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Non-Markovianity and memory enhancement in Quantum Reservoir Computing

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Markovian quantum reservoir computers are proven to lose past-input memory exponentially; switching to non-Markovian dynamics revives it and improves forecasting.

desk verdict A genuinely interesting non-Markovian reservoir construction, but the paper's central memory theorem is not proven as written—the capacity bound does not follow from the Volterra coefficient decay. read the letter →

arxiv 2505.02491 v2 pith:W5633IRQ submitted 2025-05-05 quant-ph

Antonio Sannia , Ricard Ravell Rodríguez , Gian Luca Giorgi , Roberta Zambrini This is my paper · ORCID
classification quant-ph
keywords quantumreservoircomputingnon-MarkoviandynamicsmemorycapacityVolterraseriesfadingechostatepropertychaotictime-seriesforecastingopensystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper targets a fundamental limit of quantum reservoir computers: any reservoir whose update is Markovian—the rule used by every architecture proposed so far—loses information about past inputs exponentially with the delay. The authors prove that the Volterra coefficients of a Markovian reservoir output scale as $O(e^{-a\tau_n})$ and that the capacity to reconstruct a target depending on an input $\tau$ steps back scales as $O(e^{-a\tau})$. They then show that non-Markovian update rules, in which the new reservoir state also depends on an older state $\rho_{k-\tau_E}$, revive the memory at the delay $\tau_E$ and can outperform Markovian reservoirs on tasks that need both short- and long-term correlations. A concrete embedding scheme using auxiliary qubits and partial swaps gives a tunable knob between Markovian and non-Markovian operation, and the simulations show intermediate non-Markovianity gives the best chaotic forecasting.

What carries the argument

The machine doing the work is the decomposition of the Markovian propagator into a stationary-state projector and a contraction: $e^{\mathcal{L}(s)\Delta t}=S(s)+T(s)$, with $S(s)S(w)=S(s)$ and $S(s)T(w)=0$. That identity forces every term in the Volterra expansion that reaches back $\tau_n$ steps to contain $\tau_n$ factors of $T$, each bounded by $\max_s\|T(s)\|<1$, hence the exponential bound. On the non-Markovian side, the load-bearing object is the residual update rule $\rho_{k+1}=e^{\mathcal{L}(s_{k+1})\Delta t}[\lambda\rho_k+(1-\lambda)\rho_{k-\tau_E}]$, a quantum analogue of residual and dilated skip connections, with $\lambda$ controlling how much stale state $\rho_{k-\tau_E}$ is mixed back; the embedding variant uses partial swaps with auxiliary qubits and depolarizing noise $\Omega$ to realize the same effect in a larger Markovian model, giving a physical tuning knob for non-Markovianity.

What would settle it

Measure the short-term memory capacity $C[\hat{y}(\tau)]$ of any Markovian reservoir satisfying the paper's assumptions (unique stationary state and $\|T(s)\|<1$) out to large delays $\tau$; if the capacity decays polynomially or stays flat instead of exponentially, the bound in Eq. (6) is false. A second check is to exhibit a Markovian reservoir from the published literature that violates the unique-stationary-state assumption yet still shows exponential decay, which would narrow the claimed universality.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that memory in quantum reservoir computing is not a fixed resource but is controlled by the dynamical character of the reservoir. For a Markovian reservoir with update rule $\rho_{k+1}=e^{\mathcal{L}(s_{k+1})\Delta t}\rho_k$, the evolution map splits into a stationary-state projector $S(s)$ and a contracting part $T(s)$ with $\|T(s)\|<1$; because $S(s)T(w)=0$, any memory term involving an input $\tau_n$ steps in the past must carry a product of $T$'s, forcing $|f_n(\ldots)|=O(e^{-a\tau_n})$ and, for delay-$\tau$ targets, capacity $C[\hat{y}(\tau)]=O(e^{-a\tau})$. Replacing the update by a non-Markovian rule such as $\rho_{k+1}=e^{\mathcal{L}(s_{k+1})\Delta t}[\lambda\rho_k+(1-\lambda)\rho_{k-\tau_E}]$ introduces a controlled revival of old states at delay $\tau_E$, so information that would be untraceable in the Markovian window becomes retrievable again. The same tunability is realized in an embedding with auxiliary qubits (partial swap followed by depolarizing noise of strength $\Omega$), and the numerics show capacity revivals and superior chaotic forecasting at intermediate $\Omega$.

