pith. sign in

arxiv: 2606.04686 · v1 · pith:CXGR6HQQnew · submitted 2026-06-03 · 🪐 quant-ph · cs.CE

Digital Quantum Reservoir Computing for ATM Time Series Prediction

Chiara Vercellino , Giacomo Vitali , Valeria Zaffaroni , Francesca Cibrario , Emanuele Dri , Paolo Viviani , Olivier Terzo , Davide Corbelletto This is my paper

Pith reviewed 2026-06-28 05:53 UTC · model grok-4.3

classification 🪐 quant-ph cs.CE
keywords quantum reservoir computingtime series forecastingATM cash demandnear-term quantum devicesdynamic time warpingridge regression readout
0
0 comments X

The pith

Digital quantum reservoir computing achieves competitive Dynamic Time Warping scores on ATM cash demand forecasts but does not beat the classical Prophet benchmark on MAE or NMSE.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests a digital quantum reservoir computing approach for multi-step prediction of ATM cash withdrawals using four-qubit circuits on near-term hardware. Data enters the circuit through rotation angles, measurements supply features, and only the final classical readout layer is trained with ridge regression. Across noiseless simulation, noisy emulation, and runs on an IQM Spark processor, the quantum models trail the Prophet baseline on mean absolute error and normalized mean squared error yet match or exceed it on dynamic time warping. This pattern is presented as evidence that the quantum reservoirs capture some temporal shape even if point-wise accuracy remains limited. The work supplies an empirical check of whether such circuits can handle realistic financial time series today.

Core claim

Although the QRC models do not outperform the classical Prophet benchmark in terms of Mean Absolute Error and Normalized Mean Squared Error metrics, they achieve more competitive results in Dynamic Time Warping metric, indicating a partial ability to capture temporal structure.

What carries the argument

Parametrized four-qubit reservoirs with fixed structure that use partial measurement and reset, with temporal data encoded in rotation angles and training limited to classical ridge-regression readout.

If this is right

  • Varying the circuit ansatz, reservoir memory length, and choice of measurement observables changes forecasting performance.
  • Performance can be compared directly across noiseless simulation, noise-aware emulation, and execution on real quantum processors.
  • The method supplies an empirical benchmark for digital QRC on realistic financial data rather than toy problems.
  • Current results highlight both limitations and remaining potential on near-term hardware.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If DTW superiority persists under different classical baselines, QRC might suit tasks where preserving the overall shape of demand curves matters more than minimizing per-step error.
  • The fixed four-qubit size and partial-reset design could be tested on other financial or sensor time series to check whether the temporal-capture pattern generalizes.
  • Hardware improvements that reduce noise in the same circuit layout would directly test whether the observed DTW competitiveness can be turned into lower MAE.

Load-bearing premise

That the Prophet model together with the three chosen metrics (MAE, NMSE, DTW) provide a sufficient and unbiased test of whether the quantum reservoir adds value for multi-step ATM forecasting.

What would settle it

Running the same ATM series through additional classical forecasters and operational metrics such as cumulative cash-holding cost would show whether the DTW advantage produces measurable improvements in practice.

Figures

Figures reproduced from arXiv: 2606.04686 by Chiara Vercellino, Davide Corbelletto, Emanuele Dri, Francesca Cibrario, Giacomo Vitali, Olivier Terzo, Paolo Viviani, Valeria Zaffaroni.

Figure 1
Figure 1. Figure 1: Quantum circuit block for a generic t time step of the baseline ansatz. System qubits q0 and q1 store memory across time steps and interact with auxiliary qubits q2 and q3, which are measured and reset at each step. Secondly, we tested the MERA ansatz [3]. In MERA, each layer captures quantum entanglement at a different level, capturing dependencies with a hierarchical organization [20]. In our implementat… view at source ↗
Figure 2
Figure 2. Figure 2: MERA circuit block for a single time step [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two consecutive circuit blocks. Auxiliary qubits ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustrative scheme of the digital QRC. At each time [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance metrics as a function of the prediction horizon ( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance metrics as a function of the QRC memory parameter. The metrics are averaged across the prediction [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Performance metrics as a function of the prediction horizon ( [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of one-step-ahead (h = 1) predictions for the best QRC configuration and the Prophet benchmark. In our case, this constraint prevents us from testing memory values of m ≥ 28 for the MERA ansatz and m ≥ 21 for the baseline ansatz. While assessing whether a multi-output model could be less affected by the accumulation of quantum noise over increasing prediction horizons, we tested the Regressor Ch… view at source ↗
read the original abstract

