Probing Qubit Noise with a Channel-Resolved Post-Markovian Master Equation

Chun-Tse Li; Daniel A. Lidar; Jingming Tan; Vasil Gucev

REVIEW 4 major objections 5 minor 2 cited by

Probing Qubit Noise with a Channel-Resolved Post-Markovian Master Equation

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A post-Markovian master equation with a damped 13-kHz memory kernel captures the non-Markovian idle noise of a superconducting qubit, with spectator crosstalk as the dominant source.

desk verdict The non-Markovianity witnesses are plausible, but the 13 kHz memory kernel is not yet supported—treat it as an inversion artifact until the authors add error bars, a synthetic-data test, and a spectator-off control. read the letter →

arxiv 2510.12894 v3 pith:HWYFI4MN submitted 2025-10-14 quant-ph physics.comp-ph

Chun-Tse Li , Jingming Tan , Vasil Gucev , Daniel A. Lidar This is my paper

classification quant-phphysics.comp-ph

keywords non-Markoviannoisepost-MarkovianmasterequationmemorykernelsuperconductingqubitsZZcrosstalkquantumprocesstomographyinformationbackflowCP-divisibility

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

The paper aims to show that non-Markovian memory in a superconducting qubit's idle evolution can be diagnosed and modeled in a single framework. Using idle-evolution tomography on a superconducting quantum processor, the authors find violations of CP-divisibility (the requirement that each intermediate evolution step be memoryless) and revivals of trace-distance and relative-entropy measures, both signatures of information flowing back from the environment. They then fit a Lindblad baseline to the decaying Bloch trajectories and invert a post-Markovian master equation in Laplace space to reconstruct the memory kernel. The reconstructed transverse kernel is a damped oscillation at about 13 kHz and is nearly identical for four different state preparations, which they take as evidence that the memory is intrinsic to the device rather than a preparation artifact. A closed-form spectator-ZZ crosstalk model reproduces the observed revivals while keeping longitudinal relaxation Markovian, and two-qubit tomography shows mutual-information revivals on the same timescale, supporting spectator crosstalk as the dominant source.

What carries the argument

The central object is the post-Markovian master equation (PMME), in which a Lindblad generator L0 describes smooth Markovian decay while a memory kernel k(tau) weights the delayed density matrix. The carrying identity is the Laplace-domain equation in the damping basis of the Lindblad generator: k(s - lambda_i) = (1/lambda1_i)[s - lambda0_i - 1/xi_i(s)], which converts measured, normalized Bloch-mode coefficients xi_i(t) into an empirical kernel by numerical inverse Laplace transform. The damping basis - the eigenoperators of the Lindblad generator - is what makes the problem channel-resolved: the longitudinal modes have vanishing coupling to the kernel, so only the transverse coherence mode

What would settle it

Repeat the idle-evolution tomography with all spectator qubits prepared in |0> instead of |+>: if the 13-kHz kernel oscillation and the ~30-microsecond revivals vanish, spectator ZZ crosstalk is the cause; if they persist, the memory is not spectator-driven. Separately, vary the Laplace-inversion damping parameter sigma and the time window Tmax while keeping the same data: a kernel frequency that shifts with these numerical choices would indicate an inversion artifact, while a stable frequency would support the intrinsic-device claim.

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Extended reading notes

Core claim

The central claim is that a channel-resolved post-Markovian master equation (PMME) - a Lindblad equation augmented with an integral over a memory kernel acting on the delayed state - quantitatively describes the idle dynamics of a superconducting qubit. Diagonalizing the generator in its damping basis separates two longitudinal modes, which remain Markovian, from two transverse modes, whose measured Bloch coefficients determine the kernel through a closed Laplace-domain identity. Applying numerical Laplace inversion to tomographic data yields a kernel k3(tau) that is a damped oscillation with frequency about 13 kHz and is nearly state-independent. Because the same oscillation appears for all

Load-bearing premise

The central claim stands or falls on whether the 13-kHz oscillatory kernel is genuine device memory rather than an artifact of numerically inverting a Laplace transform of roughly 50 noisy Bloch-vector samples with a hand-set damping parameter; the paper gives no error bars or sensitivity analysis, and the crosstalk attribution assumes the temporal coincidence between mutual-information revivals and backflow revivals is causal.