Load-bearing premise

The memory-decay proof assumes that every input maps the reservoir to a unique stationary state and that the update map is strictly contracting with fading memory, so that the Volterra expansion converges; reservoir models with multiple stationary states or non-contracting updates are outside the proof.

Editorial extensions

If this is right

  • For any Markovian quantum reservoir satisfying a unique stationary state and fading memory, the paper's bound makes exponential decay of long-term memory unavoidable; no amount of Hamiltonian or dissipation tuning within the Markovian class can produce a polynomial or flat memory profile.
  • The Volterra and capacity bounds apply also when inputs are quantum states, so the limitation is not an artifact of classical encoding.
  • Non-Markovian updates with delay $\tau_E$ create a memory revival near $\tau_E$, giving reservoirs the coexistence of fresh and distant past information that tasks requiring both can exploit.
  • The proposed embedding with auxiliary qubits gives an experimentally accessible control parameter $\Omega$ interpolating from Markovian ($\Omega=1$) to non-Markovian ($\Omega\to0$), with optimal forecasting at intermediate values.
  • In autonomous forecasting of the chaotic benchmark series, the non-Markovian reservoir at $\Omega=0.5$ outperforms the Markovian one, while maximum non-Markovianity ($\Omega=0$) degrades performance, meaning the resource must be tuned, not maximized.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the proof's engine—the $S(s)T(w)=0$ orthogonality—does not use the details of the Lindblad structure, so the same exponential-memory wall should apply to any input-driven contracting map with a unique fixed point, quantum or classical; that would make the Markovian memory decay a general property of contractive reservoir dynamics.
  • Beyond the paper: the $\tau_E$-delayed state mixing is formally the quantum analogue of dilated skip connections; choosing $\tau_E$ and $\lambda$ to match a task's correlation timescales could turn non-Markovianity into a design parameter for long-range temporal tasks.
  • Beyond the paper: a clean experimental test of the revival mechanism would measure the capacity $C[\hat{y}(\tau)]$ for the embedded model as a function of $\Omega$, predicting that the revival peak at $\tau=\tau_E$ grows as $\Omega\to0$ while the short-delay capacity shrinks; the authors show this trend but do not derive its scaling.
  • Beyond the paper: the contrast between the autonomous forecasting task (gain from non-Markovianity) and the externally driven benchmark (no gain) suggests that non-Markovian memory matters most in closed-loop operation, where prediction errors feed back as inputs; quantifying this regime would connect quantum reservoir memory to control and forecasting theory.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims a general theorem that Markovian quantum reservoir computers (QRC) have exponentially decaying memory capacity, and argues that quantum non-Markovianity can overcome this limitation. It introduces a 'Quantum Residual Reservoir' with an explicit delayed-state recurrence and an embedding scheme based on partial swaps and depolarizing channels, and supports the claims with numerical experiments on short-term memory tasks and Mackey-Glass chaotic time-series forecasting.

Significance. If the central memory bound were established, the paper would be an important contribution: it points to a largely underexplored design axis for QRC and gives a concrete, tunable embedding that interpolates between Markovian and non-Markovian operation. The numerical work is substantial, with 100 random realizations, washout/training/test protocol, and multiple benchmarks. However, the main analytical claim is currently under-supported, so the significance is conditional on a repair or careful restatement of the theorem.

