Solving the ground state of quantum many-body Hamiltonians is crucial for understanding the properties of quantum matters. However, the time complexity of numerical methods for the ground state problem often scales exponentially with the system size, limiting the class of classically solvable Hamiltonians.

After discussing the complexity of quantum Hamiltonians, in this talk, we investigate what limits the ground state simulation in the variational Monte Carlo method. Even though this method has a long history, recent advances by applying machine learning (ML) idea made it promising for attacking longstanding Hamiltonian problems. To be specific, neural network-based Ansatz such as restricted Boltzmann machine and convolutional neural networks achieved state-of-the-art accuracy in solving one and two-dimensional transverse Ising and Heisenberg models.

However, more recent works reported that this method fails to solve more complicated Hamiltonians, such as two-dimensional frustrated spin systems. We systematically study what property of the Hamiltonian leads such a failure and show that non-stoquasticity of the Hamiltonian plays an essential role in the simulation. To be specific, we find that the sampling problem appears for simple non-stoquastic Hamiltonians, but phases deep in non-stoquastic parameter regions cause expressivity problems.