Editorial extensions

If this is right

A single, approximately state-independent memory kernel could describe idle noise for any preparation of the qubit, making device calibration and noise-aware compilation feasible.
Standard T1/T2 Markovian characterization misses a 13-kHz oscillatory memory channel, so error budgets and threshold estimates should include such kernels.
Because mutual-information revivals coincide with backflow revivals, suppressing spectator ZZ crosstalk (via couplers or decoupling sequences) should reduce the non-Markovianity itself.
The channel-resolved structure says longitudinal relaxation can be treated as Markovian while transverse coherence carries the memory, simplifying future noise models and mitigation.
The reconstructed kernel provides a quantitative reduced model that directly informs layout-aware scheduling and dynamical-decoupling placement.

Reading between the lines

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

Editorial inference: a control experiment with all spectator qubits prepared in |0> rather than |+> would test the crosstalk attribution directly; the paper does not report such a spectator-off run, so the causality of the temporal coincidence remains an inference.
Editorial inference: the 13-kHz oscillation frequency should shift with the effective spectator coupling; reconstructing the kernel under different spectator states (or with a coupler detuned) would map that dependence and sharpen the model.
Editorial inference: varying the numerical Laplace-inversion damping parameter and the time window would reveal whether the oscillation is stable or a finite-window artifact; a stable frequency across these choices would strengthen the intrinsic-device claim.
Editorial inference: extending the same reconstruction to two-qubit process tomography could yield a multi-qubit PMME kernel, but the paper only reconstructs the single-qubit transverse kernel, leaving that as a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The qualitative part of this paper is solid and worth your time: the CP-divisibility violations and the trace-distance/relative-entropy revivals are complementary witnesses, and the timescales line up with mutual-information revivals in the two-qubit crosstalk data. That is a believable demonstration that these IBM qubits are non-Markovian. The closed-form spectator-ZZ model with local dissipation is also a useful piece of work—the N-spectator Kronecker decomposition in Appendix C is elegant and the limiting behavior (product of cosines vs. pure damping) is clearly derived. The mode-resolved memory-kernel superoperator, introduced to resolve the k2/k3 ambiguity, is a genuine formal step.

But the central quantitative claim—a nearly state-independent, 13 kHz oscillatory memory kernel—is not supported by the evidence as presented. The extraction pipeline in Sec. IV C starts from a Lindblad baseline fit to the same trajectories, then Laplace-transforms roughly 50 noisy Bloch samples with a hand-set damping sigma = 1/(3Tmax), and inverts numerically. No shot counts, no error bars, no sensitivity analysis. That is exactly the setup where a mis-specified exponential envelope or a finite-window truncation artifact can masquerade as a damped oscillation. The stress-test note has it right: a synthetic-data test—invert a known kernel through the same code and see if you recover it—is necessary before anyone should believe the 13 kHz number. I also agree that the crosstalk-dominance conclusion is correlation, not causation; there is no spectator-off control.

Two smaller but real issues. First, the paper is internally inconsistent: the abstract says the kernel is reconstructed analytically from the fitted closed-form dynamics, but the body does a numerical inversion of the measured trajectories. That needs to be fixed. Second, the manuscript is rough—there is a duplicated Introduction, typos, and awkward notation. That doesn't change the science, but it makes the draft hard to referee in its current form.

Who is this for? People working on non-Markovian noise characterization and device validation will find the framework and the crosstalk model useful, provided the kernel claim is either substantiated or downgraded. I would send it to peer review—it deserves a serious referee—but I would require the uncertainty quantification, the synthetic-data test, a spectator-off control, and a cleaned-up revision before accepting. As is, I would not cite the 13 kHz kernel in my own work.

Referee Report

4 major / 5 minor

Summary. The paper develops a channel-resolved post-Markovian master equation (PMME) framework and applies it to a superconducting qubit on IBM hardware. Time-resolved state and process tomography are used to diagnose non-Markovianity through CP-divisibility violations, trace-distance/relative-entropy backflow, and two-qubit mutual information. A Lindblad baseline is fitted to the measured Bloch trajectories, and a memory kernel k(t) is reconstructed by numerical Laplace inversion. The headline quantitative claim is a nearly state-independent memory kernel k3(τ) with a damped oscillation at ω/2π ≈ 13 kHz, attributed to spectator-induced ZZ crosstalk.

