Obtaining the low-energy physics of frustrated magnets is a challenging task for all computational methods. Quantum Monte Carlo, in particular, suffers from the Monte Carlo sign problem: Roughly speaking, the signs (or complex phases) that capture interference effects in quantum wave functions cannot be captured as a probability distribution, so one cannot efficiently sample from them. A priori, neural quantum states do not have this issue; however, many NQS ansätze split the wav function into its phase and magnitude, which makes it difficult to learn the former in a training procedure with destructive interference.

Together with Chris Roth of UT Austin, I have solved this problem for the $ J_1 - J_2 $ Heisenberg model on the square lattice. We used the group-convolutional neural network (GCNN) ansatz first proposed by Chris: in these, the wave function is written as the sum of a number of terms, which makes tracking destructive interferences much easier.

Ansätze with separate wave function amplitudes and phases (left) cannot track destructive interferences. Writing the wave function as a sum of terms (right) improves the issue.

In addition, GCNNs are designed to impose the spatial symmetries of the problem on the wave function, similar to how convolutional neural networks impose translation symmetry. In these, translating entries of the input causes the entries of all subsequent layers to be translated the same way (i.e., the mapping between layers is translation equivariant). In the final layer, summing over all entries then yields an invariant wave function, which does not change upon translating the input. Furthermore, by applying the phase factors $ e^{ik \cdot r}$, we can construct ansätze with any given wave vector $\vec k$: by training the neural network in each of these wave vector sectors, we can access several low-energy states in addition to the ground state.