Learning hard quantum distributions with variational autoencoders

Studying general quantum many-body systems is one of the major challenges in modern physics because it requires an amount of computational resources that scales exponentially with the size of the system.Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a new representation of states based on variational autoencoders. Variational autoencoders are a type of generative model in the form of a neural network. We probe the power of this representation by encoding probability distributions associated with states from different classes. Our simulations show that deep networks give a better representation for states that are hard to sample from, while providing no benefit for random states. This suggests that the probability distributions associated to hard quantum states might have a compositional structure that can be exploited by layered neural networks. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterising states of the size expected in first generation quantum hardware.Comment: v2: 9 pages, 3 figures, journal version with major edits with respect to v1 (rewriting of section "hard and easy quantum states", extended discussion on comparison with tensor networks

Bayesian Cointegrated Vector Autoregression models incorporating Alpha-stable noise for inter-day price movements via Approximate Bayesian Computation

We consider a statistical model for pairs of traded assets, based on a Cointegrated Vector Auto Regression (CVAR) Model. We extend standard CVAR models to incorporate estimation of model parameters in the presence of price series level shifts which are not accurately modeled in the standard Gaussian error correction model (ECM) framework. This involves developing a novel matrix variate Bayesian CVAR mixture model comprised of Gaussian errors intra-day and Alpha-stable errors inter-day in the ECM framework. To achieve this we derive a novel conjugate posterior model for the Scaled Mixtures of Normals (SMiN CVAR) representation of Alpha-stable inter-day innovations. These results are generalized to asymmetric models for the innovation noise at inter-day boundaries allowing for skewed Alpha-stable models. Our proposed model and sampling methodology is general, incorporating the current literature on Gaussian models as a special subclass and also allowing for price series level shifts either at random estimated time points or known a priori time points. We focus analysis on regularly observed non-Gaussian level shifts that can have significant effect on estimation performance in statistical models failing to account for such level shifts, such as at the close and open of markets. We compare the estimation accuracy of our model and estimation approach to standard frequentist and Bayesian procedures for CVAR models when non-Gaussian price series level shifts are present in the individual series, such as inter-day boundaries. We fit a bi-variate Alpha-stable model to the inter-day jumps and model the effect of such jumps on estimation of matrix-variate CVAR model parameters using the likelihood based Johansen procedure and a Bayesian estimation. We illustrate our model and the corresponding estimation procedures we develop on both synthetic and actual data.Comment: 30 page

An open source MATLAB program for fast numerical Feynman integral calculations for open quantum system dynamics on GPUs

This MATLAB program calculates the dynamics of the reduced density matrix of an open quantum system modeled by the Feynman-Vernon model. The user gives the program a vector describing the coordinate of an open quantum system, a hamiltonian matrix describing its energy, and a spectral distribution function and temperature describing the environment's influence on it, in addition to the open quantum system's intial density matrix and a grid of times. With this, the program returns the reduced density matrix of the open quantum system at all (or some) moments specified by that grid of times. This overall calculation can be divided into two stages: the setup of the Feynman integral, and the actual calculation of the Feynman integral for time-propagation of the density matrix. When this program calculates this propagation on a multi-core CPU, it is this propagation that is usually the rate limiting step of the calculation, but when it is calculated on a GPU, the propagation is calculated so quickly that the setup of the Feynman integal actually becomes the rate limiting step for most cases tested so far. The overhead of transfrring information from the CPU to the GPU and back seems to have negligible effect on the overall runtime of the program. When the required information cannot fit on the GPU, the user can choose to run the entire program on a CPU.Comment: 8 pages, 2 figures, 1 table, 22 reference

Detection of the tulip breaking virus (TBV) in tulips using optical sensors

The tulip breaking virus (TBV) causes severe economic losses for countries that export tulips such as the Netherlands. Infected plants have to be removed from the field as soon as possible. There is an urgent need for a rapid and objective method of screening. In this study, four proximal optical sensing techniques for the detection of TBV in tulip plants were evaluated and compared with a visual assessment by crop experts as well as with an ELISA (enzyme immunoassay) analysis of the same plants. The optical sensor techniques used were an RGB color camera, a spectrophotometer measuring from 350 to 2500 nm, a spectral imaging camera covering a spectral range from 400 to 900 nm and a chlorophyll fluorescence imaging system that measures the photosynthetic activity. Linear discriminant classification was used to compare the results of these optical techniques and the visual assessment with the ELISA score. The spectral imaging system was the best optical technique and its error was only slightly larger than the visual assessment error. The experimental results appear to be promising, and they have led to further research to develop an autonomous robot for the detection and removal of diseased tulip plants in the open field. The application of this robot system will reduce the amount of insecticides and the considerable pressure on labor for selecting diseased plants by the crop expert. © 2010 The Author(s