Transfer Learning for Neural-Network Quantum States
Abstract:
A quantum many-body system is a system of many interacting particles at the scale of atoms. There are different quantum many-body models describing such systems. Given a quantum many-body system in a given model, described by its Hamiltonian matrix, a square matrix of order exponential in the size of the system, each possible state of the system is described by the solution of the Schrodinger equation: the wave function and the corresponding energy. The ground state is the state for which the energy is minimum. Successive excited states have respectively higher energy.
Neural-network quantum state, proposed by Giuseppe Carleo and Matthias Troyer, is a Monte Carlo method that trains a neural network to be a surrogate of the wave function. The first contribution of this thesis is an open-source neural-network quantum states library for general-purpose graphics processing units using the TensorFlow library. An important scientific question in quantum many-body physics is an identification, at the limit at infinite size or for very large systems, of the phases in which a system finds itself and the phase transitions that a system crosses as the parameters defining interactions in the model vary. Phase transitions occur at quantum critical points. Empirically, a quantum critical point can be identified by finding the inflection points of a physical quantity in the ground state.
Finding quantum critical points requires increasing the size of the systems and sweeping the space of parameters. The second and third contributions consist of several physics-informed transfer learning protocols to scale neural-network quantum states to larger sizes and to explore the space of parameters for the effective and efficient identification of quantum critical points. From the point-of-view of the architecture of the neural-network quantum states, we also devise transfer learning protocols to larger, deeper, and broader networks as the fourth contribution. In the fifth contribution, we consider excited states. Quantum critical points formally corresponding to closing energy gaps between the states, could be relevant to approximate the wave function and calculate the energy of excited states. We devise and adopt four transfer learning protocols to achieve this purpose. The four transfer learning protocols, in different ways, transfer the learning from one state to another.
These transfer learning protocols are demonstrated and evaluated for the computation of wave function, of observables, and of energy, as well as for the search of quantum critical points for systems in the quantum Ising, Ising J_1-J_2, and Heisenberg models. A comparative empirical evaluation shows that the proposed transfer learning protocols for the new neural network quantum states library are more scalable, effective, and efficient than state-of-the-art methods including the baseline neural-network quantum states original implementation or its general-purpose graphic processing unit counterpart without transfer learning