Abstract We study quantum state-space models (QSSMs) as recurrent architectures for sequence pro cessing, with an emphasis on the interplay between quantum dynamics, measurement design, and readout expressivity. We consider a delayed XOR benchmark, a canonical task for probing long-range memory and parity-based dependencies, and implement a noiseless, statevector-based quantum reservoir driven by input-conditioned unitary evolution. Using expectation values of local Pauli observables as features, we find that quantum state space dynamics alone remain near chance level on the delayed XOR task, despite the high dimensional and nonlinear nature of the underlying quantum evolution. This result highlights a fundamental observability limitation: parity information is encoded in higher-order correlations that are not linearly accessible through simple expectation-value measurements. We then demonstrate that augmenting the quantum features with a minimal tapped-delay readout, including a bilinear interaction term between delayed inputs, restores perfect gener alization. This augmentation does not modify the quantum dynamics, but instead aligns the observation model with the algebraic structure of the task. Our findings clarify the respective roles of quantum dynamics and measurement interfaces in quantum recurrent models, and suggest that performance gains arise not solely from quantum evolution, but from the coherent integration of dynamics, observability, and readout design.
Abstract We study quantum state-space models (QSSMs) as recurrent architectures for sequence pro cessing, with an emphasis on the interplay between quantum dynamics, measurement design, and readout expressivity. We consider a delayed XOR benchmark, a canonical task for probing long-range memory and parity-based dependencies, and implement a noiseless, statevector-based quantum reservoir driven by input-conditioned unitary evolution. Using expectation values of local Pauli observables as features, we find that quantum state space dynamics alone remain near chance level on the delayed XOR task, despite the high dimensional and nonlinear nature of the underlying quantum evolution. This result highlights a fundamental observability limitation: parity information is encoded in higher-order correlations that are not linearly accessible through simple expectation-value measurements. We then demonstrate that augmenting the quantum features with a minimal tapped-delay readout, including a bilinear interaction term between delayed inputs, restores perfect gener alization. This augmentation does not modify the quantum dynamics, but instead aligns the observation model with the algebraic structure of the task. Our findings clarify the respective roles of quantum dynamics and measurement interfaces in quantum recurrent models, and suggest that performance gains arise not solely from quantum evolution, but from the coherent integration of dynamics, observability, and readout design.
This paper studies temporal learning with quantum reservoirs driven by continuous-time dynamics. Our goal is to model long-range dependencies under irregular sampling while avoiding backpropagation through time and repeated numerical ODE integration. We formulate a Continuous-time Quantum Reservoir (CQR) framework in which an input-modulated Hamiltonian drives Schrödinger evolution, and a classical linear readout is fitted from sampled quantum features. We analyze three variants: (i) CQR-unitary, which uses expectation values of Hermitian observables; (ii) CQR-measurement, which uses projective-measurement probabilities; and (iii) CQR-trainable, which additionally tunes input-coupling Hamiltonian parameters via parameter-shift updates. Experiments on Lorenz forecasting, human-activity classification, and two regression benchmarks show competitive accuracy, improved memory capacity, and clear compute-accuracy trade-offs across variants. We also provide a Stone-Weierstrass-based universality result for the induced feature algebra and discuss practical design limitations.
Quantum reservoir computers (QRCs) have emerged as a promising approach to quantum machine learning, since they utilize the natural dynamics of quantum systems for data processing and are simple to train. Here, we consider n-qubit quantum extreme learning machines (QELMs) with initial-state encoding and continuous-time reservoir dynamics. Applying the Pauli transfer matrix (PTM) formalism reveals the encoding-generated features and their linear transformation by quantum channels, and thereby provides a nonlinear vector autoregression model as an interpretable classical representation of a QELM. We show that the nonlinear information processing capacity of a QELM factorizes into the ability of the encoding to represent a given target function and the decodability of the relevant Pauli features after the effective PTM. Comparing geometric and statistical decodability reveals features that are algebraically hidden yet recoverable through data correlations. We relate the decodability of Pauli features to the spreading of measurement operators under Heisenberg evolution (Krylov growth). Focusing on the unitary channel, we discuss how structured Hamiltonians, when paired with certain observables, can impair model performance, as reflected in a low rank of the readout and the effective PTMs. We trace this effect to Hamiltonian symmetries and derive asymptotic rank estimates for symmetry-resolved observable families. Our decodability analysis disentangles the contributions of encoding, reservoir dynamics, and measurement observables, and thus provides interpretable diagnostics that can inform the design of QELMs and reduce reliance on exhaustive hyperparameter search. As a specific application, we consider forecasting nonlinear dynamical systems and show that the readout of a trained QELM closely matches the expression of the underlying flow map at leading orders.
