Memory Multi-Fractional Brownian Motion (mmFBM)

• Memory multi-fractional Brownian motion is a Gaussian process defined by a time-dependent Hurst exponent and a causal memory kernel, capturing evolving local roughness and long-range dependence.
• It extends familiar fractional and multifractional Brownian motions by incorporating a Volterra-type driver, which plays a critical role in modeling noise in superconducting-qubit dynamics.
• The model is pivotal for simulating non-stationary 1/f-type noise and assessing decoherence in quantum systems, directly influencing circuit design and error estimation.

Memory multi-fractional Brownian motion (mmFBM) denotes a class of Gaussian, non-Markovian constructions that combine a time-dependent Hurst exponent with an explicit memory kernel, thereby extending both fractional Brownian motion and multifractional Brownian motion to settings in which local roughness, temporal correlation strength, and spectral slope evolve over time. In the superconducting-qubit literature, mmFBM is introduced operationally through a Volterra-type driver,

$\mathcal{M}(t)=\int_0^t K(t,s)\,dB^{H(s)}(s),$

and then used in stochastic dynamics for noisy charge offsets; closely related work also uses Riemann–Liouville-type kernels and history-preserving multifractional formulations in which past increments retain the exponent assigned at their time of origin rather than being retrospectively reweighted (Haq, 27 Jul 2025, Haq, 18 May 2026, Wang et al., 2023).

1. Terminology and scope of the concept

Across the recent literature, the label “memory multi-fractional Brownian motion” is used for several related but nonidentical objects. In the superconducting-qubit model of 2025, mmFBM is a Gaussian noise model with two defining ingredients: a time-varying Hurst exponent $H(t)$ and a causal memory kernel $K(t,s)$, with the stochastic driver entering the reduced qubit dynamics through

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$

That formulation is explicitly described as operational and oriented toward simulation and physical modeling rather than as a full mathematical treatise with a closed-form covariance theory (Haq, 27 Jul 2025).

A closely related construction appears in the anomalous-diffusion literature under the name “memory-multi-fractional Brownian motion” or “MMFBM.” There, the process is defined by

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$

so that the kernel exponent depends on the past time $s$, not on the observation time $t$. This distinction is central: the process preserves the imprint of earlier exponents $\alpha(s)$ and therefore does not reset its memory structure when the environment changes (Wang et al., 2023).

The nomenclature is further complicated by acronym overlap. In a different stochastic-process literature, “mmfBm” denotes multi-mixed fractional Brownian motion, namely an infinite superposition of independent fBms with distinct fixed Hurst indices,

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$

which is a stationary-increment Gaussian mixture rather than a time-varying-Hurst memory kernel model (Almani et al., 2021). For that reason, mmFBM is best understood as a family resemblance term rather than a uniquely standardized object.

2. Mathematical constructions and relation to Brownian models

The natural baseline is standard Brownian motion $W(t)$, a centered Gaussian process with covariance

$H(t)$0

stationary independent increments, and Markovian dynamics. Fractional Brownian motion $H(t)$1, $H(t)$2, keeps Gaussianity but replaces independent increments by correlated increments with covariance

$H(t)$3

For $H(t)$4, the process exhibits long-range dependence; for $H(t)$5, it is antipersistent; and for constant $H(t)$6 it remains self-similar (Haq, 27 Jul 2025).

Multifractional Brownian motion replaces the constant Hurst exponent by a function $H(t)$7. In the qubit-noise formulation, this appears as

$H(t)$8

so that exact self-similarity is generally lost and the increments become non-stationary. White-noise-based stochastic calculus for mBm makes the same point from another direction: $H(t)$9 is represented as a Gaussian field with time-dependent Hurst function $K(t,s)$0, and when $K(t,s)$1 its local Hölder exponent is almost surely $K(t,s)$2, while Itô-type formulas involve the derivative of the time-varying variance $K(t,s)$3 rather than a constant quadratic-variation density (Lebovits et al., 2011).

mmFBM adds an explicit memory filter to the multifractional driver. In the superconducting-qubit model, the defining noise term is

$K(t,s)$4

In the 2026 extension, the same idea is written in Riemann–Liouville form as

$K(t,s)$5

with an adaptive memory kernel $K(t,s)$6 for $K(t,s)$7 (Haq, 18 May 2026). These are Volterra-type constructions: the present value depends on a weighted integral over the entire past, and the memory kernel itself may evolve with time.

3. Statistical structure: covariance, spectra, and history dependence

The defining statistical feature of mmFBM is the coexistence of non-stationarity and long memory. In the qubit-noise model, non-stationarity arises from two mechanisms: the variable Hurst exponent $K(t,s)$8 and the non-translation-invariant kernel $K(t,s)$9. A representative simulation choice is

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$0

together with local variance scaling $$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$1, so that the increment statistics depend explicitly on absolute time (Haq, 27 Jul 2025).

The 2026 formulation provides an exact covariance for slowly varying $$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$2,

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$3

and under the adiabatic condition

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$4

it uses the locally stationary approximation

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$5

The corresponding local power spectral density for increments scales as

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$6

and the variance obeys the leading asymptotic law

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$7

(Haq, 18 May 2026).