We investigate a digital quantum reservoir computing (QRC) framework for multi-step forecasting of automated teller machine (ATM) cash demand time series on near-term quantum devices. The proposed approach uses parametrized four-qubit reservoirs with a fixed structure exploiting partial measurement and reset, where temporal data is encoded in rotation angles. Training is restricted to a classical Ridge-regression readout. We systematically analyze the impact of the circuit ansatz\"e, reservoir memory, measurement-derived observables, and the execution backend on the forecasting performance. Experiments are performed with noiseless simulation, noise-aware emulation, and a real IQM Spark quantum processor. Although the QRC models do not outperform the classical Prophet benchmark in terms of Mean Absolute Error and Normalized Mean Squared Error metrics, they achieve more competitive results in Dynamic Time Warping metric, indicating a partial ability to capture temporal structure. These findings provide an empirical assessment of digital QRC for realistic financial forecasting and highlight both its current limitations and its potential on near-term quantum hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a digital quantum reservoir computing (QRC) framework for multi-step ATM cash-demand forecasting using fixed-structure 4-qubit circuits with partial measurement/reset and rotation-angle encoding, trained via classical ridge regression. Systematic experiments on noiseless simulation, noise emulation, and real IQM Spark hardware compare QRC variants against the Prophet benchmark, reporting that QRC does not outperform on MAE or NMSE but achieves more competitive DTW scores, interpreted as evidence of partial temporal-structure capture. The work also examines effects of ansatz, memory length, observables, and backend.

Significance. If the central empirical claim holds after addressing controls, the manuscript supplies one of the first hardware-executed assessments of digital QRC on a realistic financial time-series task, with transparent reporting of mixed metric outcomes and explicit analysis of circuit and execution parameters. This contributes concrete data on near-term device applicability even while underscoring current performance gaps.

major comments (3)
  1. [Abstract] Abstract (final paragraph): The inference that competitive DTW performance indicates 'a partial ability to capture temporal structure' is load-bearing for the headline claim, yet rests solely on comparison to Prophet. No classical reservoir-computing baseline (e.g., echo-state network or liquid-state machine with matched reservoir dimension, linear readout, and memory length) is reported. Without it, any DTW advantage is equally consistent with generic RC properties rather than quantum-specific effects of the parametrized 4-qubit circuit.
  2. [Experiments] Experiments / Methods (implied by systematic analysis of ansatz, memory, observables, backend): All reported variations remain internal to the quantum-reservoir architecture. Because no matched classical RC control is included, changes in performance across these parameters cannot isolate quantum contributions from the broader reservoir-computing framework, undermining attribution of the DTW result to the quantum device.
  3. [Abstract] Abstract and results discussion: The manuscript provides no justification that Prophet plus the three chosen metrics (MAE, NMSE, DTW) constitute a sufficient test for whether the quantum reservoir adds operational value in multi-step ATM forecasting; alternative classical time-series models or domain-specific metrics are not discussed.
minor comments (2)
  1. [Abstract] Abstract: Typo in 'the circuit ansatz"e,' (likely 'ansatz,').
  2. [Methods] The manuscript should explicitly state data-split protocol, number of forecast horizons, and whether statistical significance tests were applied to the metric differences.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below, agreeing where the manuscript requires clarification or tempered claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): The inference that competitive DTW performance indicates 'a partial ability to capture temporal structure' is load-bearing for the headline claim, yet rests solely on comparison to Prophet. No classical reservoir-computing baseline (e.g., echo-state network or liquid-state machine with matched reservoir dimension, linear readout, and memory length) is reported. Without it, any DTW advantage is equally consistent with generic RC properties rather than quantum-specific effects of the parametrized 4-qubit circuit.