Significance. If the quantitative kernel claim were reliable, the paper would provide an attractive end-to-end bridge between operational non-Markovianity witnesses and a reduced memory-kernel noise model, with a useful closed-form multi-spectator ZZ model in Appendix C. The two complementary qualitative witnesses (CP-divisibility and information backflow) are a genuine strength, and the analytical diagonalization of the PMME in the damping basis is clearly presented. However, the central quantitative claim rests on an ill-conditioned numerical Laplace inversion of a small number of noisy samples, with no error bars, no shot counts, and no synthetic-data validation. As written, the 13 kHz kernel and the state-independence/conclusion about device-level memory are not established. With additional statistical and sensitivity analysis the framework could be significant; in its current form the quantitative part is not yet substantiated.

major comments (4)

[Sec. V.D and Fig. 8] The kernel reconstruction has no error bars, no shot counts, and no sensitivity analysis. Eq. (34) contains the term 1/ξ̃_i(s), which is extremely sensitive to noise wherever ξ̃_i(s) is small, and Eq. (49) inverts only a finite-window damped DFT with a hand-set σ=1/(3Tmax). The oscillatory k3(τ) in Fig. 8 may therefore be a numerical artifact of truncation or noise rather than device memory. The authors should report shot counts, provide bootstrap or Monte Carlo error bars on k3(τ), and perform a synthetic-data test: invert a known Markovian trajectory and a known damped-oscillatory kernel while varying σ, N, and Tmax to verify that the 13 kHz feature is stable.
[Sec. V.A, Eq. (51)] CP-divisibility violations are inferred from λ_min of the Choi matrix of S_t S_s^+, where S_s is a noisy, often nearly singular superoperator and the pseudoinverse amplifies tomography noise. No noise threshold, confidence interval, or comparison against a simulated Markovian process under the same tomography uncertainty is provided. Since Fig. 5 is presented as evidence of non-Markovianity, a null test or threshold is needed to ensure that the red/purple regions are not produced by the inversion of noisy matrices.
[Sec. V.D, steps 2–3] The memory kernel is reconstructed from the residual between the data and a Lindblad trajectory fitted by minimizing MSE against the same data (Eq. 64). Any systematic misfit of the Lindblad/exponential ansatz—frequency drift, stretched-exponential decay, or gate-calibration drift—is automatically projected into k(τ). The claim that k3(τ) is a physical memory kernel therefore requires a synthetic Markovian control: generate data with the fitted L0+L1, add the same sampling noise, run the identical reconstruction pipeline, and show that the recovered kernel is a delta function (or at least not an oscillatory 13 kHz feature). Without this test, the 'state-independent kernel' may be a re-expression of the baseline's incomplete fit rather than an independent measurement.
[Sec. VI and Fig. 7] The conclusion that spectator-induced crosstalk is the 'dominant physical origin' of the observed memory is not established by a control experiment. The temporal coincidence between mutual-information revivals and backflow revivals is circumstantial; no spectator-off, spectator-detuned, or spectator-state-variation control is shown for the kernel reconstruction. At minimum, the causal wording in the abstract and conclusion should be softened, or an experiment varying the spectator configuration should be added.

minor comments (5)

[General structure] The manuscript contains two consecutive 'Introduction' sections (I and II) with partially overlapping content and duplicated references ([12] and [17] are the same paper). This should be consolidated into a single introduction.
[Fig. 8 caption] Typographical errors: 'Masured' should be 'Measured', and '13,kHz' should be '13 kHz'. Also, the axis labels and units for the kernel plots are not clearly described.
[Sec. III.D] The sentence 'Note that in our case, [L0, L1]' is incomplete; it should state that the two generators commute and that this justifies simultaneous diagonalization. The eigenvalue labeling in Eq. (21) is not fully defined.
[Sec. IV.A] The text says 'sufficiently large number of shots' but no shot count is reported anywhere. Since statistical uncertainty is central to the reconstruction, the number of shots per circuit and per Pauli basis should be stated explicitly.
[Data/code availability] No data availability or code availability statement is included. For an experimental reconstruction paper, releasing tomography data and the inversion script would materially strengthen reproducibility.

