[
  {
    "citation": "\u010cindrak et al. (2026). Memory-Nonlinearity Trade-off across Quantum Reservoir Computing Frameworks. https://arxiv.org/abs/2603.21371",
    "evidence_atoms": [
      {
        "atom_type": "claim",
        "confidence": "high",
        "content": "QRC performance reflects a memory-nonlinearity trade-off across protocol families.",
        "locator": "knowledge/papers/2603.21371.txt",
        "source_ref": "https://arxiv.org/abs/2603.21371"
      }
    ],
    "source_type": "paper",
    "summary": "Analyzes how QRC performance varies with memory-vs-nonlinearity operating regimes across frameworks.",
    "title": "Memory-Nonlinearity Trade-off across Quantum Reservoir Computing Frameworks",
    "url": "https://arxiv.org/abs/2603.21371",
    "year": 2026
  },
  {
    "citation": "Brusaschi et al. (2026). Quantum inference on a classically trained quantum extreme learning machine. https://arxiv.org/abs/2603.20167",
    "source_type": "paper",
    "summary": "Studies train/infer separation for QELM with classical training and quantum inference.",
    "title": "Quantum inference on a classically trained quantum extreme learning machine",
    "url": "https://arxiv.org/abs/2603.20167",
    "year": 2026
  },
  {
    "citation": "Memory-enhanced quantum extreme learning machines for characterizing non-Markovian dynamics (2026). https://arxiv.org/abs/2603.17182",
    "source_type": "paper",
    "summary": "Introduces memory-augmented QELM designs for non-Markovian characterization tasks.",
    "title": "Memory-enhanced quantum extreme learning machines for characterizing non-Markovian dynamics",
    "url": "https://arxiv.org/abs/2603.17182",
    "year": 2026
  },
  {
    "citation": "Efficient time-series prediction on NISQ devices via time-delayed quantum extreme learning machine (2026). https://arxiv.org/abs/2602.21544",
    "source_type": "paper",
    "summary": "Proposes time-delayed QELM for NISQ-friendly sequence prediction.",
    "title": "Efficient time-series prediction on NISQ devices via time-delayed quantum extreme learning machine",
    "url": "https://arxiv.org/abs/2602.21544",
    "year": 2026
  },
  {
    "citation": "Quantum Reservoir Autoencoder: Conditions, Protocol, and Noise Resilience (2026). https://arxiv.org/abs/2602.19700",
    "source_type": "paper",
    "summary": "Defines a QRC autoencoder protocol and studies robustness under noise.",
    "title": "Quantum Reservoir Autoencoder: Conditions, Protocol, and Noise Resilience",
    "url": "https://arxiv.org/abs/2602.19700",
    "year": 2026
  },
  {
    "citation": "Spectral Phase Encoding for Quantum Kernel Methods (2026). https://arxiv.org/abs/2602.19644",
    "source_type": "paper",
    "summary": "Presents spectral phase encoding techniques for quantum kernel models.",
    "title": "Spectral Phase Encoding for Quantum Kernel Methods",
    "url": "https://arxiv.org/abs/2602.19644",
    "year": 2026
  },
  {
    "citation": "Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach (2026). https://arxiv.org/abs/2602.18377",
    "source_type": "paper",
    "summary": "Builds interpretability tools for QELM using Pauli-transfer matrix analysis.",
    "title": "Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach",
    "url": "https://arxiv.org/abs/2602.18377",
    "year": 2026
  },
  {
    "citation": "A Programmable Linear Optical Quantum Reservoir with Measurement Feedback for Time Series Analysis (2026). https://arxiv.org/abs/2602.17440",
    "source_type": "paper",
    "summary": "Introduces a programmable linear-optical reservoir with measurement feedback.",
    "title": "A Programmable Linear Optical Quantum Reservoir with Measurement Feedback for Time Series Analysis",
    "url": "https://arxiv.org/abs/2602.17440",
    "year": 2026
  },
  {
    "citation": "Kernel-based optimization of measurement operators for quantum reservoir computers (2026). https://arxiv.org/abs/2602.14677",
    "source_type": "paper",
    "summary": "Derives kernel-based measurement-operator optimization for QRC readouts.",
    "title": "Kernel-based optimization of measurement operators for quantum reservoir computers",
    "url": "https://arxiv.org/abs/2602.14677",
    "year": 2026
  },
  {
    "citation": "QuaRK: A Quantum Reservoir Kernel for Time Series Learning (2026). https://arxiv.org/abs/2602.13531",
    "source_type": "paper",
    "summary": "Defines a reservoir-derived kernel for quantum time-series learning.",
    "title": "QuaRK: A Quantum Reservoir Kernel for Time Series Learning",
    "url": "https://arxiv.org/abs/2602.13531",
    "year": 2026
  },
  {
    "citation": "A Quantum Reservoir Computing Approach to Quantum Stock Price Forecasting in Quantum-Invested Markets (2026). https://arxiv.org/abs/2602.13094",
    "source_type": "paper",
    "summary": "Applies QRC methodology to financial forecasting benchmarks.",
    "title": "A Quantum Reservoir Computing Approach to Quantum Stock Price Forecasting in Quantum-Invested Markets",
    "url": "https://arxiv.org/abs/2602.13094",
    "year": 2026
  },
  {
    "citation": "Practical Quantum Reservoir Computing in Rydberg Atom Arrays (2026). https://arxiv.org/abs/2602.00610",
    "source_type": "paper",
    "summary": "Explores practical implementation constraints for Rydberg-array QRC.",
    "title": "Practical Quantum Reservoir Computing in Rydberg Atom Arrays",
    "url": "https://arxiv.org/abs/2602.00610",
    "year": 2026
  },
  {
    "citation": "Margin-Based Generalisation Bounds for Quantum Kernel Methods under Local Depolarising Noise (2026). https://arxiv.org/abs/2601.23084",
    "source_type": "paper",
    "summary": "Provides margin-based generalization analysis for noisy quantum kernels.",
    "title": "Margin-Based Generalisation Bounds for Quantum Kernel Methods under Local Depolarising Noise",
    "url": "https://arxiv.org/abs/2601.23084",
    "year": 2026
  },
  {
    "citation": "Practical Evaluation of Quantum Kernel Methods for Radar Micro-Doppler Classification on NISQ Hardware (2026). https://arxiv.org/abs/2601.22194",
    "source_type": "paper",
    "summary": "Benchmarks quantum kernels on radar micro-Doppler classification.",
    "title": "Practical Evaluation of Quantum Kernel Methods for Radar Micro-Doppler Classification on Noisy Intermediate-Scale Quantum (NISQ) Hardware",
    "url": "https://arxiv.org/abs/2601.22194",
    "year": 2026
  },
  {
    "citation": "Quantum Wiener architecture for quantum reservoir computing (2026). https://arxiv.org/abs/2601.04812",
    "source_type": "paper",
    "summary": "Introduces Wiener-style architectural modeling for QRC.",
    "title": "Quantum Wiener architecture for quantum reservoir computing",
    "url": "https://arxiv.org/abs/2601.04812",
    "year": 2026
  },
  {
    "citation": "Image Denoising via Quantum Reservoir Computing (2025). https://arxiv.org/abs/2512.18612",
    "source_type": "paper",
    "summary": "Applies QRC to image denoising tasks.",
    "title": "Image Denoising via Quantum Reservoir Computing",
    "url": "https://arxiv.org/abs/2512.18612",
    "year": 2025
  },
  {
    "citation": "Maritime object classification with SAR imagery using quantum kernel methods (2025). https://arxiv.org/abs/2512.11367",
    "source_type": "paper",
    "summary": "Uses quantum kernels for SAR maritime object classification.",
    "title": "Maritime object classification with SAR imagery using quantum kernel methods",
    "url": "https://arxiv.org/abs/2512.11367",
    "year": 2025
  },
  {
    "citation": "Entanglement estimation of Werner states with a quantum extreme learning machine (2025). https://arxiv.org/abs/2511.01387",
    "source_type": "paper",
    "summary": "Uses QELM to estimate entanglement indicators in Werner states.",
    "title": "Entanglement estimation of Werner states with a quantum extreme learning machine",
    "url": "https://arxiv.org/abs/2511.01387",
    "year": 2025
  },
  {
    "citation": "From quantum feature maps to quantum reservoir computing: perspectives and applications (2025). https://arxiv.org/abs/2510.01797",
    "source_type": "paper",
    "summary": "Surveys links between feature-map methods and QRC applications.",
    "title": "From quantum feature maps to quantum reservoir computing: perspectives and applications",
    "url": "https://arxiv.org/abs/2510.01797",
    "year": 2025
  },
  {
    "citation": "Entanglement and Classical Simulability in Quantum Extreme Learning Machines (2025). https://arxiv.org/abs/2509.06873",
    "source_type": "paper",
