Abstract In the canonical approach of loop quantum gravity, arguably the most important outstanding problem is finding and interpreting solutions to the Hamiltonian constraint. In this work, we demonstrate that methods of machine learning are in principle applicable to this problem. We consider U(1) BF theory in three dimensions, quantised with loop quantum gravity methods. In particular, we formulate a master constraint corresponding to Hamilton and Gauß constraints using loop quantum gravity methods. To make the problem amenable for numerical simulation we fix a graph and introduce a cutoff on the kinematical degrees of freedom, effectively considering U q ( 1 ) BF theory at a root of unity. We show that the neural network quantum state ansatz can be used to numerically solve the constraints efficiently and accurately. We compute expectation values and fluctuations of certain observables and compare them with exact results or exact numerical methods where possible. We also study the dependence on the cutoff.
Abstract In the canonical approach of loop quantum gravity, arguably the most important outstanding problem is finding and interpreting solutions to the Hamiltonian constraint. In this work, we demonstrate that methods of machine learning are in principle applicable to this problem. We consider U(1) BF theory in three dimensions, quantised with loop quantum gravity methods. In particular, we formulate a master constraint corresponding to Hamilton and Gauß constraints using loop quantum gravity methods. To make the problem amenable for numerical simulation we fix a graph and introduce a cutoff on the kinematical degrees of freedom, effectively considering U q ( 1 ) BF theory at a root of unity. We show that the neural network quantum state ansatz can be used to numerically solve the constraints efficiently and accurately. We compute expectation values and fluctuations of certain observables and compare them with exact results or exact numerical methods where possible. We also study the dependence on the cutoff.
Abstract The aim of the present work is to demonstrate the applicability of machine learning techniques and in particular neural network quantum states in the context of loop quantum gravity, albeit for a simplified theory. We consider 3-dimensional Euclidean gravity in a gauge theoretical formulation in a certain weak coupling limit due to Smolin, and show that the gauge group becomes Abelian, resulting in U(1)3 BF-theory. The theory is quantised using loop quantum gravity methods. The kinematical degrees of freedom are truncated, on account of computational feasibility, by fixing a graph and deforming the algebra of the holonomies to impose a cutoff on the charge vectors. This leads to a quantum theory related to Uq(1)3 BF-theory. The effect of imposing the cutoff on the charges is examined. We also implement the quantum volume operator of 3d loop quantum gravity. Most importantly we compare two constraints for the quantum model obtained: a master constraint enforcing curvature and Gauß constraint, as well as a combination of a quantum Hamilton constraint constructed using Thiemann’s strategy and the Gauß master constraint. The two constraints are solved using the neural network quantum state ansatz, demonstrating its ability to explore models which are out of reach for exact numerical methods. The solutions spaces are quantitatively compared and although the forms of the constraints are radically different, the solutions turn out to have a surprisingly large overlap. We also investigate the behavior of the quantum volume in solutions to the constraints.
The classical simulation of quantum systems typically requires exponential resources. Recently, the introduction of a machine learning-based wavefunction ansatz has led to the ability to solve the quantum many-body problem in regimes that had previously been intractable for existing exact numerical methods. Here, we demonstrate the utility of the variational representation of quantum states based on artificial neural networks for performing quantum optimization. We show empirically that this methodology achieves high approximation ratio solutions with polynomial classical computing resources for a range of instances of the Maximum Cut (MaxCut) problem whose solutions have been encoded into the ground state of quantum many-body systems up to and including 256 qubits.
One of the main challenges of quantum many-body physics is that the dimensionality of the Hilbert space grows exponentially with the system size, which makes it extremely difficult to solve the Schrodinger equations of the system. But typically, many physical systems have a simplified internal structure which makes the parameters needed to characterize their ground states exponentially smaller. This makes many numerical methods possible in capture the physics of the system. Among these modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of the big data, are now attracting many interests. Neural network representation of quantum many-body states shows great potential in solving some traditionally difficult quantum problems involving large number of freedoms. In this work, we briefly review the progress of using the artificial neural network to build quantum many-body ansatz states. We take Boltzmann machine representation as a prototypical example to illustrate various aspects of neural network representation of quantum many-body states. We first briefly review the classical neural networks, then we illustrate how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states, including entanglement features, representational power, and the relation with tensor network states, are discussed. For applications, we briefly review the progress of many-body calculating based on neural network states, neural network state approach to tomography, and also the classical simulation of quantum computing based on Boltzmann machine states. At the end of the work, some outlooks and open problems are given.
Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with Z_{d} gauge group and non-Abelian Kitaev D(G) models on different geometries. Focusing on the special case of Z_{2} gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wave function for the Z_{2} theory away from the exactly solvable limit, and to demonstrate the confining or deconfining phase transition of the Wilson loop order parameter.
Neural networks have been proposed as efficient numerical wave function Ans\"atze, which can be used to variationally search a wide range of functional forms for ground-state solutions. These neural network methods are also advantageous in that more variational parameters and system degrees of freedom can be easily added. We benchmark the methodology by using neural networks to study several different integrable bosonic quantum systems in one dimension and compare our results to the exact solutions. While testing the scalability of the procedure to systems with many particles, we also introduce using symmetric function inputs to the neural network to enforce exchange symmetries of indistinguishable particles.
Quantum neural networks (QNNs) are a revolutionary paradigm placed at the frontier of machine learning and quantum computing. By utilizing quantum mechanical principles like superposition, entanglement, and quantum parallelism, QNNs facilitate the solution of complex machine learning problems that were thought to be impossible according to conventional models. This chapter provides a detailed study of QNNs, from elementary principles to practical applications. It begins with a rundown of the most significant concepts in machine learning and quantum computing, followed by QNN architecture, with important points including quantum gates and parameterized quantum circuits. The chapter addresses challenges in training QNNs, with special attention given to hybrid quantum-classical optimization algorithms and their utility in successful model training. It also examines multiple applications of QNNs, including simulation in quantum chemistry, optimization problems, and real-world machine learning applications, to demonstrate how they have the potential to outperform conventional algorithms in high dimensional or highly entangled data environments. This chapter provides a comparative assessment of quantum machine learning frameworks, quantum hardware platforms, and hybrid optimization methods and aids in determining the most suitable tools to design QNNs. It finally discusses possible areas of future research and research issues in the field, emphasizing the urgent need to improve quantum hardware, increase noise tolerance, and scale up. As quantum computing technology evolves, QNNs are expected to be at the center of solving some of the most complex and computationally intensive problems encountered in various industries.