A quantum computer can provide fast solution for certain computational problems (e.g. factoring and discrete logarithm) which require exponential time on an ordinary computer. Physical realization of a quantum computer is a big challenge for scientists. One important problem is decoherence and systematic errors in unitary transformations which occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to Shor’s discovery of fault-tolerant quantum computation, with subsequent improvements. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision δ. Unfortunately, the threshold value of δ is rather small 1 ; it is very difficult to achieve this precision.

Needless to say, classical computation can be also performed fault-tolerantly. However, it is rarely done in practice because classical gates are reliable enough. Why is it possible? Let us try to understand the easiest thing — why classical information can be stored reliably on a magnetic media. Magnetism arise from spins of individual atoms. Each spin is quite sensitive to thermal fluctuations. But the spins interact with each other and tend to be oriented in the same direction. If some spin flips to the opposite direction, the interaction forces it to flip back to the direction of other spins. This process is quite similar to the standard error correction procedure for the repetition code. We may say that errors are being corrected at the physical level. Can we propose something similar in the quantum case? Yes, but it is not so simple. First of all, we need a quantum code with local stabilizer operators.

A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.

**Lagged correlation-based deep learning for directional trend change prediction in financial time series**

An increased interest in deep-layered machine learning approaches for time series analysis and forecasting resultedin applications in various fields, establishing this area as a challenging topic of interest (Cao and Tay, 2003; Nesreenet al., 2010). When it comes to the effective use of deep neural networks, one of the primary concerns is a sensible approach to feature engineering for useful data representations. This process often depends on domain knowledge about the respective area of application and is, more often than not, a time-consuming part of research (Najafabadiet al., 2015). Some researchers equate applied machine learning, in an attempt to emphasize the relative importance, with the concept of feature engineering itself (Ng,2012). Such representations have to be informationally rich enough to incorporate the looked-for lagged correlations between time series while, at the same time, being constrained to a discrete number per observation and variable for input features in a feed-forward neural network.

Zhang and Qi (2005) find that such models are not able to capture the necessary information when applied to raw data from time series with seasonal and trend patterns, which opens the field for approaches to feature engineering that allow for an effective use of time series data for trend predictions in a variety of application areas.

In this paper, we test the hypothesis that deep feed-forward neural networks, combined with exponential smoothing for the training inputs, are suitable for learning lagged correlations between the step-wise trends of a large number of time series, and that such models can be success-fully applied to current research on real-world forecasting problems. In order to test this approach, we apply the proposed method to gradients computed for five years of historical stock price data of the S&P 500 stocks in one-hour intervals for daily trends, adding the complication of relatively few observations. For a more in-depth overview of soft computing methods in financial market research,interested readers are referred to Cavalcante et al. (2016),with Weng et al. (2018) providing an application of ensemble methods to financial markets using a variety of text-based and index-based features.

Trend change prediction in complex systems with a large number of noisy time series is a problem with many applications for real-world phenomena, with stock markets as a notoriously difficult to predict example of such systems. We approach predictions of directional trend changes via complex lagged correlations between them, excluding any information about the target series from the respective inputs to achieve predictions purely based on such correlations with other series. We propose the use of deep neural networks that employ step-wise linear regressions with exponential smoothing in the preparatory feature engineering for this task, with regression slopes as trend strength indicators for a given time interval. We apply this method to historical stock market data from 2011 to 2016 as a use case example of lagged correlations between large numbers of time series that are heavily influenced by externally arising new information as a random factor. The results demonstrate the viability of the proposed approach, with state-of-the-art accuracies and accounting for the statistical significance of the results for additional validation, as well as important implications for modern financial economics.

**Deep Learning Volatility**

We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models -including the rough volatility family- and a range of derivative contracts. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several numerical pricers and model families (such as rough volatility models) within the scope of applicability in industry practice. The form in which information from available data is extracted and stored influences network performance: This approach is inspired by representing the implied volatility and option prices as a collection of pixels. In a number of applications we demonstrate the prowess of this modelling approach regarding accuracy, speed, robustness and generality and also its potentials towards model recognition.