Universality conditions of unified classical and quantum reservoir computing

Reservoir computing is a versatile paradigm in computational neuroscience and machine learning, that exploits the non-linear dynamics of a dynamical system - the reservoir - to efficiently process time-dependent information. Since its introduction, it has exhibited remarkable capabilities in various applications. As widely known, classes of reservoir computers serve as universal approximators of functionals with fading memory. The construction of such universal classes often appears context-specific, but in fact, they follow the same principles. Here we present a unified theoretical framework and we propose a ready-made setting to secure universality. We test the result in the arising context of quantum reservoir computing. Guided by such a unified theorem we suggest why spatial multiplexing may serve as a computational resource when dealing with quantum registers, as empirically observed in specific implementations on quantum hardware. The analysis sheds light on a unified view of classical and quantum reservoir computing.