Memory-enhanced quantum extreme learning machines for characterizing non-Markovian dynamics

Abstract

We use a Quantum Extreme Learning Machine for characterizing and estimating parameters of quantum dynamics generated by a tunable collision model. The input to the learning protocol consists of quantum states produced by successive system environment interactions, while the reservoir is implemented as a disordered many body quantum system evolving under a fixed Hamiltonian. We systematically explore how extending the QELM feature space, through the inclusion of temporal information and additional observables, affects estimation performance. Our results demonstrate that temporal extensions of the feature vector consistently and significantly enhance estimation accuracy relative to the baseline protocol. Notably, incorporating memory from earlier time steps yields the most substantial and robust improvements, whereas extensions based solely on additional observables offer only marginal gains. Crucially, the advantage conferred by temporal memory becomes increasingly pronounced as the dynamics become more strongly non Markovian, indicating that environmental memory effects serve as a constructive resource for learning.

Figures (5)

• Figure 1: Schematic of the collision model with three system qubits $S_i$ (red) and three reservoir qubits $Q_i$ (blue). The protocol evolves in successive stages: each system qubit first exchanges information with its corresponding reservoir qubit via a PSWAP operation of strength $\chi$. The reservoir qubits are then subject to a depolarizing channel before undergoing unitary evolution governed by $H_{\text{bath}}$. Finally, the system qubits evolve under their internal interactions described by $H_{\text{chain}}$.
• Figure 2: Schematic representation of the protocol. The input states $\rho_s(t)$ are processed by a quantum reservoir, and the resulting features are evaluated by a linear regression readout. The output performance is quantified by the NMSE.
• Figure 3: NMSE as a function of the collision step for different QELM feature extensions and coupling strengths $\lambda = 0.10,\; 0.50,\; 1.00$. Solid lines correspond to the baseline QELM without extensions. Dotted lines show the inclusion of the immediate past reservoir state, dashed lines represent the use of a fixed earlier memory step $k_1$, and dash-dotted lines indicate the extension based on the additional observable $\langle\sigma_x\rangle$ (Average of 2000 realizations).
• Figure 4: Comparison of the NMSE at time step $k=4$ for different depolarizing parameter $\lambda$, using the standard QELM (blue) and three extensions: immediate past $k^{'} = k -1$ (orange), distant step $k^{'} = k_1$ (green), and additional observable $\langle\sigma_x\rangle$ (red)(Average of 2000 realizations).
• Figure 5: NMSE for the estimation of $\lambda$ with extended feature vectors. Solid lines correspond to the baseline case, dotted and dashed lines represent memory extensions, and dash-dotted lines correspond to the use of $\langle\sigma_x\rangle$.