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arXiv:2609.27072

Programmable Coherent Memory Kernels for Quantum Reservoir Computing in Waveguide QED

Hong Jiang, Yu Wu, Yue Ban, Xi Chen, Juan José García-Ripoll

quant-phSubmitted Sep 22, 2026• Announced Sep 24, 2026• 0 citations
arXivHTMLPDF

Abstract

Reservoir computing requires not only long memory, but also memory distributed over the timescales relevant to a given task. In most quantum reservoir computing architectures, however, the memory profile emerges from intrinsic dynamics and cannot be directly engineered. Here we introduce a waveguide-QED quantum reservoir in which coherent memory return is controlled by the coupling geometry. The reservoir consists of a driven Kerr-nonlinear resonator coupled to a one-dimensional waveguide at spatially separated points. An unmeasured propagating field carries information about past inputs and coherently returns it to the resonator after a finite propagation time, forming a non-Markovian feedback channel without intermediate measurement or classical reinjection. Delay-resolved memory benchmarks reveal two complementary functions: waveguide propagation determines when past information returns, whereas Kerr dynamics convert the returned field into nonlinear temporal features accessible through a linear readout. NARMA-$n$ prediction shows that the optimal feedback delay shifts systematically with task order and approximately tracks the explicit input-product lag. These results establish coherent delayed feedback as a mechanism for engineering task-relevant memory in quantum reservoir computing.