    Authors: We agree that the DTW comparison to Prophet alone does not isolate quantum-specific contributions from generic reservoir-computing behavior. The manuscript's primary benchmark was Prophet, a standard classical model for ATM forecasting. In revision we will revise the abstract to remove the phrasing that attributes the DTW result to 'partial ability to capture temporal structure' in a quantum-specific sense and will add an explicit limitations paragraph noting the lack of a matched classical RC control. revision: yes

  2. Referee: [Experiments] Experiments / Methods (implied by systematic analysis of ansatz, memory, observables, backend): All reported variations remain internal to the quantum-reservoir architecture. Because no matched classical RC control is included, changes in performance across these parameters cannot isolate quantum contributions from the broader reservoir-computing framework, undermining attribution of the DTW result to the quantum device.

    Authors: We concur that parameter sweeps confined to the quantum architecture cannot separate quantum effects from classical RC properties. The analysis was designed to characterize the digital QRC implementation on hardware rather than to prove quantum advantage. We will insert a dedicated subsection in Methods and Results that states this limitation and clarifies that observed trends cannot be attributed to the quantum device without classical RC controls. revision: yes

  3. Referee: [Abstract] Abstract and results discussion: The manuscript provides no justification that Prophet plus the three chosen metrics (MAE, NMSE, DTW) constitute a sufficient test for whether the quantum reservoir adds operational value in multi-step ATM forecasting; alternative classical time-series models or domain-specific metrics are not discussed.

    Authors: Prophet was chosen because it is a widely adopted benchmark in the ATM cash-demand literature and handles the seasonality and trend components typical of this domain. MAE, NMSE and DTW are standard metrics for point-wise accuracy and temporal alignment. We will expand the introduction and discussion sections to justify these selections, reference alternative classical models (ARIMA, LSTM), and note that domain-specific financial metrics could be explored in future work. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical comparison with standard readout

full rationale

The paper reports experimental results from running parametrized 4-qubit circuits on ATM cash-demand series, training a classical ridge-regression readout, and comparing MAE/NMSE/DTW against the Prophet benchmark. No mathematical derivation, uniqueness theorem, or first-principles prediction is claimed; performance numbers are obtained by direct execution on simulators and hardware. Ridge regression is an external linear solver whose coefficients are fitted to the reservoir states and are not renamed as a quantum-derived prediction. No self-citation chain supports a load-bearing premise, and no ansatz is smuggled via prior work. The absence of a classical RC baseline is a methodological limitation but does not create circularity in any claimed derivation. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the approach implicitly assumes standard quantum circuit semantics, the validity of ridge regression as readout, and that the chosen observables and memory length capture relevant temporal structure. No explicit free parameters or invented entities are named.

axioms (2)
  • domain assumption Standard quantum circuit evolution and partial measurement semantics hold on the target hardware.
    Required for any digital QRC claim; invoked by the description of rotation-angle encoding and partial measurement/reset.
  • domain assumption Ridge regression on the measured observables is sufficient to extract forecasting performance.
    The paper restricts training to this classical step; the assumption is stated in the abstract.

pith-pipeline@v0.9.1-grok · 5720 in / 1408 out tokens · 34353 ms · 2026-06-28T05:53:12.247641+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 7 canonical work pages · 2 internal anchors

  1. [1]

    Quantum reservoir computing with repeated measurements on superconducting devices,

    T. Yasuda, Y . Suzuki, T. Kubota, K. Nakajima, Q. Gao, W. Zhang, S. Shimono, H. I. Nurdin, and N. Yamamoto, “Quantum reservoir computing with repeated measurements on superconducting devices,” arXiv preprint arXiv:2310.06706, 2023

    work page arXiv 2023

  2. [2]

    Temporal information processing on noisy quantum computers,

    J. Chen, H. I. Nurdin, and N. Yamamoto, “Temporal information processing on noisy quantum computers,”Physical Review Applied, vol. 14, no. 2, p. 024065, 2020

