Fig. 1: Reservoir computing scheme.

a The overall input-output map. The input sequence s is mapped to a sequence of ancillary single-mode Gaussian states. These states are injected one by one into a suitable fixed quantum harmonic oscillator network by sequentially resetting the state of the oscillator chosen as the ancilla, x^A. The rest of the network—taken to be the reservoir—has operators x^R. Network dynamics maps the ancillary states into reservoir states, which are mapped to elements of the output sequence o by a trained function h of reservoir observables. Only the readout is trained whereas the interactions between the network oscillators remains fixed, which is indicated by dashed and solid lines, respectively. b The corresponding circuit. The reservoir interacts with each ancillary state through a symplectic matrix S(Δt) induced by the network Hamiltonian H during constant interaction time Δt. Output (o_k) at timestep k is extracted before each new input. \({{\bf{x}}}_{k}^{A}\) are the ancillary operators conditioned on input s_k and \({{\bf{x}}}_{k}^{R}\) are the reservoir operators after processing this input. c Wigner quasiprobability distribution of ancilla encoding states in phase space of ancilla position and momentum operators q and p. Here, the contours of the distribution are indicated by dark yellow lines. Input may be encoded in coherent states using amplitude ∣α∣ and phase \(\arg (\alpha )\) of displacement \(\alpha \in {\mathbb{C}}\), or in squeezed states using squeezing parameter r and phase of squeezing φ (where a and b are the length and height of an arbitrary contour), or in thermal states using thermal excitations n_th.