Modeling long-range dependencies in sequential data remains a central challenge in machine learning. Transformers address this challenge through attention mechanisms, but their quadratic complexity with respect to sequence length limits scalability to long contexts. State-space models (SSMs) provide an efficient alternative with linear-time computation by evolving a latent state through recurrent updates, but their memory is typically formed via additive or linear transitions, which can limit their ability to capture complex global interactions across tokens. In this work, we introduce one of the first studies to leverage the superposition property of quantum systems to enhance state-based sequence modeling. In particular, we propose Quantum Long-Attention Memory (QLAM), a hybrid quantum-classical memory mechanism that can be viewed as a quantum extension of state-space models. Instead of maintaining a classical latent state updated through additive dynamics, QLAM represents the hidden state as a quantum state whose amplitudes encode a superposition of historical information. The state evolves through parameterized quantum circuits conditioned on the input, enabling a non-classical, globally update mechanism. In this way, QLAM preserves the recurrent and linear-time structure of SSMs while fundamentally enriching the memory representation through quantum superposition. Unlike attention mechanisms that explicitly compute pairwise interactions, QLAM implicitly captures global dependencies through the evolution of the quantum state, and retrieves task-relevant information via query-dependent measurements. We evaluate QLAM on sequential variants of standard image classification benchmarks, including sMNIST, sFashion-MNIST, and sCIFAR-10, where images are flattened into token sequences. Across all tasks, QLAM consistently improves over recurrent baselines and transformer-based models.
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.
Quantum reservoir computing (QRC) uses the dynamics of a fixed or weakly tuned quantum system to transform temporal and sequential inputs into measured features, while training is typically confined to a classical readout. This separation reduces reliance on repeated quantum parameter updates and avoids the barren plateaus associated with variational circuit training. Its computational power is often attributed to the exponentially large Hilbert space of the quantum system. However, the memory, nonlinearity, and expressivity that determine what a reservoir can actually compute depend jointly on the input encoding, quantum evolution, observables, measurement, and readout, not on Hilbert space dimension alone. On hardware, these capabilities are further constrained by finite sampling, hardware noise, measurement backaction, and the cost of estimating observables, so a large state space alone does not guarantee useful computation. In this survey, we develop a common system model that connects these components and use it to organize QRC foundations, computational properties, reservoir architectures, operating protocols, and physical implementations. We examine spin, photonic, superconducting, bosonic, neutral atom, and other analog platforms, together with applications, software and high performance computing support, benchmarking, and reproducibility. The analysis distinguishes hardware demonstrations from simulations and identifies the assumptions and resources that govern comparisons across implementations. Current results do not establish a broad quantum advantage over well matched classical reservoirs. We therefore specify the resource accounting, benchmark standards, and theoretical criteria needed to evaluate claims of quantum advantage.
Quantum reservoir computing (QRC) uses fixed quantum dynamics as a high-dimensional temporal feature map and trains only a lightweight classical readout. QRC is attractive for near-term quantum machine learning, but its performance depends strongly on architecture choices such as input encoding, reservoir depth, entanglement topology, measurement features, state-reset policy, feature construction, and readout regularization. We introduce \method, a simulator-based benchmark that formulates QRC design as constrained black-box architecture search and evaluates whether large language models can act as proposal controllers for this search problem. The benchmark compares five policies under identical evaluation budgets: random search, evolutionary search, Bayesian/TPE optimization, a feedback-based LLM agent, and \hybrid, which combines LLM proposals with memory, mutation, crossover, duplicate avoidance, and exploration. On NARMA10, Mackey-Glass forecasting, and temporal parity, \hybrid{} is the most consistent policy: it ranks first on NARMA10 and temporal parity and second on Mackey-Glass, narrowly behind evolutionary search. Under a 25-evaluation budget and three seeds, \hybrid{} improves over random search on all tasks, including a 23.6\% relative reduction in Mackey-Glass error. The results do not show that LLMs are universal QRC optimizers; rather, they show that generative models can be useful high-level controllers when embedded inside validated, reproducible hybrid search loops.