    "summary": "Studies entanglement onset, transition behavior, and simulability limits in QELM image-classification settings.",
    "title": "Entanglement and Classical Simulability in Quantum Extreme Learning Machines",
    "url": "https://arxiv.org/abs/2509.06873",
    "year": 2025
  },
  {
    "citation": "Feedback Connections in Quantum Reservoir Computing with Mid-Circuit Measurements (2025). https://arxiv.org/abs/2503.22380",
    "source_type": "paper",
    "summary": "Analyzes feedback-enhanced QRC with mid-circuit measurement design.",
    "title": "Feedback Connections in Quantum Reservoir Computing with Mid-Circuit Measurements",
    "url": "https://arxiv.org/abs/2503.22380",
    "year": 2025
  },
  {
    "citation": "Feedback-enhanced quantum reservoir computing with weak measurements (2025). https://arxiv.org/abs/2503.17939",
    "source_type": "paper",
    "summary": "Examines weak-measurement feedback effects in reservoir dynamics.",
    "title": "Feedback-enhanced quantum reservoir computing with weak measurements",
    "url": "https://arxiv.org/abs/2503.17939",
    "year": 2025
  },
  {
    "citation": "Application of quantum machine learning using quantum kernel algorithms on multiclass neuron M type classification (2025). https://arxiv.org/abs/2502.06281",
    "source_type": "paper",
    "summary": "Applies quantum-kernel methods to multiclass neuronal classification.",
    "title": "Application of quantum machine learning using quantum kernel algorithms on multiclass neuron M type classification",
    "url": "https://arxiv.org/abs/2502.06281",
    "year": 2025
  },
  {
    "citation": "Robust Quantum Reservoir Computing for Molecular Property Prediction (2024). https://arxiv.org/abs/2412.06758",
    "source_type": "paper",
    "summary": "Evaluates QRC robustness for low-data molecular prediction benchmarks.",
    "title": "Robust Quantum Reservoir Computing for Molecular Property Prediction",
    "url": "https://arxiv.org/abs/2412.06758",
    "year": 2024
  },
  {
    "citation": "Harnessing Quantum Extreme Learning Machines for image classification (2024). https://arxiv.org/abs/2409.00998",
    "source_type": "paper",
    "summary": "Compares encoding and reservoir choices for QELM image classification.",
    "title": "Harnessing Quantum Extreme Learning Machines for image classification",
    "url": "https://arxiv.org/abs/2409.00998",
    "year": 2024
  },
  {
    "citation": "Large-scale quantum reservoir learning with an analog quantum computer (2024). https://arxiv.org/abs/2407.02553",
    "source_type": "paper",
    "summary": "Demonstrates analog large-scale QRC and analyzes kernel geometry/observables.",
    "title": "Large-scale quantum reservoir learning with an analog quantum computer",
    "url": "https://arxiv.org/abs/2407.02553",
    "year": 2024
  },
  {
    "citation": "Frequency- and dissipation-dependent entanglement advantage in spin-network Quantum Reservoir Computing (2024). https://arxiv.org/abs/2403.08998",
    "source_type": "paper",
    "summary": "Investigates entanglement advantages under dissipation/frequency controls.",
    "title": "Frequency- and dissipation-dependent entanglement advantage in spin-network Quantum Reservoir Computing",
    "url": "https://arxiv.org/abs/2403.08998",
    "year": 2024
  },
  {
    "citation": "Quantum reservoir computing with repeated measurements on superconducting devices (2023). https://arxiv.org/abs/2310.06706",
    "source_type": "paper",
    "summary": "Studies repeated-measurement QRC on superconducting hardware.",
    "title": "Quantum reservoir computing with repeated measurements on superconducting devices",
    "url": "https://arxiv.org/abs/2310.06706",
    "year": 2023
  },
  {
    "citation": "Several fitness functions and entanglement gates in quantum kernel generation (2023). https://arxiv.org/abs/2309.03307",
    "source_type": "paper",
    "summary": "Explores kernel construction sensitivity to fitness functions and entangling gates.",
    "title": "Several fitness functions and entanglement gates in quantum kernel generation",
    "url": "https://arxiv.org/abs/2309.03307",
    "year": 2023
  },
  {
    "citation": "Quantum Kernel for Image Classification of Real World Manufacturing Defects (2022). https://arxiv.org/abs/2212.08693",
    "source_type": "paper",
    "summary": "Applies quantum-kernel classifiers to real manufacturing-defect images.",
    "title": "Quantum Kernel for Image Classification of Real World Manufacturing Defects",
    "url": "https://arxiv.org/abs/2212.08693",
    "year": 2022
  },
  {
    "citation": "The role of entanglement for enhancing the efficiency of quantum kernels towards classification (2022). https://arxiv.org/abs/2209.05142",
    "source_type": "paper",
    "summary": "Analyzes how entanglement affects quantum-kernel efficiency.",
    "title": "The role of entanglement for enhancing the efficiency of quantum kernels towards classification",
    "url": "https://arxiv.org/abs/2209.05142",
    "year": 2022
  },
  {
    "citation": "Time Series Quantum Reservoir Computing with Weak and Projective Measurements (2022). https://arxiv.org/abs/2205.06809",
    "source_type": "paper",
    "summary": "Compares weak vs projective measurements in time-series QRC.",
    "title": "Time Series Quantum Reservoir Computing with Weak and Projective Measurements",
    "url": "https://arxiv.org/abs/2205.06809",
    "year": 2022
  },
  {
    "citation": "Computing with two quantum reservoirs connected via optimized two-qubit nonselective measurements (2022). https://arxiv.org/abs/2201.07969",
    "source_type": "paper",
    "summary": "Studies coupled-reservoir architectures with optimized nonselective measurements.",
    "title": "Computing with two quantum reservoirs connected via optimized two-qubit nonselective measurements",
    "url": "https://arxiv.org/abs/2201.07969",
    "year": 2022
  },
  {
    "citation": "Opportunities in Quantum Reservoir Computing and Extreme Learning Machines (2021). https://arxiv.org/abs/2102.11831",
    "source_type": "paper",
    "summary": "Reviews opportunities and open directions for QRC/QELM development.",
    "title": "Opportunities in Quantum Reservoir Computing and Extreme Learning Machines",
    "url": "https://arxiv.org/abs/2102.11831",
    "year": 2021
  },
  {
    "citation": "Schuld (2021). Supervised quantum machine learning models are kernel methods. https://arxiv.org/abs/2101.11020",
    "source_type": "paper",
    "summary": "Establishes kernel-method interpretation for broad classes of supervised quantum models.",
    "title": "Supervised quantum machine learning models are kernel methods",
    "url": "https://arxiv.org/abs/2101.11020",
    "year": 2021
  },
  {
    "citation": "The effect of data encoding on the expressive power of variational quantum machine learning models (2020). https://arxiv.org/abs/2008.08605",
    "source_type": "paper",
    "summary": "Characterizes encoding-induced expressivity via Fourier-spectrum analysis.",
    "title": "The effect of data encoding on the expressive power of variational quantum machine learning models",
    "url": "https://arxiv.org/abs/2008.08605",
    "year": 2020
  },
  {
    "citation": "Maass, Natschl\u00e4ger, Markram (2002). Real-time computing without stable states. https://doi.org/10.1162/089976602760407955",
    "source_type": "paper",
    "summary": "Foundational liquid-state computing formulation underlying reservoir-computing principles.",
    "title": "Real-time computing without stable states: A new framework for neural computation based on perturbations",
    "url": "https://doi.org/10.1162/089976602760407955",
    "year": 2002
  },
  {
    "citation": "Jaeger (2001). The Echo State Approach to Analysing and Training Recurrent Neural Networks. https://www.ai.rug.nl/minds/uploads/EchoStatesTechRep.pdf",
    "source_type": "report",
    "summary": "Technical report introducing echo-state networks and training perspective for reservoirs.",
    "title": "The Echo State Approach to Analysing and Training Recurrent Neural Networks",
    "url": "https://www.ai.rug.nl/minds/uploads/EchoStatesTechRep.pdf",
    "year": 2001
  },
  {
    "assumptions": [
      "QRC performance is governed by a tunable memory-nonlinearity trade-off measurable via IPC decomposition.",
      "Hamiltonian and measured observables constrain effective degrees of freedom more than encoding choice.",
      "Task suitability is benchmark-dependent; Lorenz and Mackey-Glass are used as stress tests for nonlinearity and memory."