major comments (3)
  1. [Memory bounds of Markovian reservoirs, Eq. (6)] The inference from Eq. (4) to Eq. (6) is not valid as written. Eq. (4) bounds the absolute value of each Volterra kernel f_n, whereas Eq. (5) defines capacity as a normalized squared error after an unconstrained linear readout. The optimal readout can rescale features by arbitrary factors, so an exponentially small feature can be amplified to achieve unit capacity. Concretely, if a reservoir observable has expectation e^{-aτ}s_{k-τ}, the linear readout can choose a weight of order e^{aτ} and obtain C=1 for the target s_{k-τ} for every τ, consistently with |f_1(τ)|=O(e^{-aτ}). Thus the claimed exponential decay of memory capacity is not established; a proof would need to bound the projection of the target onto the readout feature space, or impose a bound on the readout weights, a noise model, or regularization. Since this is the paper's central theoretical result, it must be fixed or explicitly scoped down.
  2. [Quantum Residual Reservoir, Eq. (8)] The memory revival at τ=τ_E is put into the evolution rule by hand. The recurrence ρ_{k+1}=e^{L(s_{k+1})Δt}[λρ_k+(1-λ)ρ_{k-τ_E}] contains an explicit ρ_{k-τ_E} term, so the numerical revival in Fig. 2(a) is a direct consequence of the recursion rather than an emergent prediction of non-Markovian dynamics. The manuscript should present Eq. (8) as a designed or toy model, and the analysis of how non-Markovianity enhances memory should be tied to the embedded model, where the memory of the auxiliary system arises from the partial-swap interaction.
  3. [Introduction and Supplemental Sec. I] The statement that the result applies to 'all reservoir models proposed so far' is broader than the assumptions used in the proof. The derivation in the Supplemental assumes a unique stationary state for every input, contractivity ∥T(s)∥<1, and a fading-memory property guaranteeing convergence of the Volterra expansion. These assumptions are asserted, not verified for the published QRC models cited in the paper. Please either verify the assumptions for the cited models or explicitly scope the theorem to the class of Markovian QRCs satisfying those assumptions.
minor comments (4)
  1. [Eq. (8) and Fig. 2] Eq. (8) states λ∈[0,1), but Fig. 2(a) labels the curve λ=1 as the Markovian case; please clarify how the boundary value is treated.
  2. [Supplemental Eq. (S4)] The formula for the non-Markovianity measure appears to contain a typo: D(ρ_{k,1},ρ_{k,1}) should presumably be D(ρ_{k,1},ρ_{k,2}).
  3. [Fig. 3 caption] The phrase 'in descending order with respect to Ω' is ambiguous; the listed MSE values are 1.8×10⁻², 6.8×10⁻³, 2×10⁻² for Ω=1, 0.5, 0, so 'in order of decreasing Ω' would be clearer.
  4. [Throughout] There are several typos: 'Sante F e' should be 'Santa Fe', and 'non-Markocian' in the reference to the Supplemental material should be 'non-Markovian'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: the Markovian memory bound is derived self-containedly, and the caveats about Eq. (6) and the constructed τ_E revival are correctness-level issues rather than circular reductions.

full rationale

The central Markovian memory bound is derived in the Supplemental Material (Sec. I) directly from the GKLS generator's spectral decomposition, Eqs. (S1)-(S3), with the decay constant e^{-a}=max_s ||T(s)||. No fitted value or target quantity enters the proof, and the argument is self-contained apart from the standard echo-state/contraction condition cited from prior published work. The non-Markovian examples are presented explicitly as constructions: Eq. (8) contains a literal ρ_{k-τ_E} term, so the revival at τ=τ_E is built into the update rule rather than derived from a general principle; the paper presents it as an illustrative example and additionally provides an embedded model whose non-Markovianity is numerically quantified, so the central conclusion does not rest on a circular equivalence. The main caveat is that Eq. (6) is asserted as a direct combination of Eqs. (4) and (5) without showing that the optimal linear readout in Eq. (5) cannot rescale an exponentially small Volterra coefficient; this is a missing mathematical step, not a self-definitional or fitted-input circularity. Self-citations [18,25] are used for background conditions such as the echo state property and fading memory, and they are not the load-bearing derivation. Accordingly, no circular step is identified; score 2 reflects only minor non-load-bearing self-citation, not actual circularity.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard reservoir-computing assumptions such as the echo state property, fading memory, and Volterra series convergence, plus hand-chosen numerical hyperparameters. No new physical particles, mediators, forces, or dimensions are postulated.

