Abstract
Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin-1/21/2J_1-J_2J1−J2 Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks, one for the amplitude and the other for the phase of the variational wavefunction. By introducing concrete mitigation strategies against inherent numerical instabilities in the stochastic reconfiguration algorithm we obtain a variational energy comparable to that reported recently with neural networks that incorporate knowledge about the physical system. Through a detailed analysis of the individual components of the algorithm, we conclude that the rugged nature of the energy landscape constitutes the major obstacle in finding a satisfactory approximation to the ground state wavefunction, and prevents learning the correct sign structure. In particular, we show that in the present setup the neural network expressivity and Monte Carlo sampling are not primary limiting factors.
Highlights
We evaluate the performance of variational quantum Monte Carlo (VQMC) by monitoring the energy per site and the density of energy variance of |ψθ 〉
We focus on deep neural network (DNN) architectures, built from layers representing affine-linear transformations parametrized by the parameters θ
While they have been demonstrated to work well in a number of problems so far [58,67,72,99], when it comes to learning the ground state of frustrated spin systems, such as the J1−J2 model, holomorphic architectures for neural network quantum states exhibit certain deficiencies: first, the holomorphic constraint correlates the phase and amplitude gradients of the variational wavefunction parameters, which means that an update to the network parameters will cause a change in both the amplitude and the phase of the output
Summary
Understanding the effect of many-body interactions in quantum systems is a long-standing challenge of modern physics: the exponential growth of the Hilbert space with the number of particles makes solving the Schrödinger equation impossible beyond a few dozen degrees of freedom. The non-stoquasticity of the Hamiltonian is alleviated by incorporating the MarshallPeierls rule and the gain in accuracy is attributed to the symmetrization of very large Restricted Boltzmann Machines with quantum number projections While these studies show that neural networks are sufficiently versatile to find a pretty good approximation to the ground state — and excited states [80, 83, 84] — in a large part of parameter space, the region J2/J1 ≈ 0.5 appears to be an intriguing exception, in particular for pure neural network states without extra physics input. These difficulties seem to be at odds with the fact that neural networks are universal function approximators in the limit of sufficiently large network size [85,86,87], and should constitute a suited variational ansatz class that does not require further physical insight
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