Quantum State Tomography via Nonconvex Riemannian Gradient Descent.

The recovery of an unknown density matrix of large size requires huge computational resources. State-of-the-art performance has recently been achieved with the factored gradient descent (FGD) algorithm and its variants since they are able to mitigate the dimensionality barrier by utilizing some of the underlying structures of the density matrix. Despite the theoretical guarantee of a linear convergence rate, convergence in practical scenarios is still slow because the contracting factor of the FGD algorithms depends on the condition number κ of the ground truth state. Consequently, the total number of iterations needed to achieve the estimation error ϵ can be as large as O(sqrt[κ]ln(1/ϵ)). In this Letter, we derive a quantum state tomography scheme that improves the dependence on κ to the logarithmic scale. Thus, our algorithm can achieve the approximation error ϵ in O(ln(1/κϵ)) steps. The improvement comes from the application of nonconvex Riemannian gradient descent (RGD). The contracting factor in our approach is thus a universal constant that is independent of the given state. Our theoretical results of extremely fast convergence and nearly optimal error bounds are corroborated by the numerical results.

Path sampling of recurrent neural networks by incorporating known physics.

Recurrent neural networks have seen widespread use in modeling dynamical systems in varied domains such as weather prediction, text prediction and several others. Often one wishes to supplement the experimentally observed dynamics with prior knowledge or intuition about the system. While the recurrent nature of these networks allows them to model arbitrarily long memories in the time series used in training, it makes it harder to impose prior knowledge or intuition through generic constraints. In this work, we present a path sampling approach based on principle of Maximum Caliber that allows us to include generic thermodynamic or kinetic constraints into recurrent neural networks. We show the method here for a widely used type of recurrent neural network known as long short-term memory network in the context of supplementing time series collected from different application domains. These include classical Molecular Dynamics of a protein and Monte Carlo simulations of an open quantum system continuously losing photons to the environment and displaying Rabi oscillations. Our method can be easily generalized to other generative artificial intelligence models and to generic time series in different areas of physical and social sciences, where one wishes to supplement limited data with intuition or theory based corrections.

Learning molecular dynamics with simple language model built upon long short-term memory neural network.

Recurrent neural networks have led to breakthroughs in natural language processing and speech recognition. Here we show that recurrent networks, specifically long short-term memory networks can also capture the temporal evolution of chemical/biophysical trajectories. Our character-level language model learns a probabilistic model of 1-dimensional stochastic trajectories generated from higher-dimensional dynamics. The model captures Boltzmann statistics and also reproduces kinetics across a spectrum of timescales. We demonstrate how training the long short-term memory network is equivalent to learning a path entropy, and that its embedding layer, instead of representing contextual meaning of characters, here exhibits a nontrivial connectivity between different metastable states in the underlying physical system. We demonstrate our model's reliability through different benchmark systems and a force spectroscopy trajectory for multi-state riboswitch. We anticipate that our work represents a stepping stone in the understanding and use of recurrent neural networks for understanding the dynamics of complex stochastic molecular systems.