free parameters (5)
  • lambda (residual mixing weight) = 0.1, 0.5, 1 (hand-chosen)
    Controls the convex combination of current and delayed reservoir state in Eq. (8); set by hand to sweep from Markovian to non-Markovian behavior, with the revival strength increasing as lambda decreases.
  • tau_E (environment memory timescale) = 10 (hand-chosen)
    Delay of the non-Markovian feedback term in Eq. (8); the capacity revival is positioned at tau = tau_E by construction in the residual model.
  • eta (partial swap angle) = pi/4 (hand-chosen)
    Interaction strength between reservoir and auxiliary qubits in the embedded model; fixed for all numerical tasks.
  • Omega (depolarizing strength) = 0, 0.5, 1 (hand-chosen)
    Tunes the degree of non-Markovianity in the embedded model; Omega=1 is Markovian, smaller values are more non-Markovian, and the best Mackey-Glass forecasting is found at Omega=0.5.
  • Reservoir hyperparameters h, gamma, dt, N = h=1, gamma=0.1, dt=10 or 0.5, N=3 or 2N=8
    Fixed simulation hyperparameters for the magnetic field, dissipation rate, evolution time, and system size; no optimization or fitting is reported.
assumptions (3)
  • domain assumption Each reservoir input s_k maps to a Liouvillian L(s_k) with a unique stationary state, and the echo state property holds for every input sequence.
    Used to identify the stationary-state projector S(s) and to require contractivity of T(s), which yields the exponential bound in Eq. (4). This is not proven for all QRC architectures, so the blanket claim about all existing architectures is narrower than stated.
  • domain assumption The reservoir output admits a converging Volterra series expansion under the fading memory property.
    Invoked in Eq. (3) following Boyd and Chua; requires continuity of L(s) in the input and fading memory, conditions stated but not verified for the examples.
  • standard math The GKLS master equation is the appropriate generator for every Markovian QRC reservoir studied.
    The Markovian bound is formulated only for GKLS-type Liouvillians of the form e^{LΔt}, excluding other Markovian discrete-time quantum channel descriptions that are not generated by a GKLS master equation.

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Pith. "Pith review of Non-Markovianity and memory enhancement in Quantum Reservoir Computing." pith.science (2026). https://pith.science/paper/W5633IRQ

@misc{pith2026250502491,
  author       = {Pith},
  title        = {Pith review of: Non-Markovianity and memory enhancement in Quantum Reservoir Computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5633IRQ}},
  note         = {Machine review of arXiv:2505.02491}
}
read the original abstract

Featuring memory of past inputs is a fundamental requirement for machine learning models processing time-dependent data. In quantum reservoir computing, all architectures proposed so far rely on Markovian dynamics, which, as we prove, inherently lead to an exponential decay of past information, thereby limiting long-term memory capabilities. We demonstrate that non-Markovian dynamics can overcome this limitation, enabling extended memory retention. By analytically deriving memory bounds and supporting our findings with numerical simulations, we show that non-Markovian reservoirs can outperform their Markovian counterparts, particularly in tasks that require a coexistence of short- and long-term correlations. We introduce an embedding approach that allows a controlled transition from Markovian to non-Markovian evolution, providing a path for practical implementations. Our results establish quantum non-Markovianity as a key resource for enhancing memory in quantum machine learning architectures, with broad implications in quantum neural networks.

Figures

Figures reproduced from arXiv: 2505.02491 by the authors.

Figure 1
Figure 1. FIG. 1. Scheme of a reservoir evolution (a). In the Markovian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quantum Residual Reservoir performance analysis. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Reservoir predictions of the Mackey-Glass time-series [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reference graph

Works this paper leans on

71 extracted references · 58 canonical work pages · cited by 3 Pith papers

  1. [1]

    Lim and S

    B. Lim and S. Zohren, Time-series forecasting with deep learning: a survey, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences 379, 20200209 (2021)

    work page 2021

  2. [2]

    Hochreiter and J

    S. Hochreiter and J. Schmidhuber, Long short-term mem- ory, Neural Computation 9, 1735 (1997)

    1997

  3. [3]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, Attention is all you need, arxiv:1706.03762 (2023)

    arXiv 2023

  4. [4]

    Desislavov, F

    R. Desislavov, F. Mart´ ınez-Plumed, and J. Hern´ andez- Orallo, Trends in AI inference energy consumption: Be- yond the performance-vs-parameter laws of deep learn- ing, Sustainable Computing: Informatics and Systems 38, 100857 (2023)

    work page 2023

  5. [5]