Circularity Check

2 steps flagged · score 4.0 of 10

Partial circularity: the 'Markovian prediction' is a fit to the same data, and the kernel extraction is 'validated' by self-consistency; the state-independent kernel itself is not a fitted parameter.

fitted input called prediction [Sec. V.D (Results), Fig. 8 discussion, after Eq. (64)]
"Fig. 8 (rows 1–4) compares the experimental Bloch coordinates (markers) with the purely Markovian prediction (solid lines). Although L0 reproduces the overall damped-oscillation trend, it fails to capture the revivals at t≈25, 60, and 95, µs"

The solid lines are not a prediction: the caption describes them as 'best-fit Markovian (Lindblad) trajectories obtained from the MSE error minimization' against the very same Bloch data (Eq. (64)). The mismatch between markers and solid lines is therefore a residual of a fit, not a failed predictive test. The memory kernel reconstructed in Sec. IV C from these same trajectories and this fitted baseline is a re-expression of the fit residual, so its non-Markovian features are not an independent confirmation of the PMME model.
self definitional [Sec. VI Conclusion; Eqs. (24) and (34)]
"Starting from a Lindblad approximation that faithfully reproduces the long-time exponential damping, we analytically calculated the PMME kernel in Laplace space and reconstructed a nearly state-independent memory kernel. This memory kernel has oscillating behavior in both the real and imaginary parts which generally deviates from the delta function kernel that results in the Markovian effect, thereby validating the kernel extraction procedure..."

Equation (34) is obtained by algebraically inverting Eq. (24): k̃(s−λ_i) is solved from the measured ξ̃_i(s), so substituting the reconstructed kernel back into Eq. (24) reproduces the input ξ_i(s) identically by construction. A kernel that reproduces the same trajectories used to extract it is therefore guaranteed, and calling this 'thereby validating the kernel extraction procedure' is a self-consistency check rather than independent validation. The state-independence comparison across initial states is an independent check, but it does not by itself rule out numerical Laplace-inversion artifacts.

full rationale

The paper's central quantitative finding — a nearly state-independent, oscillatory transverse memory kernel with ω/2π ≈ 13 kHz — is not itself a fitted parameter: the same Lindblad parameters are used for four independently tomographed initial states, and the similarity of k3(τ) across those states is not enforced by the construction. However, the surrounding validation narrative is partially circular. The 'purely Markovian prediction' in Fig. 8 is actually the best-fit Lindblad trajectory obtained by MSE minimization against the same data, so its disagreement with the markers is a residual, not a falsifiable prediction. Similarly, the conclusion's claim that the oscillatory kernel 'validates the kernel extraction procedure' is a self-consistency statement: Eq. (34) is the algebraic inverse of Eq. (24), so any extracted kernel necessarily reproduces the input ξ_i(t). The PMME formalism itself is cited to a well-known independent framework ([12]/[17]) and is not a problematic load-bearing self-citation. The absence of shot counts, error bars, and sensitivity analysis for σ and T_max is a correctness/robustness risk, not circularity. On balance, the core state-independence result has independent content, but the paper overstates the evidential value of a residual-based reconstruction, yielding partial circularity rather than a fully circular derivation.

Assumptions & free parameters 5 free parameters · 6 assumptions · 1 invented entities

The central quantitative claims depend on a phenomenological PMME ansatz, a Lindblad baseline fitted to the same data, hand-set numerical inversion parameters, and an unverified spectator-ZZ model. Per-mode kernels are introduced ad hoc to fix an ambiguity. No error propagation or independent validation is supplied.

free parameters (5)

qubit precession frequency omega_z = not reported in text
Fitted to Bloch trajectories via MSE minimization; enters L0 and the kernel reconstruction.
amplitude damping rate gamma_AD = not reported in text
Fitted to Bloch trajectories; determines longitudinal and transverse Markovian decay in L0.
pure dephasing rate gamma_PD = not reported in text
Fitted to Bloch trajectories; defines L1 and fixes the eigenvalues used in the kernel inversion.
spectator ZZ coupling strengths J0q = not reported in text
The abstract says fitted closed-form dynamics; Appendix C introduces J0q but no fitted values or uncertainties are given.
Laplace inversion damping parameter sigma = 1/(3Tmax)
Chosen by hand in Section IV.C.1; directly affects the reconstructed kernel through the damped discrete Fourier approximation.