    2020

  3. [3]

    Quantum circuit ansatz: patterns of abstraction and reuse of quantum algorithm design,

    X. Guo, T. Muta, and J. Zhao, “Quantum circuit ansatz: patterns of abstraction and reuse of quantum algorithm design,” in2024 IEEE International Conference on Quantum Software (QSW). IEEE, 2024, pp. 69–80

    2024

  4. [4]

    Large-scale quantum reservoir learning with an analog quantum computer,

    M. Kornja ˇca, H.-Y . Hu, C. Zhao, J. Wurtz, P. Weinberg, M. Hamdan, A. Zhdanov, S. H. Cantu, H. Zhou, R. A. Bravoet al., “Large-scale quantum reservoir learning with an analog quantum computer,”arXiv preprint arXiv:2407.02553, 2024

    work page arXiv 2024

  5. [5]

    Iqm spark: Quantum computing platform,

    IQM Quantum Computers, “Iqm spark: Quantum computing platform,” 2024, accessed: 2026-03-31. [Online]. Available: https://www.meetiqm. com/iqm-spark/

    2024

  6. [6]

    Forecasting at scale,

    S. J. Taylor and B. Letham, “Forecasting at scale,”The American Statistician, vol. 72, no. 1, pp. 37–45, 2018

    2018

  7. [7]

    The “echo state

    H. Jaeger, “The “echo state” approach to analysing and training recurrent neural networks-with an erratum note,”Bonn, Germany: German na- tional research center for information technology gmd technical report, vol. 148, no. 34, p. 13, 2001

    2001

  8. [8]

    Real-time computing without stable states: A new framework for neural computation based on perturbations,

    W. Maass, T. Natschl ¨ager, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,”Neural computation, vol. 14, no. 11, pp. 2531–2560, 2002

    2002

  9. [9]

    Special issue on echo state networks and liquid state machines

    H. Jaeger, W. Maass, and J. Principe, “Special issue on echo state networks and liquid state machines.” 2007

    2007

  10. [10]

    Harnessing disordered-ensemble quantum dynamics for machine learning,

    K. Fujii and K. Nakajima, “Harnessing disordered-ensemble quantum dynamics for machine learning,”Physical Review Applied, vol. 8, no. 2, p. 024030, 2017

    2017

  11. [11]

    Boosting computational power through spatial multiplexing in quantum reservoir computing,

    K. Nakajima, K. Fujii, M. Negoro, K. Mitarai, and M. Kitagawa, “Boosting computational power through spatial multiplexing in quantum reservoir computing,”Physical Review Applied, vol. 11, no. 3, p. 034021, 2019

    2019

  12. [12]

    Quantum reservoir computing for credit card default prediction on a neutral atom platform,

    G. Vitali, C. Vercellino, P. Viviani, O. Terzo, B. Montrucchio, V . Zaf- faroni, F. Cibrario, C. Mattia, G. Ranieri, A. Sabatinoet al., “Quantum reservoir computing for credit card default prediction on a neutral atom platform,”arXiv preprint arXiv:2510.04747, 2025

    work page arXiv 2025

  13. [13]

    Quantum Reservoir Computing for Realized Volatility Forecasting

    Q. Li, C. Mukhopadhyay, A. Bayat, and A. Habibnia, “Quantum reservoir computing for realized volatility forecasting,”arXiv preprint arXiv:2505.13933, 2025. 11

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  14. [14]

    A Quantum Reservoir Computing Approach to Quantum Stock Movement Forecasting in Quantum-Invested Markets

    W. Otieno, A. Zagoskin, A. G. Balanov, J. T. Gongora, and S. E. Savel’ev, “A quantum reservoir computing approach to quantum stock price forecasting in quantum-invested markets,”arXiv preprint arXiv:2602.13094, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  15. [15]

    G. E. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung,Time series analysis: forecasting and control. John Wiley & Sons, 2015

    2015

  16. [16]

    R. J. Hyndman and G. Athanasopoulos,Forecasting: principles and practice. OTexts, 2018

    2018

  17. [17]