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Reservoir computing requires not only long memory, but also memory distributed over the timescales relevant to a given task. In most quantum reservoir computing architectures, however, the memory profile emerges from intrinsic dynamics and cannot be directly engineered. Here we introduce a waveguide-QED quantum reservoir in which coherent memory return is controlled by the coupling geometry. The reservoir consists of a driven Kerr-nonlinear resonator coupled to a one-dimensional waveguide at spatially separated points. An unmeasured propagating field carries information about past inputs and coherently returns it to the resonator after a finite propagation time, forming a non-Markovian feedback channel without intermediate measurement or classical reinjection. Delay-resolved memory benchmarks reveal two complementary functions: waveguide propagation determines when past information returns, whereas Kerr dynamics convert the returned field into nonlinear temporal features accessible through a linear readout. NARMA-nnn prediction shows that the optimal feedback delay shifts systematically with task order and approximately tracks the explicit input-product lag. These results establish coherent delayed feedback as a mechanism for engineering task-relevant memory in quantum reservoir computing.
Quantum reservoir computing processes temporal information through driven many-body dynamics, but its performance is ultimately limited by how accurately past inputs can be extracted from finite measurements. Here we formulate this limitation as a local multiparameter estimation problem and introduce a delay-space quantum Fisher information matrix to quantify the distinguishability of information stored at different delays. This perspective identifies Fisher-orthogonal memory as a measurement-efficient design principle: different delays should perturb the reservoir state along mutually Fisher-orthogonal directions. We first analyze the single-qubit limit using the Gill--Massar bound, revealing an optimal write-store-routing trade-off. Guided by this structure, we construct solvable multi-qubit reservoirs based on Clifford routing orbits and Singer-cycle Pauli algebra. The resulting dynamics yield diagonal delay-space QFIMs with analytically programmable fading profiles. Under finite-shot local Pauli readout, these reservoirs retain sharp memory windows that are absent in a validation-selected Ising baseline. Their product-task behavior is governed by second-order responses inherited from the same Pauli-routing algebra. Our results provide an analytically controlled route toward measurement-efficient quantum reservoir computing.
A linear optical medium can delay, mix, and superpose light, but never make two pulses multiply: multiplication is nonlinear, and a linear system has no such operation. This roots a sharp limit on continuous-variable quantum reservoir computers (QRCs) built from Gaussian optics. Within the reservoir they cannot form genuine products of the input at different past times, the cross-time nonlinear correlations many temporal computations require; they can only fake them by storing each past input separately and multiplying in the readout, forcing an exponentially harder high-order measurement. We show that a single Kerr (intensity-dependent phase) element in a time-delayed feedback loop removes this limit. The Kerr effect makes phase depend on intensity, a true multiplication inside the medium; feedback makes the light revisit that element repeatedly, so one mode mixes its own history against itself once per round-trip. Feedback turns time into space: D passes through one nonlinear mode replace D parallel linear modes. We prove an unbounded resource separation (Theorem 3, Corollary 2): an N-mode Gaussian reservoir reaches cross-time nonlinear rank at most 2N, a hardware ceiling, while a single Kerr mode reaches rank equal to its feedback depth D, costing no extra modes. For every N, one Kerr mode performs a computation no N-mode linear reservoir can. Loss is the counterintuitive ingredient: each round-trip dims the light, so the nonlinear phase differs pass to pass, giving every echo its own fingerprint; without loss the passes would be redundant. We confirm activation on an exact open-system simulation and ground the separation in nonlinear channel equalization. Achievable D is 30 to 230 on integrated platforms, so one nonlinear mode replaces up to about 100 linear ones, at the price of measurement time.
Feedback-driven quantum reservoir computing has so far been studied primarily in gate-based architectures, motivating alternative scalable, hardware-friendly physical platforms. Here we investigate a linear-optical quantum reservoir architecture for time-series processing based on multiphoton interference in a reconfigurable interferometer network equipped with threshold detectors and measurement-conditioned feedback. The reservoir state is constructed from coarse-grained coincidence features, and the feedback updates only a structured, budgeted subset of programmable phases, enabling recurrence without training internal weights. By sweeping the feedback strength, we identify three dynamical regimes and find that memory performance peaks near the stability boundary. We quantify temporal processing via linear memory capacity and validate nonlinear forecasting on benchmarks, namely Mackey-Glass series, NARMA−n-n−n and non-integrable Ising dynamics. The proposed architecture is compatible with current photonic technology and lowers the experimental barrier to feedback-driven QRC for time-series analysis with competitive accuracy.
Featuring memory of past inputs is a fundamental requirement for machine learning models processing time-dependent data. In quantum reservoir computing, all architectures proposed so far rely on Markovian dynamics, which, as we prove, inherently lead to an exponential decay of past information, thereby limiting long-term memory capabilities. We demonstrate that non-Markovian dynamics can overcome this limitation, enabling extended memory retention. By analytically deriving memory bounds and supporting our findings with numerical simulations, we show that non-Markovian reservoirs can outperform their Markovian counterparts, particularly in tasks that require a coexistence of short- and long-term correlations. We introduce an embedding approach that allows a controlled transition from Markovian to non-Markovian evolution, providing a path for practical implementations. Our results establish quantum non-Markovianity as a key resource for enhancing memory in quantum machine learning architectures, with broad implications in quantum neural networks.
The control of open-system dynamics provides a powerful mechanism for using quantum information to process sequential tasks. While quantum reservoir computing typically relies on Markovian dissipation to process sequential data, the computational role of non-Markovian memory effects remains largely unexplored. We introduce a framework for quantum reservoir computing where non-Markovianity is explicitly modeled and regulated using fractional derivatives. By employing fractional time subordination, we generate tunable, heavy-tailed relaxation dynamics that govern the information backflow between the system and its environment. Non-Markovian information backflow regulates the overall temporal retention of the system, maximizing short-term linear memory capacity. Within an optimal operating regime, fractional non-Markovianity redistributes the reservoir memory towards recent inputs, improving both linear memory capacity and nonlinear prediction accuracy at short delays, at the expense of long-range temporal retention. System-environment interaction proves being not merely a requirement for quantum reservoir computing, but an active resource that embeds memory retention directly into the quantum evolution.
Learning with a quantum reservoir can be far simpler than simulating its dynamics. We show that a quantum reservoir governed by a fixed evolution, consisting of an input coupling followed by an independently sampled Haar unitary, admits an exact recursive kernel in the large-dimension limit at fixed history length. Memory orthogonality and factorization of propagated overlaps reduce each kernel evaluation to a scalar recurrence, linear in history length and independent of reservoir size. We establish self-averaging around this limit and connect the kernel coefficients to out-of-time-ordered correlators. Our analysis of finite-dimensional effects shows that the computational simplicity of the Haar kernel does not come at the expense of expressive power. Moreover, we demonstrate how the kernel can be tuned for different learning tasks by adjusting properties such as memory retention and temporal interaction order. On the NARMA10 benchmark, the tuned kernel achieves lower prediction error than the tested Ising and Sachdev--Ye--Kitaev reservoirs.