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arxiv: 2605.18914 · v1 · pith:NE6ADOTJnew · submitted 2026-05-18 · 🪐 quant-ph

Non-Stationary Decoherence in Superconducting Qubits: Memory Multi-Fractional Brownian Motion and a Time-Dependent Quantum Brownian Motion Extension

Mahboob Ul Haq This is my paper

Pith reviewed 2026-05-20 11:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords superconducting qubitsnon-stationary decoherencemulti-fractional Brownian motion1/f noisecoherence timesnon-Markovian dephasingCaldeira-Leggett model
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The pith

A memory multi-fractional Brownian motion model with time-dependent Hurst exponent captures non-stationary 1/f noise in superconducting qubits more accurately than constant-exponent versions.

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

The paper develops a unified stochastic drift model for superconducting charge qubits that employs memory multi-fractional Brownian motion with a time-varying Hurst exponent and adaptive kernel. This framework accounts for non-stationary 1/f^beta noise and long-range correlations that fixed-exponent models overlook. If the approach holds, it yields specific predictions for relaxation and dephasing times while exposing shortcomings in Markovian treatments of decoherence. The work also derives a time-local Lindblad equation and temperature-dependent scaling relations that experiments can test directly.

Core claim

The central claim is that memory multi-fractional Brownian motion with time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s) reproduces experimental 1/f spectra more accurately than constant exponents, while its quantum extension through a time-dependent Caldeira-Leggett spectral density J(omega;t) with s(t) = 2H(t)-1 consistently gives beta(t) = 2H(t)-1 and predicts coherence times T1 approximately 5.00 x 10^6 ns and T2 approximately 4.18 x 10^5 ns when charge noise dominates, along with stretched-exponential decay and a quantum-to-classical crossover.

What carries the argument

Memory multi-fractional Brownian motion (mmFBM) with time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s), extended to a time-dependent Caldeira-Leggett environment whose spectral density J(omega;t) has exponent s(t) = 2H(t)-1.

If this is right

  • Relaxation and noise amplitudes act independently on energy decay.
  • Time-varying H(t) matches experimental 1/f spectra more accurately than any constant exponent.
  • Adaptive kernel dynamics preserve correlations without artificial damping.
  • Simulations predict coherence times T1 around 5.00 x 10^6 ns and T2 around 4.18 x 10^5 ns when charge noise dominates.
  • The qubit exhibits stretched-exponential Ramsey and echo decay together with non-Markovian dephasing and a temperature-driven quantum-to-classical crossover.

Where Pith is reading between the lines

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

  • Fitting measured spectra to extract the explicit form of H(t) could pinpoint which physical mechanisms drive the non-stationarity in a given device.
  • The same time-dependent framework might apply to other qubit platforms such as flux or transmon qubits where similar non-stationary noise appears.
  • Measuring how the predicted coherence times change with temperature could test the quantum-to-classical crossover independently of the noise model itself.
  • Architectures that suppress the time variation of H(t) might achieve longer coherence than those optimized only for average noise strength.

Load-bearing premise

The assumption that a time-dependent spectral density with exponent s(t) = 2H(t)-1 will reproduce the observed noise exponent beta(t) and match experimental spectra more accurately than any fixed-exponent model.

What would settle it

An experiment that records the noise spectral density of a superconducting charge qubit over time and checks whether a single constant beta fits the data as well as or better than a time-varying beta(t) derived from the model's H(t).

Figures

Figures reproduced from arXiv: 2605.18914 by Mahboob Ul Haq.

Figure 1
Figure 1. Figure 1: Energy evolution ϵ(t) for different relaxation rates λ. Larger λ produces faster decay toward equilibrium. 2.6 Stochastic Simulation of Energy Fluctuations in Superconducting Charge Qubits To investigate decoherence in superconducting charge qubits, we modeled the energy evolution ϵ(t) using a stochastic integro-differential equation driven by memory multi-fractional Brownian motion (mmfBm): dϵ(t) = −λϵ(t)… view at source ↗
Figure 2
Figure 2. Figure 2: Effect of stochastic noise amplitude σ0 on the qubit energy dynamics. Higher σ0 leads to stronger fluctuations and enhanced irregularity. 2.7 Numerical Estimation of Relaxation and Dephasing Times The mmfBm framework was further used to estimate relaxation [24] and dephasing times in the presence of correlated environmental noise [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of qubit energy dynamics for different Hurst functions. Time-dependent [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energy trajectories obtained using different memory kernels. The adaptive kernel [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulated relaxation dynamics with exponential fit used to estimate the effective [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Simulated dephasing dynamics and exponential envelope fit used to estimate the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Classical mmfBm characterization: sample trajectories, time-dependent Hurst expo [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Quantum coherence dynamics under mmfBm-induced dephasing. Ramsey and echo [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Gate-error optimization. The total error arises from the competition between [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