    ],
    "citation": "\u010cindrak et al. (2026). Memory-Nonlinearity Trade-off across Quantum Reservoir Computing Frameworks. https://arxiv.org/abs/2603.21371",
    "claims": [
      "Distinct QRC protocols exhibit the same qualitative memory-nonlinearity trade-off curve.",
      "Intermediate operating regimes can outperform fully reset baselines on total IPC and prediction error.",
      "Near localization-ergodicity transition and moderate dissipation, nonlinear capacity peaks while retaining usable memory."
    ],
    "evidence_atoms": [
      {
        "atom_type": "claim",
        "confidence": "high",
        "content": "QRC performance reflects a memory-nonlinearity trade-off across protocol families.",
        "locator": "knowledge/papers/2603.21371.txt",
        "source_ref": "https://arxiv.org/abs/2603.21371"
      }
    ],
    "key_equations": [
      "dx/dt = \u03c3 (y - x), dy/dt = x(\u03c1 - z) - y, dz/dt = xy - \u03b2z (Lorenz-63; \u03c3=10, \u03c1=28, \u03b2=8/3).",
      "dx/dt = \u03b2 x(t-\u03c4_MG)/(1 + x(t-\u03c4_MG)^10) - \u03b3 x(t) (Mackey-Glass benchmark dynamic).",
      "IPC_tot = IPC_1 + IPC_{\u22652}, where IPC_1 tracks linear memory and IPC_{\u22652} tracks nonlinear response."
    ],
    "parameters": [
      "Lorenz parameters: \u03c3=10, \u03c1=28, \u03b2=8/3; integration dt=0.001; discretization \u0394t=0.1.",
      "Phase-transition control parameter: transverse field h (localized regime h<0.1, critical region near h\u22480.5, ergodic for large h).",
      "Dissipative regime control parameter: Lindblad decay rate \u03b3 with intermediate \u03b3 maximizing IPC_tot."
    ],
    "procedures": [
      "Compute IPC decomposition (linear memory IPC_1 and nonlinear IPC_{\u22652}) across protocol settings.",
      "Benchmark on Lorenz x\u2192x, Lorenz x\u2192z, and Mackey-Glass predictions using normalized RMSE.",
      "Sweep reset length, measurement strength, Ising field, and dissipation to identify IPC-performance maxima."
    ],
    "source_type": "paper",
    "summary": "Analyzes how QRC performance varies with memory-vs-nonlinearity operating regimes across frameworks.",
    "title": "Memory-Nonlinearity Trade-off across Quantum Reservoir Computing Frameworks",
    "url": "https://arxiv.org/abs/2603.21371",
    "year": 2026
  },
  {
    "citation": "Brusaschi et al. (2026). Quantum inference on a classically trained quantum extreme learning machine. https://arxiv.org/abs/2603.20167",
    "key_equations": [
      "ing M labels or values {ym }M  m=0 (e.g., an entanglement",
      "ym    \u223c    wmb xb =        wmb Tr(\u00b5b \u03c1(r) ),   (1)",
      "suitable metric, such as the mean squared error MSE =         tion implemented by the QELM to the signal and idler",
      "cally requires the estimation of expectation values, possi-   \u03b1j = (0, . . . , \u03b1j , . . . , 0), and shapes its amplitude by ap-",
      "tegration time [2, 24], the achievable speedup is modest,     Therefore, Ckj \u221d Ikj only if U\u0303i = U\u2020QELM,i , i.e., the",
      "The paradigm shift introduced here is to exploit the       The set of intensities Ib=kj replace the POVM outcomes"
    ],
    "source_type": "paper",
    "summary": "Studies train/infer separation for QELM with classical training and quantum inference.",
    "title": "Quantum inference on a classically trained quantum extreme learning machine",
    "url": "https://arxiv.org/abs/2603.20167",
    "year": 2026
  },
  {
    "citation": "Memory-enhanced quantum extreme learning machines for characterizing non-Markovian dynamics (2026). https://arxiv.org/abs/2603.17182",
    "key_equations": [
      "non-Markovian quantum dynamics. Section III QELM                                 P\u0302m (\u03c7) = cos(\u03c7)I + i sin(\u03c7)S\u0302m           (3)",
      "\u03bb (\u03c1) =         K\u0302im \u03c1K\u0302im\u2020 ,         (4)",
      "K\u03021,2,3 = \u03bb/4\u03c3\u0302x,y,z ., and 0 \u2264 \u03bb \u2264 1.",
      "channel to the bath [33]. In Ref. [34], the channel in-                  \u03bb = 1, the bath is reset to a completely mixed state",
      "due to the tunability of its non-Markovian character. In                 Conversely, \u03bb = 0 preserves bath correlations, max-",
      "\u201csystem qubits\u201d Sn with n = 1, . . . , N . The dynamics of                der their Hamiltonian eq. 1 for the same time in-"
    ],
    "source_type": "paper",
    "summary": "Introduces memory-augmented QELM designs for non-Markovian characterization tasks.",
    "title": "Memory-enhanced quantum extreme learning machines for characterizing non-Markovian dynamics",
    "url": "https://arxiv.org/abs/2603.17182",
    "year": 2026
  },
  {
    "citation": "Efficient time-series prediction on NISQ devices via time-delayed quantum extreme learning machine (2026). https://arxiv.org/abs/2602.21544",
    "key_equations": [
      "superconducting processor ibm kawasaki as well as its             state matrix Xt,k = xt,k with X \u2208 RT,K and xt =",
      "The effectiveness of TD-QELM is evaluated using the            time-multiplexing at NV times \u03c4n = nT /NV with n \u2208",
      "Nonlinear AutoRegressive Moving Average (NARMA)                   {1, 2, . . . , NV } is employed, resulting in a total of NR =",
      "QRC protocol across different noise conditions. In partic-                                  yt = xt wout ,                      (4)",
      "due to increasing noise accumulation. These results indi-         mize the loss L = (y \u2212 y\u0302)2 , and are obtained as",
      "efficient time-series forecasting on current NISQ devices.                           wout = (X \u22a4 X)\u22121 X \u22a4 y\u0302,                   (5)"
    ],
    "source_type": "paper",
    "summary": "Proposes time-delayed QELM for NISQ-friendly sequence prediction.",
    "title": "Efficient time-series prediction on NISQ devices via time-delayed quantum extreme learning machine",
    "url": "https://arxiv.org/abs/2602.21544",
    "year": 2026
  },
  {
    "citation": "Quantum Reservoir Autoencoder: Conditions, Protocol, and Noise Resilience (2026). https://arxiv.org/abs/2602.19700",
    "key_equations": [
      "noise (p = 0.005), the MSE degrades to 10\u22123 \u201310\u22121 . Asymmetric resource allocation\u201410 shots for",
      "vectors [4, 5]. A trainable linear readout layer y\u0302 = V W           tion. Unlike quantum autoencoders, the QRA keeps the",
      "the more direct input\u2013output relationship of parallel-                 provement (mean 102\u00d7, 16 seeds \u00d7 3 trials = 48",
      "making reversibility challenging, also provides a rich fea-            (d = 31) and baseline comparison, we identify",
      "ture space:\u0001 with Nq = 10 data qubits, QRC extracts                    the iterative protocol structure\u2014not the feature",
      "3Nq + N2q + 1 = 76 features per time step without in-                  dimension\u2014as the dominant noise bottleneck."
