Session E21: Machine Learning for Quantum Matter IV

Live

Precise verification and parameter estimation of quantum devices are critical for further technological progress in simulation and understanding of quantum matter. In this presentation, I will discuss how learning algorithms can be used efficiently for this task. We show how to use Bayesian learning and neural networks to reconstruct the physics of large scale out-of-equilibrium quantum systems in the context of quantum simulation with ultracold atoms. We investigate the scalability of this class of methods for efficient estimation of all local parameters with previously inaccessible levels of precision. Moreover, we show how reinforcement learning can be used to design an optimal workflow for parameter estimation protocols. Finally, we discuss how this class of learning methods can be generalised for other quantum device calibration purposes, such as the characterization of energy levels of quantum dots in bilayer graphene.   

Live

Neural-network-based machine-learning techniques are becoming ubiquitous in quantum information and computing. Some problems faced during characterization of quantum systems can be translated to machine-learning tasks where deep neural networks have proven successful in many domains. Many of these tasks are data-driven, e.g., identifying interesting properties of quantum systems, state discrimination, and tomography. We discuss the problem of quantum state characterization in the context of discriminative modelling. We show that deep-neural-network-based techniques can be adopted for quantum state classification and reconstruction under different types of noise, requiring fewer data points and converging faster than standard methods by using optical quantum states as examples. We demonstrate how convolutional neural networks can distinguish several classes of optical quantum states, including bosonic codes. We further present one possibility for adaptive data collection during tomography by analysing which data points a trained neural network considers important for state classification.   

Live

Quantum state tomography (QST) is a data-intensive task which can be connected to generative modeling problems in machine learning. Generative models based on deep neural networks attempt to learn an underlying distribution for observed data. We connect this task to learning the density matrix of a quantum state from measurement statistics. We go beyond a restricted-Boltzmann-machine approach for QST by combining variational autoencoders (VAEs) and conditional generative adversarial networks (CGANs) into a QST-CGAN architecture. Our method uses standard neural-network architectures and training to learn a quantum state description from measurement data. We compare the QST-CGAN's performance against a standard iterative-maximum-likelihood (iMLE) method for reconstructing optical quantum states. The QST-CGAN method converges faster (almost two orders of magnitude) than iMLE and works well both for pure and mixed quantum states of low rank. We also demonstrate that our QST-CGAN method can be adapted easily to deal with noise and requires much less data (up to two orders of magnitude) than iMLE to reach the same reconstruction fidelity.   

Live

Calculating the spectral function of two dimensional systems is arguably one of the most pressingchallenges in modern computational condensed matter physics. While efficient techniques are avail-able in lower dimensions, two dimensional systems present insurmountable hurdles, ranging fromthe sign problem in quantum Monte Carlo, to the entanglement area law in tensor network basedmethods. We hereby present a variational approach based on a Chebyshev expansion of the spectralfunction and a neural network representation for the wave functions. The Chebyshev moments areobtained by recursively applying the Hamiltonian and projecting on the space of variational statesusing a modified natural gradient descent method. We compare this approach with a modified oneexpanding on the Kyrlov subspace using Chebyshev polynomials. We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice, both with and withoutfrustration, and compare our results with those obtained by other methods in the literature.   

Live

Machine learning models are a powerful theoretical tool for analyzing data from quantum simulators, in which results of experiments are sets of snapshots of many-body states. Thus far, the complexity of these models has inhibited new physical insights from this approach. Here, using a novel set of nonlinearities we develop a network architecture that discovers features in the data which are directly interpretable in terms of physical observables. We demonstrate this architecture on sets of simulated snapshots produced by two candidate theories approximating the doped Fermi-Hubbard model. From the trained networks, we uncover that the key distinguishing features are fourth-order spin-charge correlators, providing a means to compare experimental data to theoretical predictions. Our approach is applicable to arbitrary lattice data, paving the way for new physical insights from machine learning studies of experimental and numerical data.   