assumptions (6)

domain assumption The PMME convolution form exactly describes the reduced qubit dynamics for the chosen L0, L1, and scalar/superoperator kernel.
Phenomenological ansatz from Shabani-Lidar; not derived from a microscopic system-bath Hamiltonian in this paper.
standard math The damping-basis decomposition with [L0, L1] = 0 and the stated eigenvalues is valid.
True for the specific L0, L1 chosen, but assumes the noise generator has this exact Lindblad form.
domain assumption Process tomography at each idle time yields CPTP maps with no unmodeled drift, leakage, or calibration errors.
Section IV.B projects raw estimates onto CPTP; no calibration or drift analysis is reported.
domain assumption Finite-window numerical Laplace and inverse Laplace transforms recover the true kernel from 50 sampled points.
Section IV.C truncates integrals and uses a Riemann sum; no error bound or convergence check is given.
domain assumption Spectator qubits act as a controlled environment described by coherent ZZ couplings plus independent local Markovian dissipators.
Appendix C assumes this model; it is not independently verified and the spectator-off control is missing.
domain assumption Negative eigenvalues of the pseudo-inverted intermediate Choi matrix are evidence of non-Markovianity rather than tomography noise.
Section V.A uses pseudo-inverse; without error bars, noise can create negative eigenvalues.

invented entities (1)

Mode-resolved memory-kernel superoperator
purpose: Lifts the scalar PMME kernel to per-mode kernels to resolve the Hermiticity ambiguity k2 != k3.
Introduced in this paper as a bookkeeping extension of the Shabani-Lidar scalar kernel; no independent falsifiable handle is provided.

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Pith. "Pith review of Probing Qubit Noise with a Channel-Resolved Post-Markovian Master Equation." pith.science (2026). https://pith.science/paper/HWYFI4MN

@misc{pith2026251012894,
  author       = {Pith},
  title        = {Pith review of: Probing Qubit Noise with a Channel-Resolved Post-Markovian Master Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWYFI4MN}},
  note         = {Machine review of arXiv:2510.12894}
}

abstract

Accurate noise characterization is essential for scaling quantum processors toward fault-tolerant operation. Although reduced qubit dynamics are often modeled with Markovian master equations, present-day devices can exhibit memory effects generated by residual qubit-qubit couplings, structured environments, and finite bath correlation times. Here we develop a channel-resolved, Post-Markovian Master Equation model for non-Markovian noise and test it in superconducting qubits. Using idle-evolution tomography on IBM Quantum processors, we identify complementary operational signatures of non-Markovianity, including violations of CP-divisibility and revivals of distinguishability-based information-backflow measures. We further derive a closed-form spectator-$ZZ$ model with local dissipation and show that it captures the observed transverse Bloch-vector revivals while leaving the longitudinal relaxation mode Markovian within the model. The fitted closed-form dynamics enable an analytical reconstruction of the transverse memory kernel, whose damped oscillatory structure captures the non-Markovian correction beyond the fitted Markovian baseline. Two-qubit tomography shows buildup and revivals of quantum mutual information on comparable timescales, supporting spectator-induced crosstalk as an important contributor to the observed memory effects. Our results connect operational non-Markovianity diagnostics, microscopic crosstalk modeling, and reduced memory-kernel reconstruction in a single experimental framework for superconducting quantum hardware.

Figures

Figures reproduced from arXiv: 2510.12894 by the authors.

**Figure 2.** Bloch vector under 5000 ID gates The choice to use entangled states is due to their inherent sensitivity to environmental disturbances. Focusing on the quantum coherence through time evolution allows one to detect non-Markovian noise. Indeed, recent study have shown that non-Markovian noise can significantly [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Layout of the IBM superconducting quantum processor in the heavy-hex lattice (partial view). Each qubit connects to at most three neighbors. In this example, we select qubits 23, 24, 25, and 34, with qubit 24 serving as the main qubit and the other three acting as spectator qubits. each probe time t. More concretely, we choose the tomographically complete POVM {Em,k} such that each outcome and corresponding probabi… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Schematic of the basic experimental setup. The circuit is composed of three parts: (1) the state-preparation unitary USP, (2) an idling phase implemented by a sequence of identity gates, and (3) the final basis rotations for tomography. By varying the number of identit…

**Figure 5.** Figure 5: Heat-maps of the minimum eigenvalue λmin χt,s for the intermediate Choi matrix as a function of start time s and idle time t, measured on IBM’s 127-qubit “Strasbourg” processor. (a) Left panel shows the case where the main qubit is number 4 and the spectator qubits are…

**Figure 6.** Figure 6: Time evolution of (a) the trace distance T [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: (a) Quantum mutual information I(A:B)ρ(t) and (b) quantum conditional entropy S(A|B)ρ(t) for four different initial product states. An initial rise of I(A:B) up to ∼ 0.7 bits within the first 10 µs demonstrates fast crosstalk-induced correlation build-up. Revivals ever…

