The state of an open quantum system is completely determined by its density matrix, which evolves according to a specific equation called the Lindblad master equation. When the system is composed by many interacting particles, the exponential complexity arising from the many-body problem makes it impossible to simulate. We recently developed a novel approach to the problem, combining the representative power of neural networks with approximate optimization schemes based on Markov-chain sampling. Unfortunately, the algorithm that emerged is somewhat different from traditional machine-learning problems, and it has been hard to exploit existing frameworks. During this talk we will detail some key concepts of this novel research direction and how Julia dispatch system empowered us to rapidly iterate on several prototypes during our research.

I will briefly present the theory of optimization algorithms on manifolds, and share my experiences implementing this functionality in Optim.jl.

Computational studies of electronic structures, that is the modelling of electrons in materials and chemical compounds, is now a standard procedure in industry and research and crucial for progress in material science, chemical manufacturing or pharmacy. Recently we have developed the DFTK.jl package, a Julia implementation of plane-wave density-functional theory, one common simulation method in the field. This talk will focus on the rationale for employing Julia for this task and illustrates our design within the Julia ecosystem. It will discuss the suitability of Julia for this interdisciplinary research field and our strategy to integrate with interesting Julia developments such as automatic differentiation, GPU acceleration and deviating numerical precision.

This talk will present how basic algorithmic blocks such as sums or dot products can be implemented in Julia (and have been developed in AccurateArithmetic.jl) in a way that is both accurate and efficient. Besides naive algorithms, compensated algorithms are implemented, which effectively double the working precision, producing much more accurate results while incurring little to no overhead, especially for large input vectors. Although the vectorization of such algorithms is no particularly simple task, Julia makes it relatively easy and straightforward. This talk will illustrate how Julia can help efficiently mix Floating-Point-related concerns with SIMD-related constraints in order to get the performance of double-precision state-of-the-art BLAS libraries and the accuracy of quadruple-precision algorithms.