    ],
    "source_type": "paper",
    "summary": "Defines a QRC autoencoder protocol and studies robustness under noise.",
    "title": "Quantum Reservoir Autoencoder: Conditions, Protocol, and Noise Resilience",
    "url": "https://arxiv.org/abs/2602.19700",
    "year": 2026
  },
  {
    "citation": "Spectral Phase Encoding for Quantum Kernel Methods (2026). https://arxiv.org/abs/2602.19644",
    "key_equations": [
      "D(\u03d5) = diag ei\u03d50 , ei\u03d51 , . . . , ei\u03d5m\u22121 ,            (1)",
      "allowing the encoding to be implemented using n = \u2308log2 m\u2309 qubits.",
      "|\u03c8(x)\u27e9 = \u221a         e  |j\u27e9 ,                        (2)",
      "same kernel-based learning framework. Given a dataset {(xi , yi )}N   i=1 , clas-",
      "noise level we subsample N = 150 examples using class-balanced sampling",
      "\u03c3 = 0.20 corresponds to perturbations with variance 0.04 in the [0, 1] do-"
    ],
    "source_type": "paper",
    "summary": "Presents spectral phase encoding techniques for quantum kernel models.",
    "title": "Spectral Phase Encoding for Quantum Kernel Methods",
    "url": "https://arxiv.org/abs/2602.19644",
    "year": 2026
  },
  {
    "citation": "Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach (2026). https://arxiv.org/abs/2602.18377",
    "key_equations": [
      "d2 ) on the d-dimensional Hilbert space H, where d = 2n in terms of the number of qubits n. Given",
      "f (x) =           wk tr[Mk \u03c1(x)],                             (2.1)",
      "where {Mk }m  k=1 are a set of measurement operators in M(H), and \u03c1(x) \u2208 M(H) is the density",
      "an initial state, \u03c1(x) = E(\u00b7 \u00b7 \u00b7 Exenc (\u03c10 ) \u00b7 \u00b7 \u00b7 ). The trainable weights wk are determined by minimizing",
      "where L is a loss function, \u03bb is a regularization parameter, and {(xi , yi )}P    i=1 are P training data",
      "points. Typical loss functions are the least-squares loss L(y\u0302, y) = 21 (y\u0302 \u2212 y)2 for regression (including"
    ],
    "source_type": "paper",
    "summary": "Builds interpretability tools for QELM using Pauli-transfer matrix analysis.",
    "title": "Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach",
    "url": "https://arxiv.org/abs/2602.18377",
    "year": 2026
  },
  {
    "citation": "A Programmable Linear Optical Quantum Reservoir with Measurement Feedback for Time Series Analysis (2026). https://arxiv.org/abs/2602.17440",
    "key_equations": [
      "through an M -mode linear optical network described by a unitary V \u2208 U (M ). Let |S\u27e9 = |s1 s2 . . . sM \u27e9 be",
      "the input Fock state with j=1 sj = N . The output Fock basis is enumerated as {|Q\u2113 \u27e9}\u2113=1          , where each",
      "|Q\u2113 \u27e9 = q1 q2 . . . qM satisfying j=1 qj = N and dpnr =           N      . The transition probability is",
      "p         (|S\u27e9 \u2192 |Q\u2113 \u27e9) = QM        QM (\u2113) ,                         (1)",
      "i=1 si !  j=1 qj !",
      "where per(A) = \u03c3\u2208SN i=1 Ai,\u03c3(i) is the matrix permanent (it appears because amplitudes over all in-"
    ],
    "source_type": "paper",
    "summary": "Introduces a programmable linear-optical reservoir with measurement feedback.",
    "title": "A Programmable Linear Optical Quantum Reservoir with Measurement Feedback for Time Series Analysis",
    "url": "https://arxiv.org/abs/2602.17440",
    "year": 2026
  },
  {
    "citation": "Kernel-based optimization of measurement operators for quantum reservoir computers (2026). https://arxiv.org/abs/2602.14677",
    "key_equations": [
      "D         Hilbert space dimension (D = 2N )",
      "Table I. Summary of notation used in this work. \u22c6 For QELMs, x = zt and d = d\u2032 , while for stateful QRCs,",
      "We consider an N -qubit model, with dimension D = 2N of its Hilbert space H, described by the",
      "density matrix is an element of the (D2 = 4N )-dimensional space M(H) of Hermitian operators",
      "f (x) =           wk tr[Mk \u03c1(x)],                             (2.2)",
      "xt := (zt\u2212L , . . . , zt ).                                (2.3)"
    ],
    "source_type": "paper",
    "summary": "Derives kernel-based measurement-operator optimization for QRC readouts.",
    "title": "Kernel-based optimization of measurement operators for quantum reservoir computers",
    "url": "https://arxiv.org/abs/2602.14677",
    "year": 2026
  },
  {
    "citation": "QuaRK: A Quantum Reservoir Kernel for Time Series Learning (2026). https://arxiv.org/abs/2602.13531",
    "key_equations": [
      "via efficient measurement of the quantum reservoir featur-                   X = (\ud835\udc4b\ud835\udc61 : \ud835\udc61 \u2208 Z\u2212 ): \ud835\udc4c0 = \ud835\udc3b \u2605 (X), where the process IO is distributed",
      "tion dimension, reservoir size, multiplexing, measurement                    \ud835\udc61 = 0) and we\u2019d like to make a prediction \ud835\udc4c0 from such historical",
      "\ud835\udc45(\ud835\udc3b ) := E\ud835\udc43 [\u2113 (\ud835\udc3b (X), \ud835\udc4c0 )].                           (1)",
      "We denote by Z\u2212 := {. . . , \u22122, \u22121, 0} the set of non-positive inte-",
      "[\ud835\udc5a] := {1, . . . , \ud835\udc5a}; for any set A, |A| denotes its cardinality. Ran-",
      "in bold roman (e.g. X = (\ud835\udc4b\ud835\udc61 : \ud835\udc61 \u2208 Z\u2212 ), IO = ((\ud835\udc4b\ud835\udc61 , \ud835\udc4c\ud835\udc61 ) : \ud835\udc61 \u2208"
    ],
    "source_type": "paper",
    "summary": "Defines a reservoir-derived kernel for quantum time-series learning.",
    "title": "QuaRK: A Quantum Reservoir Kernel for Time Series Learning",
    "url": "https://arxiv.org/abs/2602.13531",
    "year": 2026
  },
  {
    "citation": "A Quantum Reservoir Computing Approach to Quantum Stock Price Forecasting in Quantum-Invested Markets (2026). https://arxiv.org/abs/2602.13094",
    "key_equations": [
      "\u2206n (u) = \u22060 + ru,",
      "regression and output (see Fig 1). This model is moti-                                    \u2126n (u) = \u21260 + ru,",
      "H\u0302(u) =        \u2212\u2206n (u)\u03c3\u0302n +           \u03c3\u0302n +                       .   tum state evolution (i.e. modulating decoherence or co-",
      "\u03c3\u0302nd = 0.5(1\u0302 \u2212 \u03c3\u0302nz ). \u03c3\u0302nx,y,z are Pauli matrices describing                          = \u2212 [H(t), \u03c1(t)] + L[\u03c1].              (2)",
      "L[\u03c1] =     (Cn \u03c1(t)Cn+ \u2212 0.5[Cn+ Cn , \u03c1(t)]). (3)",
      "Wtrain = \uf8ec ."
