}^{\left(\text{2D}\right)}\rangle$. Similarly, we can also create a pair of z-type quasiparticles (“electric charges”) that live on the vertices by the string operator $S_{{\text{P}}_1^{\text{z}}}^z={\prod }_{j\in {\text{P}}_1^{\text{z}}}{\widehat{\sigma }}_j^z$. For a closed contractable path (such as ${\text{P}}_3^x$), two quasiparticles of the same type meet with each other and will fuse to the vacuum. If two closed loops of different types form a nontrivial link (such as the Hopf link formed by ${\text{P}}_2^{\text{x}}$ and ${\text{P}}_2^{\text{z}}$), the many-body states will acquire a global phase factor $-1$ due to the nontrivial mutual statistics between the x-type and z-type quasiparticles. As shown in the main text and Appendix pp2, we find that all the quantum states evolved in these process can be described precisely and efficiently by further-restricted restricted Boltzmann machines.

Figure 4

A neural-network representation for the 3D toric code states. Similar to the 2D model, the 3D Hamiltonian is also defined as a summation of vertex and face operators (depicted by the shaded green and red regions, respectively) with an extra minus sign in front. However, for the 3D case, each vertex operator will be a product of six, rather than four, ${\widehat{\sigma }}^x$ operators. The ground states of $H_{\text{3D}}$ can also be represented by restricted Boltzmann machines, where the hidden neurons (denoted by green or red cubes) only connect to their nearest visible neurons. The weight parameters $W_{\mathcal{F},i}$ and $W_{\mathcal{V},j}$ can be chosen the same as in the 2D case.

Figure 5

Reinforcement learning of SPT states and topological phase transitions. (a) The variational ground-state energy density as a function of the iteration number of the learning process. As the iteration number increases, the energy converges smoothly to the exact values (denoted by the dashed red lines). (b) The convergence of the string-order parameter as the increasing of the iteration number for $V=0.1$. the red dashed line denotes the exact value. (c) The real and imaginary parts of the learned feature maps for representing the ground state of $H_{\text{1D}}$ with a restricted Boltzmann machine for $V=0.1$. In (a)–(c), the lattice size is $N=20, h_x=0.02$, and the hidden-unit density $\gamma =4$. (d) Reinforceent learning of a topological phase transition. As we increasing $V$, the system will go through a phase transition (at $V\sim 0.9$) from a symmetry-protected topological phase to a ferromagnetic phase. Here the lattice size is $N=50$ and the other omitted parameters are chosen the same as in (b).

Figure 6

Affected region for acting the vertex operator $A_{\mathcal{V}}$. $A_{\mathcal{V}}$ flips four spins belonging to $\mathcal{V}$ (denoted by the red balls). The shaded region stands for the region that are affected by the spin flip.

Figure 7

Anyons created by string operators and their mutual statistics. (a) Creating a pair of x-type quasiparticles living at faces ${\mathcal{F}}_1$ and ${\mathcal{F}}_7$ by apply string operator $S_{{\text{P}}_1^{\text{x}}}^x$ on the ground state $|G_{\text{toric}}^{\text{2D}}\rangle$. (b) Creating a pair of z-type quasiparticles by $S_{{\text{P}}_1^{\text{z}}}^z$. (c) A trivial contractible loop that leave the ground state unchanged. (d) A Hopf link formed by two loops ${\text{P}}_2^{\text{x}}$ and ${\text{P}}_2^{\text{z}}$. Acting the corresponding string operators on the ground state will give rise to an overall phase $-1$, which manifests the nontrivial mutual statistics between the x- and z-type quasiparticles.