Quantum reservoir computing using Jaynes-Cummings model

TL;DR

This work addresses time-series processing with quantum substrates by evaluating JaynesCummings and dispersive JaynesCummings reservoirs. It employs driving-mediated input encoding and reads out higher order bosonic observables, training a fixed reservoir with Ridge regression. The key findings show an unusual dominance of nonlinear memory over linear memory and competitive Mackey-Glass forecasting in both JC and DJC regimes, with performance enhanced by time multiplexing and higher bosonic excitations. The study demonstrates a viable route to tunable, higher capacity quantum machine learning units and paves the way for scalable quantum reservoir architectures in circuit QED and related platforms.

Abstract

We investigate quantum reservoir computing (QRC) using a hybrid qubit-boson system described by the Jaynes-Cummings (JC) Hamiltonian and its dispersive limit (DJC). These models provide high-dimensional Hilbert spaces and intrinsic nonlinear dynamics, making them powerful substrates for temporal information processing. We systematically benchmark both reservoirs through linear and nonlinear memory tasks, demonstrating that they exhibit an unusual superior nonlinear over linear memory capacity. We further test their predictive performance on the Mackey-Glass time series, a widely used benchmark for chaotic dynamics and show comparable forecasting ability. We also investigate how memory and prediction accuracy vary with reservoir parameters, and show the role of higher-order bosonic observables and time multiplexing in enhancing expressivity, even in minimal spin-boson configurations. Our results establish JC- and DJC-based reservoirs as versatile platforms for time-series processing and as elementary units that overcome the setting of equivalent qubit pairs and offer pathways towards tunable, high-performance quantum machine learning architectures.

Figures (13)

• Figure 1: The physical setup for quantum reservoir computing using the JC system. The reservoir is constituted of a qubit (here shown as an atom) interacting with a single bosonic mode inside a cavity. The discrete input time-series $\{\beta_{i}\}$ is encoded in the time-dependent amplitude $\beta$(t) of the cavity driving field. The qubit is driven by a classical field with fixed amplitude $\alpha$. The bosonic mode undergoes photon loss at a rate $\kappa$. The field reflected from the cavity is measured using a detector. The observables, in our case, the higher-order moments of the bosonic operators, are mapped to the target output $y_{i}$ using a linear regression. We show a snapshot of the Wigner distribution of the bosonic state $\rho^{b}$ in the output obtained corresponding Jaynes-Cummings Hamiltonian. The magenta color denotes negative values of the Wigner distribution, which is a signature of nonclassicality.
• Figure 2: Wigner distribution $W(\Hat{X}, \Hat{P})$ of the bosonic reduced density matrix $\rho^{b}$ for (a) JC model and (b) DJC model, in the presence of driving and dissipation. The distribution is obtained after the washout phase for a series of uniformly sampled random inputs. The magenta color denotes negative values of $W(\Hat{X}, \Hat{P})$, which is the signature of nonclassicality of $\rho^{b}$. (c) The nonlinear response of a set of higher-order moments of the bosonic mode to input value $\beta$ for JC and DJC models. The expectation values have been scaled to lie in the range $[-1, 1]$. For all figures, $dt=10$ and $\kappa=0.1$
• Figure 3: The memory capacity $C$ with respect to varying $\tau$ for the STM task using JC Hamiltonian (red line) when $dt=10$ and $\kappa=0.1$. The other parameters are $\Delta_{b}=1$, $\Delta=0$, $\chi=1$, and $\alpha=0$. The grey dashed line shows the capacity when using a two-qubit reservoir with the same interaction as shown in Eq. \ref{['2qubit_eq2']}. In the inset, we show the capacity for $\tau=0, 1, 2$ when increasing the number of virtual nodes $V$ in the readout layer.
• Figure 4: The $C$ vs. $\tau$ curve for the STM task using the DJC model, both with and without the qubit driving field $\alpha$. The other parameters are $dt=10$, $\kappa=0.1$, and $\chi=1$. The grey dashed line shows the capacity when using a two-qubit reservoir with the same interaction as shown in Eq. \ref{['2qubit_eq3']}. In the inset, we show the effect on $C$ due to varying $\alpha$, keeping the other parameters unchanged.
• Figure 5: (a) Capacity $C$ for PC task vs. delay $\tau$ using JC model when $dt=10$, $\kappa=0.1$ (red curve). The other parameters are the same as in Fig. \ref{['stm_C']}. The grey dashed line shows the capacity when using a two-qubit reservoir with similar interaction, as shown in Eq. \ref{['2qubit_eq2']}. In the inset, we show the capacity with respect to increasing the virtual nodes $V$ for different delays $\tau$. (b) The effect of varying the parameters $\Delta_{b}$, $\Delta$, $\chi$ and $\alpha$ on $C$ for $\tau=1, 2$.