    ],
    "source_type": "paper",
    "summary": "Applies QRC methodology to financial forecasting benchmarks.",
    "title": "A Quantum Reservoir Computing Approach to Quantum Stock Price Forecasting in Quantum-Invested Markets",
    "url": "https://arxiv.org/abs/2602.13094",
    "year": 2026
  },
  {
    "citation": "Practical Quantum Reservoir Computing in Rydberg Atom Arrays (2026). https://arxiv.org/abs/2602.00610",
    "key_equations": [
      "multiple local observables. Our results demonstrate a                           atom array system is (\u210f = 1) [23, 27, 28]",
      "sensitive to dynamical phases of matter and decoherence                                   H\u0302 =          \u2206i n\u0302i +        \u03c3\u0302 +      6 n\u0302i n\u0302j ,              (1)",
      "2 i=1 i       Rij",
      "noise due to its single-step nature. Additionally, in the                                        i=1                         i<j",
      "convergence property in MS-QRC leads to a severe dete-                          where N denotes the total number of atoms, n\u0302i = |ri \u27e9\u27e8ri |",
      "suggest that the SS-QRC is a more suitable architecture                         duces the qubit state flip (Pauli X) \u03c3\u0302ix = |ri \u27e9\u27e8gi | + |gi \u27e9\u27e8ri |"
    ],
    "source_type": "paper",
    "summary": "Explores practical implementation constraints for Rydberg-array QRC.",
    "title": "Practical Quantum Reservoir Computing in Rydberg Atom Arrays",
    "url": "https://arxiv.org/abs/2602.00610",
    "year": 2026
  },
  {
    "citation": "Margin-Based Generalisation Bounds for Quantum Kernel Methods under Local Depolarising Noise (2026). https://arxiv.org/abs/2601.23084",
    "key_equations": [
      "line, with the equation w \u00b7 x + b = 0. Here, w is termed          calculated using hypotheses returned as f \u2192  \u2212 w \u00b7 x + b.",
      "hyperplanes, with equations w \u00b7 x + b = \u00b11 are shown",
      "Figure 2: Diagram depicting the optimal linear hyper-                                                 i=1",
      "plane returned by the SVM to separate the two classes             where r = 1 for the hinge loss function, which is widely",
      "perplane but with distances 1 \u2212 \u03bei and 1 \u2212 \u03bej from the            Here, \u03be = (\u03be1 . . . \u03bem )T represents the slack variable",
      "\u03b3 = max                           (1)                          X              1X"
    ],
    "source_type": "paper",
    "summary": "Provides margin-based generalization analysis for noisy quantum kernels.",
    "title": "Margin-Based Generalisation Bounds for Quantum Kernel Methods under Local Depolarising Noise",
    "url": "https://arxiv.org/abs/2601.23084",
    "year": 2026
  },
  {
    "citation": "Practical Evaluation of Quantum Kernel Methods for Radar Micro-Doppler Classification on NISQ Hardware (2026). https://arxiv.org/abs/2601.22194",
    "key_equations": [
      "i=1 , where yi \u2208 {+1, \u22121}, the Support Vector Machine (SVM) solves the following",
      "\u03bei \u2265 0,     i = 1, . . . , m",
      "K(x, x\u2032 ) = \u27e8\u03d5(x), \u03d5(x\u2032 )\u27e9.",
      "K(x, x\u2032 ) = exp \u2212\u03b3\u2225x \u2212 x\u2032 \u22252 ,",
      "|\u03d5(x)\u27e9 = Ux |0n \u27e9.",
      "Kq (x, x\u2032 ) = |\u27e8\u03d5(x)|\u03d5(x\u2032 )\u27e9| ."
    ],
    "source_type": "paper",
    "summary": "Benchmarks quantum kernels on radar micro-Doppler classification.",
    "title": "Practical Evaluation of Quantum Kernel Methods for Radar Micro-Doppler Classification on Noisy Intermediate-Scale Quantum (NISQ) Hardware",
    "url": "https://arxiv.org/abs/2601.22194",
    "year": 2026
  },
  {
    "citation": "Quantum Wiener architecture for quantum reservoir computing (2026). https://arxiv.org/abs/2601.04812",
    "key_equations": [
      "f (x1 ), . . . , f (xn ) follow a multivariate normal distribu-          Then, given the parameters {\u03b1i , \u03c9i }ni=1",
      ".           .       .                E[h(t)h(s)] =        e\u2212\u03b1i (t+s) cos(\u03c9i (t \u2212 s)),    (5)",
      "\u00b5\u03b8 (xn )     K\u03b8 (xn , x1 ) \u00b7 \u00b7 \u00b7 K\u03b8 (xn , xn )                        nc i=1",
      "where \u00b5\u03b8 (x) = E[f (x)] is the mean function and                     for all time instants t, s. Then, in the limit as nc \u2192 \u221e,",
      "K\u03b8 (x, x\u2032 ) = E[(f (x) \u2212 \u00b5(x))(f (x\u2032 ) \u2212 \u00b5(x\u2032 ))] is the co-         E[h(t)h(s)] tends to:",
      "K\u03b8 (t, s) =           \u03d5\u03b8 (\u03b1, \u03c9)e\u2212\u03b1(t+s) cos(\u03c9(t \u2212 s))d\u03b1d\u03c9."
    ],
    "source_type": "paper",
    "summary": "Introduces Wiener-style architectural modeling for QRC.",
    "title": "Quantum Wiener architecture for quantum reservoir computing",
    "url": "https://arxiv.org/abs/2601.04812",
    "year": 2026
  },
  {
    "citation": "Image Denoising via Quantum Reservoir Computing (2025). https://arxiv.org/abs/2512.18612",
    "key_equations": [
      "Let the dataset be denoted by D = {(Xi , Yi )}N",
      "i=1 , a collection of N image pairs. For each",
      "Yi = N (Xi ) = Xi \u2299 (1 + \u03f5i ),                             (1)",
      "increases up to the maximum simulated dimension of d =18 atoms.",
      "For processing, each image is unrolled into a vector, xi = vec(Xi ) \u2208 RM , where M = H \u00d7 W is",
      "zi = Ud\u22a4 (yi \u2212 y\u0304).                                   (2)"
    ],
    "source_type": "paper",
    "summary": "Applies QRC to image denoising tasks.",
    "title": "Image Denoising via Quantum Reservoir Computing",
    "url": "https://arxiv.org/abs/2512.18612",
    "year": 2025
  },
  {
    "citation": "Maritime object classification with SAR imagery using quantum kernel methods (2025). https://arxiv.org/abs/2512.11367",
    "key_equations": [
      "Overall, while quantum approaches have begun to ap-         To formalise this, consider a training dataset D =",
      "i=1 \u2282 X \u00d7 {\u00b11} for some binary classification",
      "The focus has been on accelerating classical processes      task. Here X \u2261 Fd (with F = C or F = R) is the input",
      "K(x, x\u2032 ) = \u27e8\u03d5(x), \u03d5(x\u2032 )\u27e9F ,            (1)",
      "i=1 \u03b1i \u03d5(xi ) where the xi \u2019s are the",
      "imisers of the regularised empirical risk can be written                embedded argument with the vector i=1 \u03b1i \u03d5(xi )."
