Reservoir computing in quantum systems

Reservoir computing (RC) is a machine learning paradigm that exploits dynamical systems to solve temporal tasks. This technique finds applications in very diverse fields such as weather forecasting, stock market predictions, and communications. Similar to other unconventional computing paradigms inspired by the capabilities of the human brain, RC deals with hardware implementations that aim at overcoming the challenges confronted by digital computation. These challenges include the reduction of the energy budget of digital computation and the speedup of machine learning algorithms. This thesis explores the emerging field of quantum RC. We studied, either by analytical or numerical methods, which are the requirements of complex quantum systems to perform as useful reservoirs, with special attention to quantum spin models. Useful reservoir systems will be defined as those that meet the fundamental requirements that ensure a minimum level of performance from the reservoir. Our main findings are the identification of the dynamical features and input injection favoring quantum reservoirs, and the theoretical conditions for useful reservoirs. In particular, computational capabilities and input response of reservoirs displaying many-body localization are totally degraded by the presence of local integrals of motion while in thermal phases (or at the edge of transition) the operation is optimal. These computational capabilities are characterized by means of the information processing capacity, obtained for the first time for quantum reservoirs, and other benchmark tools such as the short-term memory and nonlinear autoregressive moving average tasks. Characterization of the performance through all these tools allows one to assess the linear and nonlinear contributions of a specific reservoir. Moreover, the input codification mechanism determines the nonlinear response of the reservoir together with the dynamical regime and the election of observable. We demonstrate this relation by showing explicit analytical formulas of the input-output map of the studied reservoir models. Beyond ideal conditions, we explore how all these factors can be affected by the implementation of a quantum reservoir computing experiment with an online protocol, where the extraction of information through measurements is accounted for. Weak measurements are introduced as a possible route to achieve a competitive performance for online temporal processing while keeping a high control over the required experimental resources. Finally, on a theoretical general side, all finite-dimensional quantum reservoir computing models with classical inputs must fulfill the following condition to be, at least, operational: convergent dynamics towards input-dependent fixed points.