Live

The variational method has been a cornerstone approach to tackling the quantum many-body problem since the beginnings of quantum mechanics. Throughout this history, wave-functions have grown in number of parameters and generality. The eventual conclusion to this arc is to consider the variational space of all computer programs. Using tools and inspiration from AI, we have developed an approach to represent this class (computational graph states); a novel way to optimize tens of thousands of parameters within this space (supervised wave-function optimization); and multiple novel variational ansatz (neural net backflow, etc). We will describe these advancements and our effort to push forward, in the age of AI, the variational approach to the quantum many body problem.   

Live

Approximate solutions to the Schrödinger equation typically impose a fixed functional form on the wavefunction. Neural networks have shown impressive power as accurate practical function approximators[1] and have been recently used in bosonic[2] and lattice systems[3]. We show that deep neural networks can learn the ground state wavefunction of chemical systems given only the positions and charges of the nuclei using variational Monte Carlo[4]. We find it crucial to use the second-order optimization algorithm KFAC[5], which uses curvature information from the wavefunction distribution. The neural network Ansatz, FermiNet, is compact yet flexible and gives more accurate energies than conventional Ansätze. We obtain ground state energies, ionisation potentials and electron affinities to within chemical accuracy on a variety of atoms, small molecules, and model reactions.

JS and DP contributed equally to this work.

[1] A. Krizhevsky, I. Sutskever, G.E. Hinton, NIPS’12 1097-1105 (2012).
[2] G. Carleo, M. Troyer, Science 356, 602-606 (2017).
[3] D. Luo, B.K. Clark Phys. Rev. Lett. 122, 226401 (2019).
[4] D. Pfau, J. S. Spencer, A. G. de G. Matthews, W. M. C. Foulkes, Phys. Rev. Research 2, 033429 (2020).
[5] J. Martens, R. Grosse, ICML 2408–2417 (2015).   

Live

Variational simulation with Neural-Network quantum states (NQS) is a successful approach to solve challenging quantum spin and fermionic Hamiltonians. In fermionic systems NQS were used in the second quantized formalism, where the the fermionic Hamiltonian is mapped to a nonlocally interacting spin model.

In this talk I will describe first-quantized deep Neural-Network techniques for analyzing strongly coupled fermionic systems on the lattice. The advantage of this approach is that it preserves the locality of the physical interactions. Using a Slater-Jastrow inspired ansatz, which exploits deep residual networks with convolutional residual blocks, we approximate the ground state of spinless fermions on a square lattice with nearest-neighbor interactions and study its phase diagram. In large systems, we obtain accurate estimates of the boundaries between metallic and charge ordered phases as a function of the interaction strength and the particle density.   

Live

The electronic Schrödinger equation describes fundamental properties of molecules and materials, but cannot be solved exactly for larger systems than a hydrogen atom. Recently, deep variational quantum Monte Carlo has been established as a viable path towards highly accurate solutions with favorable scaling of the computational cost with system size [1,2]. Here, we present PauliNet, a deep-neural-network architecture that includes the Hartree–Fock solution and exact cusp conditions as a baseline, and uses the Jastrow factor and backflow transformation as entry points for a graph neural network which ensures permutational antisymmetry. PauliNet outperforms comparable state-of-the-art trial wave functions on atoms, small molecules, and a strongly correlated model system, and standard quantum chemistry methods on a difficult multireferential molecule. Convergence of the solution to a given fixed-node limit is analyzed with respect to the basis-set size and the Jastrow network size [3]. Numerically exact fixed-node solutions are obtained for LiH and H₄.

[1] J. Hermann, Z. Schätzle & F. Noé, Nat. Chem. 12, 891–897 (2020)
[2] D. Pfau, J. S. Spencer, A. G. D. G. Matthews & W. M. C. Foulkes, Phys. Rev. Research 2, 033429 (2020)
[3] Z. Schätzle, J. Hermann & F. Noé, arXiv:2010.05316   

Live

The unitary dynamics of a quantum system implicitly defines its characterizing Hamiltonian. However, quantum process tomography methods are resource intensive in general. In this work, we investigate an approach based on the Takens and Ruelle time-delay embedding to learn the Hamiltonian from quantum measurements. By minimizing the Kullback-Leibler divergence between the experimental probabilities and the output of a model ansatz, we achieve convergence to the true Hamiltonian for toy model problems. Furthermore, a model parametrization inspired by the topology of the target system could help avoid local minima during learning and reduce the amount of required samples.