    Samsi, D

    S. Samsi, D. Zhao, J. McDonald, B. Li, A. Michaleas, M. Jones, W. Bergeron, J. Kepner, D. Tiwari, and V. Gadepally, in 2023 IEEE High Performance Extreme Computing Conference (HPEC) (IEEE, 2023) p. 1–9

    work page 2023

  6. [6]

    L. F. W. Anthony, B. Kanding, and R. Selvan, Carbon- tracker: Tracking and predicting the carbon footprint of training deep learning models, arXiv:2007.03051 (2020)

    arXiv 2020

  7. [7]

    J. Park, M. Naumov, P. Basu, S. Deng, A. Kalaiah, D. Khudia, J. Law, P. Malani, A. Malevich, S. Nadathur, J. Pino, M. Schatz, A. Sidorov, V. Sivakumar, A. Tul- loch, X. Wang, Y. Wu, H. Yuen, U. Diril, D. Dzhulgakov, K. Hazelwood, B. Jia, Y. Jia, L. Qiao, V. Rao, N. Rotem, S. Yoo, and M. Smelyanskiy, Deep learning inference in facebook data centers: Charac...

    arXiv 2018

  8. [8]

    Markovi´ c, A

    D. Markovi´ c, A. Mizrahi, D. Querlioz, and J. Grollier, Physics for neuromorphic computing, Nature Reviews Physics 2, 499–510 (2020). 6

    work page 2020

Show all 71 references
  1. [9]

    L. G. Wright, T. Onodera, M. M. Stein, T. Wang, D. T. Schachter, Z. Hu, and P. L. McMahon, Deep physi- cal neural networks trained with backpropagation, Nature 601, 549–555 (2022)

    2022

  2. [10]

    M. Hu, C. E. Graves, C. Li, Y. Li, N. Ge, E. Montgomery, N. Davila, H. Jiang, R. S. Williams, J. J. Yang, Q. Xia, and J. P. Strachan, Memristor-based analog computation and neural network classification with a dot product en- gine, Advanced Materials 30 (2018)

    2018

  3. [11]

    Nakajima, Physical reservoir computing—an introduc- tory perspective, Japanese Journal of Applied Physics 59, 060501 (2020)

    K. Nakajima, Physical reservoir computing—an introduc- tory perspective, Japanese Journal of Applied Physics 59, 060501 (2020)

    2020

  4. [12]

    I. F. Kohei Nakajima, Reservoir Computing: Theory, Physical Implementations, and Applications (Springer Singapore, 2021)

    2021

  5. [13]

    Cucchi, S

    M. Cucchi, S. Abreu, G. Ciccone, D. Brunner, and H. Kleemann, Hands-on reservoir computing: a tutorial for practical implementation, Neuromorphic Computing and Engineering 2, 032002 (2022)

    2022

  6. [14]

    Fujii and K

    K. Fujii and K. Nakajima, Harnessing disordered- ensemble quantum dynamics for machine learning, Phys- ical Review Applied 8 (2017)

    2017

  7. [15]

    Mujal, R

    P. Mujal, R. Mart´ ınez-Pe˜ na, J. Nokkala, J. Garc´ ıa-Beni, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Op- portunities in quantum reservoir computing and extreme learning machines, Advanced Quantum Technologies 4 (2021)

    2021

  8. [16]

    Mart´ ınez-Pe˜ na, G

    R. Mart´ ınez-Pe˜ na, G. L. Giorgi, J. Nokkala, M. C. So- riano, and R. Zambrini, Dynamical phase transitions in quantum reservoir computing, Phys. Rev. Lett. 127 (2021)

    2021

  9. [17]

    Kobayashi and Y

    K. Kobayashi and Y. Motome, Quantum reservoir probing of quantum phase transitions, arXiv:2402.07097 (2024)

    2024 arXiv

  10. [18]

    Sannia, R

    A. Sannia, R. Mart´ ınez-Pe˜ na, M. C. Soriano, G. L. Giorgi, and R. Zambrini, Dissipation as a resource for quantum reservoir computing, Quantum 8, 1291 (2024)