    ],
    "source_type": "paper",
    "summary": "Uses quantum kernels for SAR maritime object classification.",
    "title": "Maritime object classification with SAR imagery using quantum kernel methods",
    "url": "https://arxiv.org/abs/2512.11367",
    "year": 2025
  },
  {
    "citation": "Entanglement estimation of Werner states with a quantum extreme learning machine (2025). https://arxiv.org/abs/2511.01387",
    "key_equations": [
      "iness of the training process, reservoir computers and ex-                                 xk = f (uk ) .                (1)",
      "\u03f1W (p) =       I + p |\u03c8\u2212 \u27e9 \u27e8\u03c8\u2212 |           (3)",
      "of generalizability in our QELM. We train the network          of input density matrices \u03f1W (pk ) = \u03f1W (pk ) where pk is",
      "formance on a target domain not included in the training       density matrix of the whole system at t = 0",
      "pability to learn and apply its knowledge to unseen data,                        \u03c1k (0) = \u03f1W (pk ) \u2297 Rk                 (4)",
      "\u03c1k (t) = e\u2212iHt \u03c1k (0)eiHt ,             (5)"
    ],
    "source_type": "paper",
    "summary": "Uses QELM to estimate entanglement indicators in Werner states.",
    "title": "Entanglement estimation of Werner states with a quantum extreme learning machine",
    "url": "https://arxiv.org/abs/2511.01387",
    "year": 2025
  },
  {
    "citation": "From quantum feature maps to quantum reservoir computing: perspectives and applications (2025). https://arxiv.org/abs/2510.01797",
    "key_equations": [
      "rt = R(ut , rt\u22121 ; W ) \u2208 RN .                             (1)",
      "rt = rt ({ui }tt\u2212tFM ),                               (2)",
      "yt = Wout rt ,                                     (3)",
      "\u27e8O\u27e9 = \u27e8\u03c8(u)| O |\u03c8(u)\u27e9 ,                                        (5)",
      "time-series, i.e. dim(u) = 3, and values of various colour-encoded observables as a function of time.",
      "is transformed with respect to a single-qubit rotation RZ (\u03b1) = exp(\u2212i\u03b1Z/2), where Z is"
    ],
    "source_type": "paper",
    "summary": "Surveys links between feature-map methods and QRC applications.",
    "title": "From quantum feature maps to quantum reservoir computing: perspectives and applications",
    "url": "https://arxiv.org/abs/2510.01797",
    "year": 2025
  },
  {
    "citation": "Entanglement and Classical Simulability in Quantum Extreme Learning Machines (2025). https://arxiv.org/abs/2509.06873",
    "key_equations": [
      "|\u03c8\u27e9 = cos( ) |0\u27e9 + ei\u03d5 sin( ) |1\u27e9             (1)",
      "In this work, we investigate a Quantum Extreme                 tem, composed of N = \u2308M/2\u2309 qubits for a",
      "Learning Machine (QELM) architecture that inte-                feature vector \u20d7x = [x1 , ..., xM ]T \u2208 RM is given",
      "x\u27e9 =          cos(         ) |0\u27e9 + eix2i sin(       ) |1\u27e9",
      "the proposed model is provided in Fig.1. The key                        i=1",
      "H=         (\u03c3 \u03c3     + \u03c3y(i) \u03c3y(i+1) ).   (3)"
    ],
    "source_type": "paper",
    "summary": "Studies entanglement onset, transition behavior, and simulability limits in QELM image-classification settings.",
    "title": "Entanglement and Classical Simulability in Quantum Extreme Learning Machines",
    "url": "https://arxiv.org/abs/2509.06873",
    "year": 2025
  },
  {
    "citation": "Feedback Connections in Quantum Reservoir Computing with Mid-Circuit Measurements (2025). https://arxiv.org/abs/2503.22380",
    "key_equations": [
      "U (sk , mk\u22121 )        =         R(ain sk )    R(af b m0k\u22121 )     R(af b m1k\u22121 )   UHaar",
      "A regression model is then built using the next ltr steps (the                               C\u03a3 =        R\u03c42                      (7)",
      "= n                \u2212 \u03b3P (t)             (8)",
      "We now create a linear model wopt such that ytr = Xtr wopt       We set \u03b20 = 0.2, \u03b8 = 1, n = 10, \u03c4 = 17 and \u03b3 = 0.1. To",
      "compute wopt :                                                      step of \u2206t = 1. The initial values are set to a constant for the",
      "wopt = (Xtr Xtr )\u22121 Xtr"
    ],
    "source_type": "paper",
    "summary": "Analyzes feedback-enhanced QRC with mid-circuit measurement design.",
    "title": "Feedback Connections in Quantum Reservoir Computing with Mid-Circuit Measurements",
    "url": "https://arxiv.org/abs/2503.22380",
    "year": 2025
  },
  {
    "citation": "Feedback-enhanced quantum reservoir computing with weak measurements (2025). https://arxiv.org/abs/2503.17939",
    "key_equations": [
      "nonlinear function yk = f ({sl }kl=1 ). The protocol con-",
      "state |\u03c80 \u27e9 = |00\u27e9. The density matrix of the whole",
      "Ri,j (\u03b8) = CXi,j RZj (\u03b8)CXi,j RXi (\u03b8)RXj (\u03b8).      (2)",
      "zk\u22121 = \u27e8Z1 \u27e9k\u22121 , \u00b7 \u00b7 \u00b7 , \u27e8ZN \u27e9k\u22121     (3)",
      "models. Notably, our protocol is most effective in sys-                                Ures = exp(\u2212iH\u2206t).                    (4)",
      "H=       Jij Xi Xj + h   Zi ,           (5)"
    ],
    "source_type": "paper",
    "summary": "Examines weak-measurement feedback effects in reservoir dynamics.",
    "title": "Feedback-enhanced quantum reservoir computing with weak measurements",
    "url": "https://arxiv.org/abs/2503.17939",
    "year": 2025
  },
  {
    "citation": "Application of quantum machine learning using quantum kernel algorithms on multiclass neuron M type classification (2025). https://arxiv.org/abs/2502.06281",
    "key_equations": [
      "applies PauliFeatureMap (paulis = [\u2019ZI\u2019,\u2019IZ\u2019,\u2019ZZ\u2019]) with a default data mapping \u03c6S . In addition, we used five encod-",
      "applied to sample 5. The classical algorithm that provided the best score (score = 0.91 \u00b1 0.001) was the SVM",
      "components = 2 running on 2 qbits) was applied to reduce the number of dimensions. The different algorithms",
      "polynomial, and sigmoid using five selected features on all of the samples described in Fig. 1 (sample 1 = 260",
      "neurons, sample 2 = 626, sample 3 = 1143, sample 4 = 2080, sample 5 = 22,691). (c) Five-fold cross-validation",
      "samples described in Fig. 1 (sample 1 = 260 neurons, sample 2 = 626, sample 3 = 1143, sample 4 = 2080, sample"
    ],
    "source_type": "paper",
    "summary": "Applies quantum-kernel methods to multiclass neuronal classification.",
    "title": "Application of quantum machine learning using quantum kernel algorithms on multiclass neuron M type classification",
    "url": "https://arxiv.org/abs/2502.06281",
    "year": 2025
  },
  {
    "citation": "Robust Quantum Reservoir Computing for Molecular Property Prediction (2024). https://arxiv.org/abs/2412.06758",
    "key_equations": [
      "H(t) =           (|gj \u27e9 \u27e8rj | + |rj \u27e9 \u27e8gj |)",
      "dynamics) with maintaining the most relevant molecu-          cited state of an atom (|rj \u27e9), nj = |rj \u27e9 \u27e8rj |, while",
      "lar information. We find that the difference in classical     Vjk = C/\u2225rj \u2212 rk \u22256 describes the interactions between",
      "lished molecular descriptors from the MMACD dataset.          the local detuning pattern, fj , with constant \u2206l =",
      "We then proceed to assign data to clusters of differ-      6 rad/\u00b5s. We set \u2126 = 2\u03c0 rad/\u00b5s, set the atoms in the lin-"
    ],
    "source_type": "paper",
    "summary": "Evaluates QRC robustness for low-data molecular prediction benchmarks.",
