Extensible Lindblad Tomography

• Extensible Lindblad Tomography is a deterministic protocol that reconstructs time-independent, Markovian Lindblad generators using complete product Pauli preparations and measurements.
• It employs short-time linearization and transfer matrix inversion to extract coherent and incoherent parameters with improved scalability under k-local assumptions.
• Extensions of ELT incorporate non-Markovian dynamics, realistic measurement models, and advanced inference methods like neural networks to enhance quantum process characterization.

Extensible Lindblad tomography denotes a deterministic, tomographically complete implementation of Lindblad tomography built from product Pauli eigenstate preparations and Pauli measurements, used to reconstruct the parameters of a time-independent, Markovian Lindblad generator from short-time data. In "Demonstrating and Benchmarking Classical Shadows for Lindblad Tomography," it functions as the baseline against which a randomized variant, Shadow Lindblad Tomography, is benchmarked on a five-qubit superconducting transmon processor (Birke et al., 16 Feb 2026). In a broader research sense, the term also refers to a program of extending GKSL-based learning beyond the basic time-homogeneous setting, including locality-structured models, time-dependent generators, non-Markovian time-local master equations, realistic measurement models, cyclic gate tomography, and model selection over nested Lindblad families (Varona et al., 2024).

1. Definition and conceptual scope

In the narrow experimental usage introduced by Birke and coauthors, Extensible Lindblad Tomography (ELT) is a deterministic scheme that estimates Lindblad parameters using a complete tomographic dataset. It is "extensible" because, absent locality constraints, it can be extended to fully reconstruct all parameters of a general Lindblad model with exponential resources, while under $k$-local assumptions it reduces to polynomial resource scaling (Birke et al., 16 Feb 2026).

The underlying dynamics are assumed to obey a time-independent, Markovian Lindblad master equation on the computational subspace,

$$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$

with Hamiltonian $H$, jump operators $\{L_k\}$, and Lindbladian $\mathcal{L}$. Expanding in the Pauli basis $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$,

$$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$

yields the equivalent parameterization

$$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$

where $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$. The Hamiltonian parameters $\{a_P\}$ are real, and in the fully unrestricted case the parameter count is $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$0: $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$1 coherent and $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$2 incoherent (Birke et al., 16 Feb 2026).

A broader usage appears in later literature. "Lindblad-like quantum tomography" generalizes standard GKSL tomography to time-local master equations with time-dependent and potentially negative decay rates, while LIMINAL organizes extensibility as data-driven selection over nested Lindblad models with differing locality and hidden degrees of freedom (Varona et al., 2024). This suggests that "extensible" has evolved from describing a deterministic reconstruction protocol to describing a methodological principle: begin with a compact physically structured model and enlarge it only when the data require.

2. Short-time recovery and linear inversion

ELT and its shadow variant are based on the short-time recovery framework of Franca et al., in which Lindblad parameters are inferred from intercept slopes of observable time series (Birke et al., 16 Feb 2026). The coefficients of the generator are collected into a vector $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$3 and the Lindbladian is written as

$$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$4

where each $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$5 is either a coherent term $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$6 or a dissipator term

$$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$7

The measured signal is constructed from Hermitian "input" operators $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$8 and "output" observables $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$9,

$H$0

with $H$1 obtained by idling evolution. Differentiation at $H$2 gives

$H$3

Collecting all configurations yields the linear system

$H$4

For any matrix $H$5 satisfying $H$6, one defines

$H$7

The learnability condition is full rank of the relevant columns of $H$8, and the condition number of $H$9 controls statistical stability (Birke et al., 16 Feb 2026). Under $\{L_k\}$0-locality, $\{L_k\}$1 becomes sparse and each parameter depends only on a few low-weight Pauli correlators. This locality reduction is central to the practical meaning of extensibility.

This short-time linearization differs from maximum-likelihood reconstructions that fit the full matrix exponential $\{L_k\}$2 over a time window, as in the earlier two-qubit superconducting demonstration of Lindblad tomography (Samach et al., 2021). The two approaches are compatible in spirit: both reconstruct a continuous-time generator, but ELT emphasizes slope recovery from deliberately chosen short-time traces rather than global nonlinear fitting.

3. Deterministic ELT protocol

ELT uses all product eigenstates of single-qubit Pauli operators across the subsystem of interest as inputs and all product Pauli measurement bases across that subsystem as outputs. The times are short-time idling traces, uniformly spaced in $\{L_k\}$3, with $\{L_k\}$4 time steps in all reported experiments (Birke et al., 16 Feb 2026).

