Abstract--This study compares Quantum Reservoir Computing (QRC) with classical (Echo State Networks, LSTMs) and hybrid quantum-classical methods (QLSTM) for the nonlinear autoregressive moving average task (NARMA-10). We evaluate forecasting accuracy (NRMSE), computational cost, and evaluation time. Results show QRC achieves competitive accuracy while offering potential sustainability advantages, particularly in resource-constrained settings, highlighting its promise for sustainable, time-series AI applications. Time-series forecasting is fundamental across science and engineering; the NARMA-10 benchmark probes temporal memory and nonlinear processing. We present a systematic comparison of Quantum Reservoir Computing (QRC) [4] against classical and hybrid baselines--Echo State Networks (ESN) [1], Long Short-Term Memory (LSTM) [2], and a quantum-inspired LSTM (QLSTM)--on NARMA-10.

Quantum reservoir computing (QRC) offers a hardware-friendly approach to temporal learning, yet most studies target univariate signals and overlook near-term hardware constraints. This work introduces a gate-based QRC for multivariate time series (MTS-QRC) that pairs injection and memory qubits and uses a Trotterized nearest-neighbor transverse-field Ising evolution optimized for current device connectivity and depth. On Lorenz-63 and ENSO, the method achieves a mean square error (MSE) of 0.0087 and 0.0036, respectively, performing on par with classical reservoir computing on Lorenz and above learned RNNs on both, while NVAR and clustered ESN remain stronger on some settings. On IBM Heron R2, MTS-QRC sustains accuracy with realistic depths and, interestingly, outperforms a noiseless simulator on ENSO; singular value analysis indicates that device noise can concentrate variance in feature directions, acting as an implicit regularizer for linear readout in this regime. These findings support the practicality of gate-based QRC for MTS forecasting on NISQ hardware and motivate systematic studies on when and how hardware noise benefits QRC readouts.

The rapid development of machine learning and quantum computing has placed quantum machine learning at the forefront of research. However, existing quantum machine learning algorithms based on quantum variational algorithms face challenges in trainability and noise robustness. In order to address these challenges, we introduce a gradient-free, noise-robust quantum reservoir computing algorithm that harnesses discrete time crystal dynamics as a reservoir. We first calibrate the memory, nonlinear, and information scrambling capacities of the quantum reservoir, revealing their correlation with dynamical phases and non-equilibrium phase transitions. We then apply the algorithm to the binary classification task and establish a comparative quantum kernel advantage. For ten-class classification, both noisy simulations and experimental results on superconducting quantum processors match ideal simulations, demonstrating the enhanced accuracy with increasing system size and confirming the topological noise robustness. Our work presents the first experimental demonstration of quantum reservoir computing for image classification based on digital quantum simulation. It establishes the correlation between quantum many-body non-equilibrium phase transitions and quantum machine learning performance, providing new design principles for quantum reservoir computing and broader quantum machine learning algorithms in the NISQ era.

Scrambling quantum systems have attracted attention as effective substrates for temporal information processing. Here we consider a quantum reservoir processing framework that captures a broad range of physical computing models with quantum systems. We examine the scalability and memory retention of the model with scrambling reservoirs modelled by high-order unitary designs in both noiseless and noisy settings. In the former regime, we show that measurement readouts become exponentially concentrated with increasing reservoir size, yet strikingly do not worsen with the reservoir iterations. Thus, while repeatedly reusing a small scrambling reservoir with quantum data might be viable, scaling up the problem size deteriorates generalization unless one can afford an exponential shot overhead. In contrast, the memory of early inputs and initial states decays exponentially in both reservoir size and reservoir iterations. In the noisy regime, we also prove that memory decays exponentially in time for local noisy channels. These results required us to introduce new proof techniques for bounding concentration in temporal quantum models.

Quantum reservoir computing uses the dynamics of quantum systems to process temporal data, making it particularly well-suited for learning with noisy intermediate-scale quantum devices. Early experimental proposals, such as the restarting and rewinding protocols, relied on repeating previous steps of the quantum map to avoid backaction. However, this approach compromises real-time processing and increases computational overhead. Recent developments have introduced alternative protocols that address these limitations. These include online, mid-circuit measurement, and feedback techniques, which enable real-time computation while preserving the input history. Among these, the feedback protocol stands out for its ability to process temporal information with comparatively fewer components. Despite this potential advantage, the theoretical foundations of feedback-based quantum reservoir computing remain underdeveloped, particularly with regard to the universality and the approximation capabilities of this approach. This paper addresses this issue by presenting a recurrent quantum neural network architecture that extends a class of existing feedforward models to a dynamic, feedback-driven reservoir setting. We provide theoretical guarantees for variational recurrent quantum neural networks, including approximation bounds and universality results. Notably, our analysis demonstrates that the model is universal with linear readouts, making it both powerful and experimentally accessible. These results pave the way for practical and theoretically grounded quantum reservoir computing with real-time processing capabilities.