Building upon our prior work [1], we present a unified stochastic drift model (SdM) for superconducting charge qubits based on memory multi-fractional Brownian motion (mmFBM). The classical sector employs a time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s), capturing non-stationary 1/f^beta noise and long-range temporal correlations inaccessible to conventional models. The quantum extension is formulated via a time-dependent Caldeira--Leggett environment with spectral density J(omega;t) = eta(t) omega_c^{1-s(t)} omega^{s(t)} exp(-omega/omega_c), where s(t) = 2H(t)-1, consistently reproducing beta(t) = 2H(t)-1. Four central results emerge: (1) relaxation and noise amplitudes act independently on energy decay; (2) time-varying H(t) matches experimental 1/f spectra more accurately than any constant exponent; (3) adaptive kernel dynamics preserve correlations without artificial damping; and (4) simulations predict coherence times (T1 ~ 5.00 x 10^6 ns, T2 ~ 4.18 x 10^5 ns) consistent with theory when charge noise dominates. The qubit exhibits stretched-exponential Ramsey and echo decay, non-Markovian dephasing, and a temperature-driven quantum-to-classical crossover. We derive the effective time-local Lindblad master equation, establish the classical mmFBM limit at high temperatures, and provide experimentally testable scaling relations. The non-exponential decay patterns reveal fundamental limitations of Markovian approaches, and the framework guides the design of noise-resilient qubit architectures.

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

2 major / 2 minor

Summary. The manuscript proposes a unified stochastic drift model (SdM) for superconducting charge qubits based on memory multi-fractional Brownian motion (mmFBM) with time-dependent Hurst exponent H(t) and adaptive kernel K(t,s) in the classical sector. It extends this to a quantum setting via a time-dependent Caldeira-Leggett spectral density J(ω;t) = η(t) ω_c^{1-s(t)} ω^{s(t)} exp(-ω/ω_c) where s(t) = 2H(t)-1, asserted to reproduce β(t) = 2H(t)-1. Central claims include independent action of relaxation and noise on energy decay, superior matching of time-varying H(t) to experimental 1/f spectra versus constant exponents, preservation of correlations by the adaptive kernel, and predicted coherence times T1 ≈ 5.00 × 10^6 ns and T2 ≈ 4.18 × 10^5 ns when charge noise dominates, along with a derived effective time-local Lindblad master equation and testable scaling relations.

Significance. If the asserted consistency between the time-dependent kernel and the local noise exponent holds without extraneous contributions from dH/dt, the framework could provide a useful approach for modeling non-stationary decoherence in qubits, offering improved spectral matching and experimentally testable predictions for coherence times and decay patterns beyond standard Markovian or constant-exponent models.

major comments (2)
  1. [Abstract] Abstract (quantum extension paragraph): The claim that J(ω;t) with s(t) = 2H(t)-1 'consistently reproducing beta(t) = 2H(t)-1' via the adaptive kernel K(t,s) is not supported by any derivation or explicit calculation of the power spectrum from the covariance structure of non-stationary mmFBM. For time-varying H(t), the Fourier transform of the covariance generally acquires additional terms proportional to dH/dt; without showing these vanish or are controlled by the specific form of K(t,s), the reproduction of the local exponent β(t) = 2H(t)-1 is not guaranteed and undermines the unification and accuracy claims.
  2. [Results (simulations)] The reported coherence times T1 ~ 5.00 × 10^6 ns and T2 ~ 4.18 × 10^5 ns are stated to be simulation outputs 'consistent with theory when charge noise dominates,' but no details on the numerical implementation, validation against the kernel, error bars, or data exclusion criteria are referenced, making it impossible to assess whether these values follow from the model or are shaped by the choice of H(t) and η(t).
minor comments (2)
  1. [Abstract] The abstract references 'our prior work [1]' but the manuscript should ensure the full reference list is complete and that any new contributions relative to [1] are clearly delineated.
  2. [Model definition] Notation for the adaptive kernel K(t,s) and the functions H(t), η(t) should be defined with explicit functional forms or evolution equations in the main text to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each major comment below in a point-by-point manner, indicating where revisions will be incorporated to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract (quantum extension paragraph): The claim that J(ω;t) with s(t) = 2H(t)-1 'consistently reproducing beta(t) = 2H(t)-1' via the adaptive kernel K(t,s) is not supported by any derivation or explicit calculation of the power spectrum from the covariance structure of non-stationary mmFBM. For time-varying H(t), the Fourier transform of the covariance generally acquires additional terms proportional to dH/dt; without showing these vanish or are controlled by the specific form of K(t,s), the reproduction of the local exponent β(t) = 2H(t)-1 is not guaranteed and undermines the unification and accuracy claims.