The implemented procedure has three stages. First, one fixes a parameterization $\{L_k\}$5, computes the transfer matrix

$\{L_k\}$6

and determines any $\{L_k\}$7 such that $\{L_k\}$8 over the chosen parameter set. Second, for each configuration $\{L_k\}$9, one prepares $\mathcal{L}$0 via $\mathcal{L}$1, idles for time $\mathcal{L}$2, applies $\mathcal{L}$3 to rotate into the Pauli measurement basis, measures, and averages $\mathcal{L}$4 outcomes to estimate $\mathcal{L}$5. Third, one computes $\mathcal{L}$6, fits each entry of $\mathcal{L}$7 to a low-order polynomial in $\mathcal{L}$8 — linear fits in the linear-response regime used here — and extracts

$\mathcal{L}$9

For one qubit, ELT used $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$0 configurations, $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$1 shots per configuration per time, $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$2, and $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$3 shots per time step. For three qubits, ELT used $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$4, $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$5, $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$6, and $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$7 shots per time step (Birke et al., 16 Feb 2026).

Statistically, coherent parameters are real and incoherent parameters complex. Magnitudes are reported with uncertainties from the Rice distribution, using 16%/84% quantiles as $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$8-equivalent intervals derived from $\mathcal{P}_n=\{I,X,Y,Z\}^{\otimes n}$9 fits. The appendix of the Birke study compares robust low-polynomial fitting from the short-time recovery framework against $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$0 regression and finds statistically consistent parameters on two-qubit data (Birke et al., 16 Feb 2026).

A common misconception is that tomographic completeness implies scalability. ELT is tomographically complete, but without locality assumptions its cost remains exponential in qubit number. Its extensibility is therefore structural rather than automatic: the protocol can reconstruct more general models, but only locality or other priors make that reconstruction experimentally tractable.

4. Resource scaling and the shadow comparison

The comparison protocol, Shadow Lindblad Tomography (SLT), uses randomized state-preparation and measurement data together with classical-shadow post-processing to estimate the same Lindblad parameters. Its observables are Pauli transfer matrix elements of the idling channel,

$$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$1

followed by the same transformation $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$2 and the same slope extraction at $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$3 (Birke et al., 16 Feb 2026).

For each qubit $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$4, SLT prepares a random eigenstate of a random single-qubit Pauli in $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$5, records the eigenvalue sign $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$6, measures in a random Pauli basis $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$7, and records the measurement sign $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$8. If the randomization agrees with the supports of $$H=\sum_{P\in\mathcal{P}_n} a_P P, \qquad L_k=\sum_{Q\in\mathcal{P}_n} c_{k,Q}Q,$$9 and $$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$0, the single-shot estimator is

$$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$1

and otherwise it is zero. Averaging over $$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$2 randomized shots gives an unbiased estimator with variance bounded by

$$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$3

Hoeffding concentration then gives

$$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$4

This bounded-tail behavior is operationally important. The Birke implementation states that it avoids the need for median-of-means and is compatible with standard $$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$5 regression and Gaussian error propagation, unlike heavy-tailed shadow estimators in other settings (Birke et al., 16 Feb 2026). A second misconception is therefore that all shadow-tomography-based estimators require non-Gaussian robustification; in this specific PTM-based setting, the single-shot estimator is bounded.

The experimentally reported resource comparison is summarized below.

Subsystem	ELT resources	SLT outcome
1 qubit	$$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$6, $$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$7, $$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$8 shots/time	Residuals saturate near ELT uncertainty floor around $$\mathcal{L}(\rho) = -i\sum_{P\in\mathcal{P}_n} a_P [P,\rho] + \sum_{P,Q\in\mathcal{P}_n} D_{P,Q} \Big( P\rho Q - \tfrac{1}{2}\{QP,\rho\} \Big),$$9
3 qubits	$D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$0, $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$1, $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$2 shots/time	Agreement to a few kilohertz around $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$3
5 qubits, 2-local model	ELT estimated $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$4–$D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$5 h under restricted model	SLT used $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$6 randomizations in $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$7 h

Without locality assumptions, ELT requires $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$8 parameters and an informationally complete set of product settings. Under physically motivated $D_{P,Q}=\sum_k c_{k,Q}c_{k,P}^*$9-locality, ELT becomes polynomial, whereas SLT is logarithmic because randomized data are recycled across many low-weight PTM elements (Birke et al., 16 Feb 2026). This logarithmic claim is empirical only under the bounded-support assumptions that were tested on the device.

5. Experimental realizations and recovered dynamics

The experimental platform for the Birke study is a five-qubit superconducting transmon processor with fixed-frequency qubits and tunable-frequency couplers, mounted in a BlueFors XLD1000 with OPX1000 control hardware and orchestrated with the Pelagic platform (Birke et al., 16 Feb 2026). Protocols were deployed on one-qubit, three-qubit, and full five-qubit subsystems.