A distinct but closely allied memory-preserving formulation makes the dependence on history especially transparent. For the step protocol

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$8

the mean-squared displacement is

$$d\chi(t)=F(t,\chi(t))\,dt+G(t,\chi(t))\,\mathcal{M}(t).$$9

with long-time asymptotic

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$0

Accordingly, if $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$1, the post-switch scaling exponent can be determined by the pre-switch value $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$2; the response to a short perturbation in $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$3 is nonzero and decays as a power law, whereas in the corresponding memory-resetting MFBM the response is identically zero (Wang et al., 2023). This is one of the clearest demonstrations that mmFBM is not merely mBm with a time-dependent roughness parameter.

4. Superconducting-qubit noise modeling

In superconducting charge-qubit modeling, mmFBM is used as a phenomenological representation of non-stationary $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$4-type charge noise. The effective Hamiltonian is written as

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$5

where the stochastic charge offset $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$6 is driven either by the generic mmFBM equation

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$7

or by the explicit colored-noise form

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$8

The numerical kernel used for mmFBM is

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{[\alpha(s)-1]/2}\,dB(s),$$9

and representative parameter choices include $s$0, $s$1, $s$2, $s$3, and

$s$4

For those settings, the simulated fidelity decays approximately as

$s$5

while the coherence and fidelity curves in Lindblad simulations plateau at levels $s$6–$s$7, with a crossover from noise-dominated decoherence to $s$8-limited decay around $s$9; non-Markovian coherence revivals are also reported (Haq, 27 Jul 2025).

The 2026 extension embeds the same classical mmFBM sector into a time-dependent Caldeira–Leggett environment with spectral density

$t$0

thereby enforcing the same low-frequency exponent $t$1 at the quantum-bath level. In that framework, the qubit obeys a time-local pure-dephasing master equation

$t$2

with

$t$3

The Ramsey and echo envelopes become stretched exponentials,

$t$4

and simulations produce effective coherence times $t$5 and $t$6 under charge-noise-dominated conditions (Haq, 18 May 2026).

5. Relation to neighboring models and recurrent misconceptions

A first recurrent confusion is to equate time-varying roughness with explicit memory. Standard mBm changes local Hölder regularity through $t$7, but mmFBM, as used in the qubit literature, adds a separate causal kernel $t$8. That distinction matters because the “memory” then has two meanings: the inherent long-range dependence associated with fractional-type processes and the additional non-Markovian structure produced by explicit integration over the full past trajectory (Haq, 27 Jul 2025).

A second misconception is to treat all memory-preserving time-varying-Hurst models as interchangeable. Switching fractional Brownian motion,

$t$9

is close in spirit to mmFBM because it uses a single Brownian driver and a power-law kernel, so changing $\alpha(s)$0 or $\alpha(s)$1 does not create a renewal at switching times. Under heavy-tailed switching, both the MSD and PSD can acquire explicit measurement-time dependence, with effective exponents converging to those of the long-sojourn state (Balcerek et al., 2023). That mechanism resembles mmFBM, but it is not identical to the deterministic-kernel qubit constructions.

A third source of confusion is acronym collision. The multi-mixed process

$\alpha(s)$2

has stationary increments, exact Gaussian covariance obtained by superposition, and long-range dependence if and only if some $\alpha(s)$3. Its small-scale roughness is governed by $\alpha(s)$4, while long-lag dependence is governed by $\alpha(s)$5. This is a different object from memory-kernel multifractional models, even though the abbreviation “mmfBm” is sometimes visually similar (Almani et al., 2021).

Finally, mmFBM is not simply a reformulation of traditional $\alpha(s)$6 noise or random-telegraph/TLS ensembles. In the qubit application, the point of mmFBM is precisely to provide a continuous Gaussian field with $\alpha(s)$7 spectrum, adjustable long-memory, and time-varying Hurst parameter, thereby addressing non-stationary and multifractal scaling features that simpler white-noise, OU, or fixed-$\alpha(s)$8 fBM models do not capture simultaneously (Haq, 27 Jul 2025).

6. Limitations, inference, and directions of development

The present mmFBM literature is explicitly phenomenological. In the superconducting-qubit model, the main assumptions are Gaussianity, linear coupling of the noise to the qubit Hamiltonian, an ad hoc Hurst function $\alpha(s)$9, finite-time numerical implementation with an exponential cutoff, and a classical stochastic environment rather than a fully quantum bath. Regimes with dominant discrete fluctuators, strong coupling, or multi-qubit cross-correlations are identified as settings where the model may be insufficient. The same work lists natural extensions: non-Gaussian mmFBM, multidimensional mmFBM, data-driven estimation of $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$0 and $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$1, quantum mmFBM baths, and improved multi-scale estimation procedures (Haq, 27 Jul 2025).

The 2026 extension adds experimentally testable scaling relations. In a window where $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$2 is approximately constant, the model predicts

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$3

It also proposes an inference protocol based on sliding-window fits of

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$4

with

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$5

and the consistency check $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$6. Design implications are stated in terms of gate-error estimates,

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k},$7

which tie dynamical-decoupling performance directly to the inferred Hurst exponent (Haq, 18 May 2026).

A broader methodological implication is that mmFBM sits at the intersection of multifractional processes, Gaussian Volterra processes, and long-memory noise models. The literature already contains adjacent templates for several of these components: white-noise stochastic calculus for mBm, history-preserving kernels in anomalous diffusion, and explicit scale-dependent Gaussian network constructions for fBM and multifractal processes (Lebovits et al., 2011, Wang et al., 2023, Descamps, 2016). This suggests that future mmFBM work is likely to proceed along two complementary lines: tighter probabilistic foundations for covariance and regularity, and more data-driven calibration against non-stationary spectra and decoherence observables.