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arXiv:2409.12693

Coherence influx is indispensable for quantum reservoir computing

Shumpei Kobayashi, Quoc Hoan Tran, Kohei Nakajima

Published Sep 19, 2024• 0 citations
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Abstract

Echo state property (ESP) is a fundamental property that allows an input-driven dynamical system to perform information processing tasks. Recently, extensions of ESP to potentially nonstationary systems and subsystems, that is, nonstationary ESP and subset/subspace ESP, have been proposed. In this paper, we theoretically and numerically analyze the sufficient and necessary conditions for a quantum system to satisfy nonstationary ESP and subset/subspace nonstationary ESP. Based on extensive usage of the Pauli transfer matrix (PTM) form, we find that (1) the interaction with a quantum-coherent environment, termed $\textit{coherence influx}$, is indispensable in realizing nonstationary ESP, and (2) the spectral radius of PTM can characterize the fading memory property of quantum reservoir computing (QRC). Our numerical experiment, involving a system with a Hamiltonian that entails a spin-glass/many-body localization phase, reveals that the spectral radius of PTM can describe the dynamical phase transition intrinsic to such a system. To comprehensively understand the mechanisms under ESP of QRC, we propose a simplified model, multiplicative reservoir computing (mRC), which is a reservoir computing (RC) system with a one-dimensional multiplicative input. Theoretically and numerically, we show that the parameters corresponding to the spectral radius and coherence influx in mRC directly correlates with its linear memory capacity (MC). Our findings about QRC and mRC will provide a theoretical aspect of PTM and the input multiplicativity of QRC. The results will lead to a better understanding of QRC and information processing in open quantum systems.

DOI
10.48550/arxiv.2409.12693

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Quantum reservoir computing (QRC) provides a hardware-efficient paradigm for temporal information processing on near-term quantum devices. Despite rapid experimental progress, a rigorous understanding of the structural conditions required for its scalable quantum-enhanced performance remains lacking. Here, we develop a theoretical framework in Pauli-Liouville space that provides a unified analytical treatment of the echo state property (ESP), nonlinear expressive power, and quantum resources. We first analyze the widely used qubit-resetting scheme and establish that quantum magic generated by reservoir dynamics is a necessary condition for effective computation, a requirement more fundamental than ESP. However, we prove that this architecture faces inherent expressivity limitations: all nonlinear processing originates exclusively from the classical encoding map, imposing an unavoidable trade-off between nonlinearity and memory capacity. To circumvent this structural bottleneck, we rigorously analyze Hamiltonian encoding, in which temporal inputs are embedded directly into the continuous dynamics generator. We show that the ESP is natively guaranteed by the Liouvillian spectral gap, decoupling it from quantum magic. Crucially, for any non-trivial drive Hamiltonian, the discrete-time update map exhibits a transcendental, infinite-order nonlinear dependence on the instantaneous input. Moreover, the intrinsic non-commutativity of the open-system generators governs the temporal coupling of these nonlinearities, producing highly non-separable processing of the input history. Our results establish a rigorous theoretical hierarchy of QRC architectures and provide prescriptive design principles for experiments targeting genuine quantum advantages in temporal processing.
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Quantum Reservoir Computing (QRC) offers potential advantages over classical reservoir computing, including inherent processing of quantum inputs and a vast Hilbert space for state exploration. Yet, the relation between the performance of reservoirs based on complex and many-body quantum systems and non-classical state features is not established. Through an extensive analysis of QRC based on a transverse-field Ising model we show how different quantum effects, such as quantum coherence and correlations, contribute to improving the performance in temporal tasks, as measured by the Information Processing Capacity. Additionally, we critically assess the impact of finite measurement resources and noise on the reservoir’s dynamics in different regimes, quantifying the limited ability to exploit quantum effects for increasing damping and noise strengths. Our results reveal a monotonic relationship between reservoir performance and coherence, along with the importance of quantum effects in the ergodic regime. Quantum Reservoir Computing leverages the quantum properties of physical systems for solving temporal tasks. This study shows the importance of quantum effects, such as coherence and superposition, in the reservoir’s performance for different dynamical regimes, while considering the impact of finite measurements and noisy environments.
A distinctive feature of Quantum Reservoir Computing (QRC) is the ability to directly embed quantum input states into the reservoir dynamics. However, the resulting output is fundamentally linear for a single input state, preventing QRC from naturally computing nonlinear functionals such as purity or entropy. We overcome this limitation with a quantum-classical hybrid architecture combining a qubit quantum reservoir with a classical echo state network (ESN), allowing both nonlinear functional approximation and effective temporal processing. We systematically study performance under two information regimes: full-tomography and partial information (single-axis measurements), with the latter demonstrating that the hybrid system outperforms its standalone components in both linear and nonlinear tasks due to the enhanced information retrieval provided by the quantum reservoir. Building on these results, we apply an online monitoring protocol that explicitly accounts for measurement back-action and finite measurement ensembles, enabling a realistic assessment of performance under experimental conditions. These results establish hybrid quantum-classical reservoir computing (HRC) architectures as a practical and scalable route for enhanced quantum machine learning on near-term qubit hardware.