    "title": "Robust Quantum Reservoir Computing for Molecular Property Prediction",
    "url": "https://arxiv.org/abs/2412.06758",
    "year": 2024
  },
  {
    "citation": "Harnessing Quantum Extreme Learning Machines for image classification (2024). https://arxiv.org/abs/2409.00998",
    "key_equations": [
      "|\u03c8\u27e9 = cos( ) |0\u27e9 + ei\u03d5 sin( ) |1\u27e9              (1)",
      "|\u20d7x\u27e9 =                                       1 \u2212 x2i\u22121 |1\u27e9",
      "|\u20d7x\u27e9 =             cos(    ) |0\u27e9 + eix2i sin(       ) |1\u27e9                       \u2308M/2\u2309",
      "|\u20d7x\u27e9 =                               (x2i\u22121 |0\u27e9 + x2i |1\u27e9) .",
      "i=1        x22i\u22121 + x22i",
      "vector \u20d7x = [x1 , ..., xM ]T \u2208 RM and N = \u2308M/2\u2309                                                                        (5)"
    ],
    "source_type": "paper",
    "summary": "Compares encoding and reservoir choices for QELM image classification.",
    "title": "Harnessing Quantum Extreme Learning Machines for image classification",
    "url": "https://arxiv.org/abs/2409.00998",
    "year": 2024
  },
  {
    "citation": "Large-scale quantum reservoir learning with an analog quantum computer (2024). https://arxiv.org/abs/2407.02553",
    "key_equations": [
      "H(t) =           (|gj \u27e9 \u27e8rj | + |rj \u27e9 \u27e8gj |)",
      "Rydberg state of an atom (|rj \u27e9), nj = |rj \u27e9 \u27e8rj |, while            site-dependent local detunings with \u03b1i \u2206l (t) = \u2206max l  xi ;",
      "Vjk = C/\u2225rj \u2212 rk \u22256 describes the van der Waals inter-               an Nq -qubit system can thus encode Nq features.",
      "a global detuning pulse, \u2206g (ti ) = \u2206max",
      "(2) Position encoding that proceeds by engineer-                  where Zj = 2nj \u2212Ij . The local observables then form the",
      "with Vil = V (0) (1 + \u03bbxi ), where V (0) represents the bare         [5, 7, 36]. In fact, the reservoir embeddings define a ker-"
    ],
    "source_type": "paper",
    "summary": "Demonstrates analog large-scale QRC and analyzes kernel geometry/observables.",
    "title": "Large-scale quantum reservoir learning with an analog quantum computer",
    "url": "https://arxiv.org/abs/2407.02553",
    "year": 2024
  },
  {
    "citation": "Frequency- and dissipation-dependent entanglement advantage in spin-network Quantum Reservoir Computing (2024). https://arxiv.org/abs/2403.08998",
    "key_equations": [
      "The quantum reservoir we employ is a network of N =",
      "H\u0302 =           Jij \u03c3\u0302ix \u03c3\u0302jx + h         \u03c3\u0302iz ,   (1)",
      "prove accuracy without increasing the memory footprint                              i>j=1                       i=1",
      "and should be considered with caution, as the compar-          where \u03c3\u0302ia (a = x, y, z) are the Pauli operators, h is the",
      "over quantum networks [43], spin-network QRC [44], or                = L\u0302\u03c1\u0302 = \u2212i[H\u0302, \u03c1\u0302] + \u0393      L\u0302i \u03c1\u0302L\u0302i \u2212 {L\u0302i L\u0302i , \u03c1\u0302} ,",
      "oscillator-based QRC [45]. However, there remains much           dt                          i=1"
    ],
    "source_type": "paper",
    "summary": "Investigates entanglement advantages under dissipation/frequency controls.",
    "title": "Frequency- and dissipation-dependent entanglement advantage in spin-network Quantum Reservoir Computing",
    "url": "https://arxiv.org/abs/2403.08998",
    "year": 2024
  },
  {
    "citation": "Quantum reservoir computing with repeated measurements on superconducting devices (2023). https://arxiv.org/abs/2310.06706",
    "key_equations": [
      "superconducting quantum computers, it can be adapted                       \u03c1t t =          Tra Mmt U\u0302 (ut )(\u03c1t\u22121t\u22121 \u2297\u03c3a )U\u0302 \u2020 (ut )Mm",
      "qubits, like silicon.                                                      where \u03c3a = (|0\u27e9a \u27e80|)\u2297n and",
      "In this paper, we provide an experimental demonstra-                                      Mmt = Is \u2297            |mi,t \u27e9a \u27e8mi,t |.      (2)",
      "tion of the proposed QRC scheme on the IBM super-                                                            i=1",
      "memory are required. These experimental studies will                       the bit-string mt = m1,t \u00b7 \u00b7 \u00b7 mn,t is the measurement re-",
      "p(mt ) = Tr(Mmt U\u0302 (ut )(\u03c1t\u22121t\u22121 \u2297 \u03c3a )U\u0302 \u2020 (ut )Mm"
    ],
    "source_type": "paper",
    "summary": "Studies repeated-measurement QRC on superconducting hardware.",
    "title": "Quantum reservoir computing with repeated measurements on superconducting devices",
    "url": "https://arxiv.org/abs/2310.06706",
    "year": 2023
  },
  {
    "citation": "Several fitness functions and entanglement gates in quantum kernel generation (2023). https://arxiv.org/abs/2309.03307",
    "key_equations": [
      "max L(\u03b1) =                    \u03b1i \u2212             \u03b1i \u03b1j yi yj xi \u00b7 xj",
      "2 i=1 j=1",
      "s.t.          \u03b1i yi = 0, \u03b1i \u2265 0",
      "max L(\u03b1) =                  \u03b1i \u2212             \u03b1i \u03b1j yi yj \u03a6(xi ) \u00b7 \u03a6(xj )",
      "2 i=1 j=1",
      "s.t.           \u03b1i yi = 0, \u03b1i \u2265 0"
    ],
    "source_type": "paper",
    "summary": "Explores kernel construction sensitivity to fitness functions and entangling gates.",
    "title": "Several fitness functions and entanglement gates in quantum kernel generation",
    "url": "https://arxiv.org/abs/2309.03307",
    "year": 2023
  },
  {
    "citation": "Quantum Kernel for Image Classification of Real World Manufacturing Defects (2022). https://arxiv.org/abs/2212.08693",
    "key_equations": [
      "Table 1: Quantum Kernel Angle Encoding Results with Image Data (N=60, Train=42, Test=18)",
      "Table 2: Quantum Kernel IQP Encoding Results with Image Data (N=60, Train=42, Test=18)"
    ],
    "source_type": "paper",
    "summary": "Applies quantum-kernel classifiers to real manufacturing-defect images.",
    "title": "Quantum Kernel for Image Classification of Real World Manufacturing Defects",
    "url": "https://arxiv.org/abs/2212.08693",
    "year": 2022
  },
  {
    "citation": "The role of entanglement for enhancing the efficiency of quantum kernels towards classification (2022). https://arxiv.org/abs/2209.05142",
    "key_equations": [
      "comparison to the classical support vector machine. Surpris-                    Accuracy =                                       (1)",
      "P recision(P ) =",
      "[21, 22] for maximum number of features, however in both                            P recision(N ) =                             (2)",
      "In addition to the IMDb review data, we also validate our                          Recall(P ) =",
      "proposed fully entangled circuit as the most efficient circuit                       Recall(N ) =                                (3)",
      "F 1 \u2212 Score = 2 \u2217                                     (4)"
    ],
    "source_type": "paper",
    "summary": "Analyzes how entanglement affects quantum-kernel efficiency.",
    "title": "The role of entanglement for enhancing the efficiency of quantum kernels towards classification",
    "url": "https://arxiv.org/abs/2209.05142",
    "year": 2022
  },
  {
    "citation": "Time Series Quantum Reservoir Computing with Weak and Projective Measurements (2022). https://arxiv.org/abs/2205.06809",
    "key_equations": [
      "ther on repeating (part of) the reservoir dynamics or on          tum system state in discrete time steps, with k = t/\u2206t",