    Authors: We acknowledge the referee's point that an explicit derivation of the power spectrum for the non-stationary mmFBM covariance, accounting for potential dH/dt contributions in the Fourier transform, is not provided in the current manuscript. The adaptive kernel K(t,s) is constructed to generalize the stationary fractional Brownian motion case and enforce local exponent matching, but we agree that demonstrating control over extraneous terms is necessary to fully support the claim. In the revised manuscript, we will add a new subsection in the Methods or Supplementary Material deriving the relevant Fourier transform under the time-dependent kernel, showing that for the chosen form of K(t,s) and slowly varying H(t), the dH/dt terms are suppressed to higher order. This will be presented as a supporting calculation rather than an assertion. revision: yes

  2. Referee: [Results (simulations)] The reported coherence times T1 ~ 5.00 × 10^6 ns and T2 ~ 4.18 × 10^5 ns are stated to be simulation outputs 'consistent with theory when charge noise dominates,' but no details on the numerical implementation, validation against the kernel, error bars, or data exclusion criteria are referenced, making it impossible to assess whether these values follow from the model or are shaped by the choice of H(t) and η(t).

    Authors: The referee is correct that the main text and abstract summary do not include sufficient implementation details for the reported coherence times. These values are obtained by numerically integrating the derived time-local Lindblad master equation with the time-dependent J(ω;t) and H(t) profiles. In the full manuscript, the 'Numerical Methods' section outlines the use of an adaptive-step Runge-Kutta scheme and ensemble averaging, but we agree that explicit references to validation (e.g., matching simulated spectra to the mmFBM covariance), error estimation, and convergence criteria are insufficiently highlighted. We will revise the Results section to include these specifics, along with a brief description of the parameter fitting procedure for H(t) and η(t), to enable independent assessment and reproduction. revision: yes

Circularity Check

2 steps flagged

Time-dependent H(t), eta(t) defined to reproduce beta(t) make coherence predictions circular by construction

specific steps
  1. self definitional [Abstract]
    "the quantum extension is formulated via a time-dependent Caldeira--Leggett environment with spectral density J(omega;t) = eta(t) omega_c^{1-s(t)} omega^{s(t)} exp(-omega/omega_c), where s(t) = 2H(t)-1, consistently reproducing beta(t) = 2H(t)-1."

    s(t) is defined equal to 2H(t)-1 and the functional form is selected so that beta(t) equals 2H(t)-1 by direct substitution; the reproduction is therefore tautological rather than derived from the adaptive kernel K(t,s) for varying H(t).

  2. fitted input called prediction [Abstract]
    "simulations predict coherence times (T1 ~ 5.00 x 10^6 ns, T2 ~ 4.18 x 10^5 ns) consistent with theory when charge noise dominates."

    The quoted T1 and T2 values are simulation outputs obtained after choosing H(t) and eta(t) to match experimental 1/f^beta spectra; the reported consistency is therefore a direct consequence of the fitting procedure rather than an a-priori prediction.

full rationale

The paper introduces time-dependent H(t) and eta(t) in the spectral density J(ω;t) with s(t) explicitly set to 2H(t)-1, then asserts this 'consistently reproducing beta(t)=2H(t)-1' and uses the resulting simulations to 'predict' specific T1/T2 values stated as consistent with theory. This reduces the central claim and numerical outputs to the input choices rather than an independent derivation from the mmFBM kernel. The abstract's reference to prior work [1] does not supply the missing non-stationary Fourier transform validation, leaving the unification and accuracy claims load-bearing on the ansatz itself.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 2 invented entities

The central claim rests on several time-dependent functions introduced without independent derivation and on the consistency relation between classical noise exponent and quantum spectral density.

free parameters (3)
  • H(t)
    Time-dependent Hurst exponent fitted or chosen to reproduce non-stationary 1/f^beta spectra.
  • eta(t)
    Time-dependent coefficient in the spectral density J(omega;t).
  • K(t,s)
    Adaptive memory kernel introduced to preserve long-range correlations.
axioms (1)
  • domain assumption The spectral density J(omega;t) with s(t) = 2H(t)-1 consistently reproduces beta(t) = 2H(t)-1
    Invoked directly in the formulation of the quantum extension.
invented entities (2)
  • memory multi-fractional Brownian motion (mmFBM) no independent evidence
    purpose: Capture non-stationary noise and long-range temporal correlations
    New stochastic process defined for the classical sector.
  • stochastic drift model (SdM) no independent evidence
    purpose: Unify classical mmFBM with quantum time-dependent Caldeira-Leggett
    Framework constructed around the new mmFBM process.

pith-pipeline@v0.9.0 · 5845 in / 1678 out tokens · 64678 ms · 2026-05-20T11:42:07.468431+00:00 · methodology

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Reference graph

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