**Figure 8.** Figure 8: Reconstruction of the post-Markovian memory kernel for four different initial states of the main qubit. Each column corresponds to a different preparation, indicated above the panels. (Rows 1–3) Masured Bloch-vector components vx(t) (yellow), vy(t) (orange), and vz(t) …

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Reference graph

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projecting any negative eigenvalues of ˆρ(t) to zero and renormalizing, or
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solving the constrained optimization min ρ⪰0,Trρ=1 X m,k Tr(Em,kρ)−ˆpm,k(t) 2 , to ensure the final ˆρ(t) is a valid density matrix that best fits our observed data. B. Quantum Process T omography Quantum process tomography is an extension of the methods for quantum state tomography characterizing the unknown quantum channelErather than a fixed state ρ. T...
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ZZ crosstalk between two qubits The simplest model of ZZ crosstalk considers the closed-systemevolution generated by the Hamiltonian H= J 2 Z0Z1.(C1) To monitor how this coherent interaction influences themainqubit we track the time evolution of its Bloch-vector components. In...
[37]

ZZ crosstalk between two qubits with dissipator We now supplement the coherent ZZ interaction with two Markovian noise channels—amplitude damping (rate γ↓q, governing theT 1 time) and pure dephasing (rateγ ϕq, governing theT 2 time). With the standard dissipator D[L](ρ) =LρL †...
[38]

ZZ crosstalk withNspectator qubits We now generalize the single-spectator case toNspec- tators. Consider the ZZ crosstalk Hamiltonian: H=− ω0 2 Z0 + NX q=1 J0q 2 Z0Zq,L † H (·) =i[H,·],(C27) and the dissipators NX q=0 γ↓qD[σ− q ] +γ ϕqD[Zq] (C28) Close operator set.A key obser...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

projecting any negative eigenvalues of ˆρ(t) to zero and renormalizing, or

[2] [2]

solving the constrained optimization min ρ⪰0,Trρ=1 X m,k Tr(Em,kρ)−ˆpm,k(t) 2 , to ensure the final ˆρ(t) is a valid density matrix that best fits our observed data. B. Quantum Process T omography Quantum process tomography is an extension of the methods for quantum state tomography characterizing the unknown quantum channelErather than a fixed state ρ. T...

[3] [3]

For each time point, we do the quantum state tomography to get the estimated quantum state ˆρ(t)

We uniformly sample the time points [0, T max] with time step ∆tand total number of pointsN= Tmax/∆t+ 1. For each time point, we do the quantum state tomography to get the estimated quantum state ˆρ(t)

[4] [4]

We calculate the coefficientµ i(t) byµ i(t) = Tr(Li ˆρ(t)) andξ i(t) =µ i(t)/µi(0)

[5] [5]

For the mode withλ 1 i ̸= 0, we do the numerical Laplace transform to get ˜ξi(s)

[6] [6]

Below, we discuss how to do the numerical Laplace and inverse Laplace transform

We perform the numerical inverse Laplace transform on 1 λ1 i h s−λ 0 i − 1 ˜ξi(s) i and multiply the resulting func- tion bye −λit. Below, we discuss how to do the numerical Laplace and inverse Laplace transform. 7

[7] [7]

We samplef(t) on a uniform time grid tn =n∆t, n= 0,

Numerical Laplace Transform Letf(t) be a causal function (f(t) = 0 fort <0). We samplef(t) on a uniform time grid tn =n∆t, n= 0, . . . , N−1,∆t= Tmax N−1 . (43) Because the data have are finite in duration, we approx- imate the Laplace integral with an upper limitT max: ˜f(σ+iω)≈ Z Tmax 0 f(t)e −σt e−iωt dt.(44) With only a finite set of samples, the inte...

[8] [8]

, N−1 without changing the value

Numerical Inverse Laplace Transform The inverse Laplace transform can be evaluated via the Bromwich integral, f(t) = 1 2π Z ∞ −∞ ˜f(σ+iω)e σt eiωt dω,(47) which we approximate by again truncating the frequency range and discretising: f(t n) = eσtn 2π ∆ω N/2−1X k=−N/2 ˜f(σ+iω k)e iωktn ,(48) with ∆ω= 2π N∆t .Because the sum is periodic, we may shift the in...