    2024

  11. [19]

    G¨ otting, S

    N. G¨ otting, S. Wilksen, A. Steinhoff, F. Lohof, and C. Gies, Connection between memory performance and optical absorption in quantum reservoir computing, arXiv:2501.15580 (2025)

    2025 arXiv

  12. [20]

    Cheamsawat and T

    K. Cheamsawat and T. Chotibut, Dissipation alters modes of information encoding in small quantum reser- voirs near criticality, Entropy 27, 88 (2025)

    2025

  13. [21]

    Llodr` a, P

    G. Llodr` a, P. Mujal, R. Zambrini, and G. L. Giorgi, Quantum reservoir computing in atomic lattices, Chaos, Solitons & Fractals 195, 116289 (2025)

    2025

  14. [22]

    Garc´ ıa-Beni, G

    J. Garc´ ıa-Beni, G. Luca Giorgi, M. C. Soriano, and R. Zambrini, Squeezing as a resource for time series pro- cessing in quantum reservoir computing, Optics Express 32, 6733 (2024)

    2024

  15. [23]

    Sannia, G

    A. Sannia, G. L. Giorgi, S. Longhi, and R. Zambrini, Li- ouvillian skin effect in quantum neural networks, Optica Quantum 3, 189 (2025)

    2025

  16. [24]

    G¨ otting, F

    N. G¨ otting, F. Lohof, and C. Gies, Exploring quantum- ness in quantum reservoir computing, Physical Review A 108 (2023)

    2023

  17. [25]

    Mujal, R

    P. Mujal, R. Mart´ ınez-Pe˜ na, G. L. Giorgi, M. C. Sori- ano, and R. Zambrini, Time-series quantum reservoir computing with weak and projective measurements, npj Quantum Information 9 (2023)

    2023

  18. [26]

    Franceschetto, M

    G. Franceschetto, M. P lodzie´ n, M. Lewenstein, A. Ac´ ın, and P. Mujal, Harnessing quantum back-action for time- series processing, arXiv:2411.03979 (2024)

    2024 arXiv

  19. [27]

    Palacios, R

    A. Palacios, R. Mart´ ınez-Pe˜ na, M. C. Soriano, G. L. Giorgi, and R. Zambrini, Role of coherence in many-body quantum reservoir computing, Communications Physics 7 (2024)

    2024

  20. [28]

    F. Hu, S. A. Khan, N. T. Bronn, G. Angelatos, G. E. Rowlands, G. J. Ribeill, and H. E. T¨ ureci, Overcoming the coherence time barrier in quantum machine learning on temporal data, Nature Communications 15 (2024)

    2024

  21. [29]

    Breuer, E.-M

    H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Reviews of Modern Physics 88 (2016)

    2016

  22. [30]

    Rivas and S

    A. Rivas and S. F. Huelga, Open quantum systems , Vol. 10 (Springer, 2012)

    2012

  23. [31]

    De Vega and D

    I. De Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Reviews of Modern Physics 89, 015001 (2017)

    2017

  24. [32]

    Chru´ sci´ nski,Dynamical maps beyond Markovian regime, Physics Reports 992, 1 (2022)

    D. Chru´ sci´ nski,Dynamical maps beyond Markovian regime, Physics Reports 992, 1 (2022)

    2022

  25. [33]

    B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, Experimental control of the transition from Markovian to non-Markovian dynam- ics of open quantum systems, Nature Physics 7, 931–934 (2011)

    2011

  26. [34]

    Breuer, E.-M

    H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the degree of non-Markovian behavior of quantum processes in open systems, Phys. Rev. Lett. 103, 210401 (2009)

    2009

  27. [35]

    Xu, C.-F

    J.-S. Xu, C.-F. Li, M. Gong, X.-B. Zou, C.-H. Shi, G. Chen, and G.-C. Guo, Experimental demonstration of photonic entanglement collapse and revival, Phys. Rev. Lett. 104, 100502 (2010)

    2010

  28. [36]