    Long short-term memory,

    S. Hochreiter and J. Schmidhuber, “Long short-term memory,”Neural computation, vol. 9, no. 8, pp. 1735–1780, 1997

    1997

  18. [18]

    Greedy function approximation: a gradient boosting machine,

    J. H. Friedman, “Greedy function approximation: a gradient boosting machine,”Annals of statistics, pp. 1189–1232, 2001

    2001

  19. [19]

    Reservoir computing approaches to recurrent neural network training,

    M. Luko ˇseviˇcius and H. Jaeger, “Reservoir computing approaches to recurrent neural network training,”Computer science review, vol. 3, no. 3, pp. 127–149, 2009

    2009

  20. [20]

    Simulation of time evolution with multiscale entanglement renormalization ansatz,

    M. Rizzi, S. Montangero, and G. Vidal, “Simulation of time evolution with multiscale entanglement renormalization ansatz,”Physical Review A—Atomic, Molecular, and Optical Physics, vol. 77, no. 5, p. 052328, 2008

    2008

  21. [21]

    Qiskit ugate api reference,

    IBM Quantum, “Qiskit ugate api reference,” 2026, accessed: 2026- 03-31. [Online]. Available: https://quantum.cloud.ibm.com/docs/en/api/ qiskit/qiskit.circuit.library.UGate

    2026

  22. [22]

    Scikit-learn: Machine learning in python,

    F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V . Dubourget al., “Scikit-learn: Machine learning in python,”the Journal of machine Learning research, vol. 12, pp. 2825–2830, 2011

    2011

  23. [23]

    Machine learning made easy: a review of scikit- learn package in python programming language,

    J. Hao and T. K. Ho, “Machine learning made easy: a review of scikit- learn package in python programming language,”Journal of educational and behavioral statistics, vol. 44, no. 3, pp. 348–361, 2019

    2019

  24. [24]

    Quantumreser- voirpy: A software package for time series prediction,

    S. Miao, O. T. Kulseng, A. Stasik, and F. G. Fuchs, “Quantumreser- voirpy: A software package for time series prediction,”arXiv preprint arXiv:2401.10683, 2024

    work page arXiv 2024

  25. [25]

    Qiskit aer: High performance quantum circuit simulators,

    Qiskit Development Team, “Qiskit aer: High performance quantum circuit simulators,” 2024, accessed: 2026-03-31. [Online]. Available: https://qiskit.org/ecosystem/aer/

    2024

  26. [26]

    Iqm adonis: Superconducting quantum processor,

    IQM Quantum Computers, “Iqm adonis: Superconducting quantum processor,” 2025, accessed: 2026-03-31. [Online]. Available: https:// iqm-finland.github.io/cirq-on-iqm/api/iqm.cirq iqm.devices.adonis.html

    2025

  27. [27]

    Noise-robust estimation of quantum observables in noisy hardware,

    A. Hosseinkhaniet al., “Noise-robust estimation of quantum observables in noisy hardware,”arXiv preprint arXiv:2503.06695, 2025

    work page arXiv 2025

  28. [28]

    Dynamic time warping algorithm review,

    P. Senin, “Dynamic time warping algorithm review,”Information and Computer Science Department University of Hawaii at Manoa Honolulu, USA, vol. 855, no. 1-23, p. 40, 2008

    2008

  29. [29]

    Tslearn, a machine learning toolkit for time series data,

    R. Tavenard, J. Faouzi, G. Vandewiele, F. Divo, G. Androz, C. Holtz, M. Payne, R. Yurchak, M. Rußwurm, K. Kolar, and E. Woods, “Tslearn, a machine learning toolkit for time series data,”Journal of Machine Learning Research, vol. 21, no. 118, pp. 1–6, 2020. [Online]. Available: http://jmlr.org/papers/v21/20-091.html

    2020

  30. [30]

    Taking advantage of noise in quantum reservoir computing,

    L. Domingo, G. Carlo, and F. Borondo, “Taking advantage of noise in quantum reservoir computing,”Scientific Reports, vol. 13, no. 1, p. 8790, 2023

    2023