After many years of development, quantum computing hardware is rapidly developing towards commercialization. This has led to an explosion of interest in quantum algorithms and applications [1]. In the games industry the intersection of quantum computing and games has been explored for almost a decade [2]. Although most examples are currently within an educational context [3], the potential of quantum computing for procedural generation has started to be explored [4-7]. Here we go beyond this previous proof-of-principle work by developing an example of quantum procedural generation that can provide real-time generation of levels within a live game. Specifically, we explore a generative system based on Quantum Reservoir Computing (QRC) [8] to generate game levels.

We propose a way to bound the generalisation errors of several classes of quantum reservoirs using the Rademacher complexity. We give specific, parameter-dependent bounds for two particular quantum reservoir classes. We analyse how the generalisation bounds scale with growing numbers of qubits. Applying our results to classes with polynomial readout functions, we find that the risk bounds converge in the number of training samples. The explicit dependence on the quantum reservoir and readout parameters in our bounds can be used to control the generalisation error to a certain extent. It should be noted that the bounds scale exponentially with the number of qubits $n$. The upper bounds on the Rademacher complexity can be applied to other reservoir classes that fulfill a few hypotheses on the quantum dynamics and the readout function.

There is a growing interest in the development of artificial neural networks that are implemented in a physical system. A major challenge in this context is that these networks are difficult to train since training here would require a change of physical parameters rather than simply of coefficients in a computer program. For this reason, reservoir computing, where one employs high-dimensional recurrent networks and trains only the final layer, is widely used in this context. In this chapter, I introduce the basic concepts of reservoir computing. Moreover, I present some important physical implementations coming from electronics, photonics, spintronics, mechanics, and biology. Finally, I provide a brief discussion of quantum reservoir computing.

In the current Noisy Intermediate Scale Quantum (NISQ) era, the presence of noise deteriorates the performance of quantum computing algorithms. Quantum Reservoir Computing (QRC) is a type of Quantum Machine Learning algorithm, which, however, can benefit from different types of tuned noise. In this paper, we analyse the effect that finite-sampling noise has on the chaotic time-series prediction capabilities of QRC and Recurrence-free Quantum Reservoir Computing (RF-QRC). First, we show that, even without a recurrent loop, RF-QRC contains temporal information about previous reservoir states using leaky integrated neurons. This makes RF-QRC different from Quantum Extreme Learning Machines (QELM). Second, we show that finite sampling noise degrades the prediction capabilities of both QRC and RF-QRC while affecting QRC more due to the propagation of noise. Third, we optimize the training of the finite-sampled quantum reservoir computing framework using two methods: (a) Singular Value Decomposition (SVD) applied to the data matrix containing noisy reservoir activation states; and (b) data-filtering techniques to remove the high-frequencies from the noisy reservoir activation states. We show that denoising reservoir activation states improve the signal-to-noise ratios with smaller training loss. Finally, we demonstrate that the training and denoising of the noisy reservoir activation signals in RF-QRC are highly parallelizable on multiple Quantum Processing Units (QPUs) as compared to the QRC architecture with recurrent connections. The analyses are numerically showcased on prototypical chaotic dynamical systems with relevance to turbulence. This work opens opportunities for using quantum reservoir computing with finite samples for time-series forecasting on near-term quantum hardware.

Artificial intelligence (AI) systems of autonomous systems such as drones, robots and self-driving cars may consume up to 50% of total power available onboard, thereby limiting the vehicle's range of functions and considerably reducing the distance the vehicle can travel on a single charge. Next-generation onboard AI systems need an even higher power since they collect and process even larger amounts of data in real time. This problem cannot be solved using the traditional computing devices since they become more and more power-consuming. In this review article, we discuss the perspectives of development of onboard neuromorphic computers that mimic the operation of a biological brain using nonlinear-dynamical properties of natural physical environments surrounding autonomous vehicles. Previous research also demonstrated that quantum neuromorphic processors (QNPs) can conduct computations with the efficiency of a standard computer while consuming less than 1% of the onboard battery power. Since QNPs is a semi-classical technology, their technical simplicity and low, compared with quantum computers, cost make them ideally suitable for application in autonomous AI system. Providing a perspective view on the future progress in unconventional physical reservoir computing and surveying the outcomes of more than 200 interdisciplinary research works, this article will be of interest to a broad readership, including both students and experts in the fields of physics, engineering, quantum technologies and computing.