      "mation in the system. In this work, we analyze the effect                                    \u03c1uk = Lk [\u03c1uk\u22121 ]                      (1)",
      "tocol (RSP), is sketched in Fig. 1(b). It was used in the                       \u03c1 Vk =                            \u0001,         (3)",
      "conditions, so that the reservoir computer depends effec-           \u03c1k = dVk \u03c1Vk P (Vk ) = dVk (MVk \u25e6 Lk ) [\u03c1k\u22121 ],",
      "l = 1, ..., Nmeas , allow one to approximate the expec-             In order to account for the measurements during the",
      "proportional to 1/ Nmeas . In other words, the statis-                                   MVk [\u03c1] = \u03a6\u0302Vk \u03c1\u03a6\u0302\u2020Vk .             (5)"
    ],
    "source_type": "paper",
    "summary": "Compares weak vs projective measurements in time-series QRC.",
    "title": "Time Series Quantum Reservoir Computing with Weak and Projective Measurements",
    "url": "https://arxiv.org/abs/2205.06809",
    "year": 2022
  },
  {
    "citation": "Computing with two quantum reservoirs connected via optimized two-qubit nonselective measurements (2022). https://arxiv.org/abs/2201.07969",
    "key_equations": [
      "R p , p = 1, . . . M with similar structures, each consisting of n p qubits. A quantum state of a particular QR at the timestep i is",
      "described by the positive Hermitian density operator \u03c1\u0302 p [i] with the unit trace tr (\u03c1\u0302 p [i]) = 1, \u2200i. In the case of multiple reservoirs,",
      "we assume that all reservoirs are completely uncorrelated at an initial time: \u03c1\u0302[i] = \u2297M       p=1 \u03c1\u0302 p [i]. The unitary evolution dynamics",
      "of each QR is governed by a fully connected Ising-type Hamiltonian H\u0302R p = \u2211 Ji j X\u0302i X\u0302 j + hip Z\u0302i , where X\u0302, Z\u0302 are Pauli operators.",
      "{\u03a6e } : \u03a6e [\u03c1\u0302] = \u03c1\u0302(\u2022 p ) (~s[i]) \u2297 tr(\u2022 p ) (\u03c1\u0302) .                                                                             (1)",
      "Ntotal = \u2211Mp=1 n p to be the total number of qubits in the QR system. We use a random binary signal as an input in the present"
    ],
    "source_type": "paper",
    "summary": "Studies coupled-reservoir architectures with optimized nonselective measurements.",
    "title": "Computing with two quantum reservoirs connected via optimized two-qubit nonselective measurements",
    "url": "https://arxiv.org/abs/2201.07969",
    "year": 2022
  },
  {
    "citation": "Opportunities in Quantum Reservoir Computing and Extreme Learning Machines (2021). https://arxiv.org/abs/2102.11831",
    "key_equations": [
      "xk = f (sk , xk\u22121 ),                (1)",
      "labels each time step or instance. The sub-index i is asso-                                     yk = h(xout",
      "also on past inputs. This is seen recurrently iterating Eq. 1,                                xl = f (sl ),                     (3)",
      "e.g. for three steps xk = f (sk , f (sk\u22121 , f (sk\u22122 , xk\u22123 ))) . As",
      "given as a series of real numbers, {sk }, with k = 0, 1, ... ,        In this section, we will describe recently proposed compu-",
      "\u2206t, so that time is given by t = k\u2206t, whereas for a QELM              whether a given task is classical or quantum. We consider"
    ],
    "source_type": "paper",
    "summary": "Reviews opportunities and open directions for QRC/QELM development.",
    "title": "Opportunities in Quantum Reservoir Computing and Extreme Learning Machines",
    "url": "https://arxiv.org/abs/2102.11831",
    "year": 2021
  },
  {
    "citation": "Schuld (2021). Supervised quantum machine learning models are kernel methods. https://arxiv.org/abs/2101.11020",
    "key_equations": [
      "\u03c1(x) = \u2223\u03c6(x)\u27e9\u27e8\u03c6(x)\u2223 as the feature \u201cvectors\u201d2 instead of the Dirac vectors \u2223\u03c6(x)\u27e9 (see Section V A). This was first",
      "\u03ba(x, x\u2032 ) = tr [\u03c1(x\u2032 )\u03c1(x)], or, for pure states, \u03ba(x, x\u2032 ) = \u2223 \u27e8\u03c6(x\u2032 ) \u2223\u03c6(x) \u27e9 \u22232 (see in particular Ref. [12]). I will call these",
      "space of functions x \u2192 gx (\u22c5) = \u03ba(x, \u22c5), which are constructed from the kernel. The RKHS contains one such function",
      "fopt (x) = \u2211 \u03b1m tr [\u03c1(xm )\u03c1(x)] = tr [( \u2211 \u03b1m \u03c1(xm )) \u03c1(x)] ,                                 (1)",
      "m=1                             m=1",
      "where xm , m = 1, . . . , M is the training data and \u03b1m \u2208 R (see Section VI B). Looking at the expression in the round"
    ],
    "source_type": "paper",
    "summary": "Establishes kernel-method interpretation for broad classes of supervised quantum models.",
    "title": "Supervised quantum machine learning models are kernel methods",
    "url": "https://arxiv.org/abs/2101.11020",
    "year": 2021
  },
  {
    "citation": "The effect of data encoding on the expressive power of variational quantum machine learning models (2020). https://arxiv.org/abs/2008.08605",
    "key_equations": [
      "encode data inputs x = (x1 , . . . , xN ) as well as train-               .",
      "able weights \u03b8 = (\u03b81 , . . . , \u03b8M ). The circuit is measured",
      "body of literature, motivated by the dilemma of inves-            ing of layers of trainable circuit blocks W = W (\u03b8) and data",
      "In this paper we investigate these questions in a                            f\u03b8 (x) =     c\u03c9 (\u03b8)ei\u03c9x ,            (1)",
      "f\u03b8 (x) =      cn (\u03b8)einx ,             (2)    mally distinguish between data classes [27]. A central",
      "tors under a special kind of classical data pre-processing.                f\u03b8 (x) = h0| U \u2020 (x, \u03b8)M U (x, \u03b8) |0i ,           (3)"
    ],
    "source_type": "paper",
    "summary": "Characterizes encoding-induced expressivity via Fourier-spectrum analysis.",
    "title": "The effect of data encoding on the expressive power of variational quantum machine learning models",
    "url": "https://arxiv.org/abs/2008.08605",
    "year": 2020
  },
  {
    "assumptions": [
      "High-dimensional transient dynamics can be linearly decoded."
    ],
    "citation": "Maass, Natschl\u00e4ger, Markram (2002). Real-time computing without stable states. https://doi.org/10.1162/089976602760407955",
    "claims": [
      "Untrained recurrent dynamics with linear readout can solve temporal tasks."
    ],
    "key_equations": [
      "y_k(t)=sum_i w_ki x_i(t)"
    ],
    "source_type": "paper",
    "summary": "Foundational liquid-state computing formulation underlying reservoir-computing principles.",
    "title": "Real-time computing without stable states: A new framework for neural computation based on perturbations",
    "url": "https://doi.org/10.1162/089976602760407955",
    "year": 2002
  },
  {
    "assumptions": [
      "Echo-state property ensures input-driven state stability."
    ],
    "citation": "Jaeger (2001). The Echo State Approach to Analysing and Training Recurrent Neural Networks. https://www.ai.rug.nl/minds/uploads/EchoStatesTechRep.pdf",
    "claims": [
      "Linear readout training over fixed recurrent core is effective."
    ],
    "key_equations": [
      "x(t+1)=f(Wx(t)+W_in u(t+1)+W_fb y(t))"
    ],
    "source_type": "report",
    "summary": "Technical report introducing echo-state networks and training perspective for reservoirs.",
    "title": "The Echo State Approach to Analysing and Training Recurrent Neural Networks",
    "url": "https://www.ai.rug.nl/minds/uploads/EchoStatesTechRep.pdf",
    "year": 2001
  }
]