Universality conditions of unified classical and quantum reservoir computing

Reservoir computing is a versatile paradigm in computational neuroscience and machine learning, that exploits a recurrent neural network to efficiently process time-dependent information. The power of many neural network architectures resides in their universality approximation property. 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, based on the minimal sufficient conditions for a class of reservoir computers to be universal, namely the fading memory and the polynomial algebra structure of the set of their associated functionals. We test the result in the arising context of quantum reservoir computing. Guided by such a unified theorem we suggest why spatial multiplexing serves 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.