    Wang, Z.-Y

    Y. Wang, Z.-Y. Hao, J.-K. Li, Z.-H. Liu, K. Sun, J.- S. Xu, C.-F. Li, and G.-C. Guo, Observation of non- Markovian evolution of Einstein-Podolsky-Rosen steer- ing, Phys. Rev. Lett. 130, 200202 (2023)

    2023

  29. [37]

    Buscemi, R

    F. Buscemi, R. Gangwar, K. Goswami, H. Badhani, T. Pandit, B. Mohan, S. Das, and M. N. Bera, Causal and noncausal revivals of information: A new regime of non-Markovianity in quantum stochastic processes, PRX Quantum 6, 020316 (2025)

    2025

  30. [38]

    A. W. Chin, S. F. Huelga, and M. B. Plenio, Quantum metrology in non-Markovian environments, Phys. Rev. Lett. 109, 233601 (2012)

    2012

  31. [39]

    D. M. Reich, N. Katz, and C. P. Koch, Exploiting non- Markovianity for quantum control, Scientific Reports 5 (2015)

    2015

  32. [40]

    Laine, H.-P

    E.-M. Laine, H.-P. Breuer, and J. Piilo, Nonlocal mem- ory effects allow perfect teleportation with mixed states, Scientific Reports 4 (2014)

    2014

  33. [41]

    Bylicka, D

    B. Bylicka, D. Chru´ sci´ nski, and S. Maniscalco, Non- Markovianity and reservoir memory of quantum chan- nels: a quantum information theory perspective, Scien- tific Reports 4 (2014)

    2014

  34. [42]

    Y. Dong, Y. Zheng, S. Li, C.-C. Li, X.-D. Chen, G.-C. Guo, and F.-W. Sun, Non-Markovianity-assisted high- fidelity Deutsch–Jozsa algorithm in diamond, npj Quan- tum Information 4 (2018)

    2018

  35. [43]

    Thorwart, J

    M. Thorwart, J. Eckel, J. Reina, P. Nalbach, and S. Weiss, Enhanced quantum entanglement in the non- Markovian dynamics of biomolecular excitons, Chemical Physics Letters 478, 234–237 (2009)

    2009

  36. [44]

    Karpat, I

    G. Karpat, I. Yal¸ cinkaya, B. C ¸ akmak, G. L. Giorgi, and R. Zambrini, Synchronization and non-Markovianity in open quantum systems, Physical Review A 103 (2021). 7

    2021

  37. [45]

    M. A. C. Rossi, M. Cattaneo, M. G. A. Paris, and S. Maniscalco, Non-Markovianity is not a resource for quantum spatial search on a star graph subject to general- ized percolation, Quantum Measurements and Quantum Metrology 5, 40–49 (2018)

    2018

  38. [46]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n-level sys- tems, Journal of Mathematical Physics 17, 821 (1976)

    1976

  39. [47]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

    1976

  40. [48]

    See supplemental material for the proofs of the analytical results presented and for further details on the numerical methods and on the embedded non-Markocian model

  41. [49]

    I. B. Yildiz, H. Jaeger, and S. J. Kiebel, Re-visiting the echo state property, Neural Networks 35, 1–9 (2012)

    2012

  42. [50]

    Boyd and L

    S. Boyd and L. Chua, Fading memory and the problem of approximating nonlinear operators with Volterra se- ries, IEEE Transactions on Circuits and Systems 32, 1150–1161 (1985)

    1985

  43. [51]

    Dambre, D

    J. Dambre, D. Verstraeten, B. Schrauwen, and S. Mas- sar, Information processing capacity of dynamical sys- tems, Scientific Reports 2 (2012)

    2012

  44. [52]

    Ceni and C

    A. Ceni and C. Gallicchio, Residual echo state networks: Residual recurrent neural networks with stable dynamics and fast learning, Neurocomputing 597, 127966 (2024)

    2024

  45. [53]

    Chang, Y

    S. Chang, Y. Zhang, W. Han, M. Yu, X. Guo, W. Tan, X. Cui, M. Witbrock, M. Hasegawa-Johnson, and T. S. Huang, Dilated recurrent neural networks, arXiv:1710.02224 (2017)

    2017 arXiv

  46. [54]