On one qubit, the dominant recovered parameters were detuning $\{a_P\}$0 and dephasing $\{a_P\}$1, while relaxation channels were captured by $\{a_P\}$2. All SLT parameters agreed with ELT within uncertainties. Diagonalization of $\{a_P\}$3 yielded three jump operators with rates resembling relaxation, dephasing, and temperature-induced excitation, and the Hamiltonian ground state and splitting were consistent with a slightly detuned drive (Birke et al., 16 Feb 2026).

On three qubits, ordering by ELT magnitude showed predominantly single-qubit terms — detuning, dephasing, and relaxation — matching prior superconducting device reports. Every 3-local coherent and incoherent parameter was reported as $\{a_P\}$4 from zero, indicating negligible 3-body dynamics within uncertainties (Birke et al., 16 Feb 2026). This is one of the key empirical locality diagnostics.

On five qubits, unrestricted ELT over all $\{a_P\}$5 parameters was estimated to require $\{a_P\}$6 days under the reported conditions, so the full-device reconstruction used SLT under a restricted 2-local model. In that setting the protocol recovered all single-qubit dissipators and two-qubit interactions in about 9 hours of acquisition, compared with an estimated 58 hours for ELT under the same restricted model (Birke et al., 16 Feb 2026). The recovered single-qubit parameters were typically two orders of magnitude larger than the two-qubit parameters, and the nearest- and next-nearest-neighbor interactions were at most a few kilohertz with couplers idled.

These observations connect directly to earlier superconducting LT results. The 2021 two-qubit experiment reconstructed an always-on $\{a_P\}$7 shift $\{a_P\}$8, showed that restricted single-qubit models missed coupling-induced effects, and demonstrated that reduced single-qubit non-Markovianity could disappear once the coupled neighbor was explicitly included in the modeled subsystem (Samach et al., 2021). A plausible implication is that extensibility is not only about more parameters; it is also about enlarging the modeled subsystem until apparent memory effects are absorbed into a Markovian joint description.

6. Extensions beyond the baseline protocol

Several later directions broaden the meaning of extensible Lindblad tomography beyond deterministic short-time ELT. One extension is to non-Markovian time-local dynamics. "Lindblad-like quantum tomography" estimates time-local master equations

$\{a_P\}$9

explicitly allowing $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$00, while enforcing CPTP of the sampled dynamical maps $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$01. It emphasizes that multiple time snapshots are essential for identifiability of time dependences and analyzes optimal time-grid selection through Fisher information and Bayesian adaptive design (Varona et al., 2024).

A second extension concerns realistic measurements and SPAM. Maximum-likelihood tomography with non-instantaneous measurements replaces fixed POVM elements by trajectory-conditioned effective operators $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$02 obtained from backward adjoint filtering, so that

$$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$03

This framework incorporates imperfections, finite detection efficiency, decoherence during measurement, and continuous-time records, and reduces to ordinary MaxLike tomography when the measurement window is negligible (Six et al., 2015). It broadens Lindblad tomography from instantaneous experiments to record-conditioned inference.

A third extension is stroboscopic or minimal-observable tomography. For generators built from double commutators, the minimal number of distinct observables needed for stroboscopic reconstruction is the index of cyclicity,

$$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$04

which can be much smaller than $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$05 when repeated measurements in time are used (Czerwiński, 2015). More radically, recent genericity results show that for almost all Lindblad semigroups one non-trivial binary measurement evolved at sufficiently many times can be injective on the full state space, with a quantum-channel analog of Takens’ embedding theorem for low-dimensional priors (Rall et al., 17 Jan 2025). This suggests that extensibility may also proceed by trading measurement diversity for dynamical richness.

Further extensions target new inference engines and new experimental settings. Robust Lindbladian Tomography for cyclic gates uses error amplification circuits and first-order linearization of gate composition and repetition, reducing data processing to linear positive-semidefinite programming in the small-error regime (Sugiyama, 16 Mar 2025). Physics-Informed Neural Networks embed the Lindblad master equation directly into a learning objective to infer $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$06, $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$07, and $$\dot{\rho} = \mathcal{L}(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right),$$08 from sparse time-series measurements, yielding a fully differentiable digital twin of the noisy system (Sulc, 15 Sep 2025). LIMINAL, finally, turns extensibility into a statistical workflow: it fits nested candidate models to time-resolved tomography and uses likelihood-ratio tests to determine whether higher-locality Hamiltonian terms, higher-locality dissipation, time dependence, or hidden degrees of freedom are warranted. On a five-qubit superconducting processor, it identified an idling model with three-local Hamiltonian terms and two-local dissipation, while finding no support for three-local dissipation (Severin et al., 1 May 2026).

Taken together, these lines of work indicate that extensible Lindblad tomography is less a single algorithm than a family of reconstruction strategies centered on the same problem: learning open-system generators while controlling experimental cost, enforcing physical structure, and making model complexity responsive to data rather than fixed in advance.