    Fette and J

    G. Fette and J. Eggert, Short term memory and pattern matching with simple echo state networks, in Artificial Neural Networks: Biological Inspirations – ICANN 2005 (Springer Berlin Heidelberg, 2005) p. 13–18

    2005

  47. [55]

    S. Bay, P. Lambropoulos, and K. Mølmer, Atom-atom interaction in strongly modified reservoirs, Physical Re- view A 55, 1485–1496 (1997)

    1997

  48. [56]

    Imamo˘ glu,Stochastic wave-function approach to non- Markovian systems, Physical Review A 50, 3650–3653 (1994)

    A. Imamo˘ glu,Stochastic wave-function approach to non- Markovian systems, Physical Review A 50, 3650–3653 (1994)

    1994

  49. [57]

    B. M. Garraway, Nonperturbative decay of an atomic sys- tem in a cavity, Physical Review A 55, 2290–2303 (1997)

    1997

  50. [58]

    B. M. Garraway and P. L. Knight, Cavity modified quan- tum beats, Physical Review A 54, 3592–3602 (1996)

    1996

  51. [59]

    Breuer, Genuine quantum trajectories for non- Markovian processes, Physical Review A 70 (2004)

    H.-P. Breuer, Genuine quantum trajectories for non- Markovian processes, Physical Review A 70 (2004)

    2004

  52. [60]

    Arrigoni, M

    E. Arrigoni, M. Knap, and W. von der Linden, Nonequi- librium dynamical mean-field theory: An auxiliary quan- tum master equation approach, Phys. Rev. Lett. 110 (2013)

    2013

  53. [61]

    Dorda, M

    A. Dorda, M. Nuss, W. von der Linden, and E. Ar- rigoni, Auxiliary master equation approach to nonequilib- rium correlated impurities, Physical Review B 89 (2014)

    2014

  54. [62]

    Rijavec and G

    S. Rijavec and G. Di Pietra, Tunable non-Markovian dy- namics in a collision model: an application to coherent transport, New Journal of Physics 27, 043003 (2025)

    2025

  55. [63]

    M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197, 287–289 (1977)

    1977

  56. [64]

    J. D. Farmer and J. J. Sidorowich, Predicting chaotic time series, Phys. Rev. Lett. 59, 845 (1987)

    1987

  57. [65]

    Jaeger and H

    H. Jaeger and H. Haas, Harnessing nonlinearity: Predict- ing chaotic systems and saving energy in wireless commu- nication, Science 304, 78 (2004)

    2004

  58. [66]

    Ort´ ın, M

    S. Ort´ ın, M. C. Soriano, L. Pesquera, D. Brunner, D. San- Mart´ ın, I. Fischer, C. R. Mirasso, and J. M. Guti´ errez, A unified framework for reservoir computing and extreme learning machines based on a single time-delayed neuron, Scientific Reports 5 (2015)

    2015

  59. [67]

    Weigend and N

    A. Weigend and N. Gershenfeld, in IEEE International Conference on Neural Networks (1993) pp. 1786–1793 vol.3

    1993

  60. [68]

    Senanian, S

    A. Senanian, S. Prabhu, V. Kremenetski, S. Roy, Y. Cao, J. Kline, T. Onodera, L. G. Wright, X. Wu, V. Fatemi, and P. L. McMahon, Microwave signal processing using an analog quantum reservoir computer, Nature Commu- nications 15 (2024)

    2024

  61. [69]

    J. Chen, H. I. Nurdin, and N. Yamamoto, Temporal in- formation processing on noisy quantum computers, Phys- ical Review Applied 14 (2020)

    2020

  62. [70]

    Verstraete, M

    F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quan- tum computation and quantum-state engineering driven by dissipation, Nature Physics 5, 633–636 (2009)

    2009

  63. [71]

    NON-MARKOVIANITY AND MEMOR Y ENHANCEMENT IN QUANTUM RESER VOIR COMPUTING

    L. Buitinck, G. Louppe, M. Blondel, F. Pedregosa, A. Mueller, O. Grisel, V. Niculae, P. Prettenhofer, A. Gramfort, J. Grobler, R. Layton, J. Vanderplas, A. Joly, B. Holt, and G. Varoquaux, API design for ma- chine learning software: experiences from the scikit-learn project, a...

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