Near-Optimal Learning of Local Lindbladians

Abstract: We study the problem of learning local Lindbladians from black-box access to the physical evolution, and the goal is to estimate all Hamiltonian and dissipative coefficients. We give an algorithm built directly from finite-time channel probes, which runs the unknown evolution for short times, estimates the corresponding Pauli transfer matrices from classical shadows, and converts these estimates into Lindbladian coefficients by stable local Fourier inversions. For fixed locality and bounded dissipative site degree, the uses of the dynamical evolution and total evolution time scale as $\widetilde{O}(Λ^2/\varepsilon^2)$ and $\widetilde{O}(Λ/\varepsilon^2)$ respectively, in the local dynamical strength bound $Λ$ and target accuracy $\varepsilon$, with only logarithmic dependence on the number of qubits. The algorithm is non-adaptive, uses no ancillas, and uses only random product states as inputs followed by random Pauli measurements. The method does not require knowing the support of the Lindbladian in advance. We complement the algorithm with matching lower bounds, showing that the learning algorithm is near-optimal both in physical dynamics accesses and in total evolution time. We construct a single-qubit dephasing Lindbladian family that already requires $Ω(Λ^2/\varepsilon^2)$ channel uses and $Ω(Λ/\varepsilon^2)$ total evolution time, even for adaptive algorithms with arbitrary ancillas and measurements. In particular, the lower bounds imply that the Heisenberg-limited scaling achievable for Hamiltonian learning is information-theoretically impossible once dissipative coefficients must be estimated.

• The paper introduces a non-adaptive, ancilla-free protocol for learning local Lindbladians, achieving near-optimal scaling in sample and evolution time.
• It employs direct channel probing, classical shadow tomography, and endpoint differentiation to efficiently recover Lindbladian coefficients.
• Theoretical lower bounds confirm the optimal scaling of Ω(Λ²/ε²) channel queries and Ω(Λ/ε²) total evolution time, setting a standard for dissipative parameter estimation.

Near-Optimal Learning of Local Lindbladians: Formal Summary

Problem Setting and Motivation

The paper "Near-Optimal Learning of Local Lindbladians" (2606.20535) addresses the estimation of all Hamiltonian and dissipative coefficients of a local Lindbladian generator describing continuous-time, Markovian open-system dynamics, given only black-box access to the physical evolution. Lindbladian learning is critical for characterizing noisy quantum devices, error diagnosis, and validation of engineered dissipation. The challenge is to devise learning protocols that are not only theoretically rigorous, but also experimentally favorable: minimal quantum resources (no ancillas, simple input states, random Pauli measurements), non-adaptive, and independent of prior knowledge about Lindbladian support.

The exponential parameter space of generic Lindbladians precludes efficient learning unless physically realistic structure—locality and bounded dissipative site degree—is assumed. For $k$-local Lindbladians and bounded degree $d = O(1)$, both the number of coefficients and the relevant Pauli transfer matrix (PTM) entries scale polynomially in system size.

Main Algorithm and Contributions

The proposed algorithm operates by:

1. Direct Channel Probing: Executes the unknown evolution at selected short times, preparing random product states and performing random single-qubit Pauli measurements. No ancillas or entanglement are used.
2. Shadow Process Tomography: Estimates local PTM entries for all relevant Pauli pairs via classical shadow techniques. These entries encode dynamical information necessary for coefficient recovery.
3. Endpoint Differentiation: Applies stable Chebyshev interpolation to extract the derivative at $t=0$, thereby estimating PTM generator entries that directly relate to Lindbladian coefficients.
4. Local Fourier Inversion: Converts PTM generator estimates to coefficient space via local Walsh–Hadamard transforms. Recovery is stable (condition number 1), and errors are not amplified.
5. De-aliasing: Employs a thresholded peeling recursion to resolve the ambiguity arising from local extension sums, suppressing error amplification to a constant multiplicative factor.

Strong numerical results include:

• Channel-use and evolution-time scaling: Both scale as $\widetilde{O}(\Lambda^2/\epsilon^2)$ and $\widetilde{O}(\Lambda/\epsilon^2)$ respectively (where $\Lambda$ is the local dynamical strength and $\epsilon$ the target accuracy), with only logarithmic dependence on the number of qubits and failure probability.
• Non-adaptive, ancilla-free protocol: Only random product states and single-qubit Pauli measurements required.
• Support-agnostic recovery: Does not require knowing the Lindbladian support in advance.

Matching Lower Bounds

The paper provides information-theoretic lower bounds, proven by reduction to one-qubit dephasing Lindbladian instances, even for adaptive protocols with arbitrary ancillas and measurements:

• Channel use: Any algorithm must use $\Omega(\Lambda^2/\epsilon^2)$ channel accesses.
• Total evolution time: At least $\Omega(\Lambda/\epsilon^2)$ total evolution time is required.
• Quantum limit: These lower bounds establish that, unlike the Heisenberg-limited scaling of Hamiltonian learning ($O(1/\epsilon)$), Lindbladian learning for dissipative parameters cannot surpass the standard quantum limit ($d = O(1)$0). The gap persists even for full learning, although dissipation detection alone can be Heisenberg-limited.

Technical Details

Stability and Error Amplification

The recovery from PTM generator entries to Lindbladian coefficients via local Walsh--Hadamard inversions exhibits constant condition number (orthogonal transform), eliminating instance-dependent amplification. The de-aliasing recursion is guaranteed to suppress error amplification strictly to constant factors due to bounded site degree.

Complexity Analysis

For fixed locality and dissipative site degree, per-coefficient estimation requires $d = O(1)$1 queries and $d = O(1)$2 total evolution time. Processing time scales as $d = O(1)$3, where $d = O(1)$4 is the locality.

Comparison with Prior Work

This protocol attains stronger guarantees and practical simplicity in the physically local regime compared to previous approaches [franca2024efficient, ivashkov2026ansatz, romanov2026learning]. Notably, it achieves optimal scaling without ancillary systems and with non-adaptive strategies, in contrast to error-correction-inspired transcending protocols.

Implications and Outlook

Practical: The method facilitates robust, scalable Lindbladian identification for the characterization of noisy quantum hardware, enabling systematic validation and debugging of quantum devices where only local access is feasible.

Theoretical: The impossibility of Heisenberg scaling for dissipative parameters fundamentally separates Lindbladian learning from Hamiltonian learning and dissipation detection. This aligns with quantum metrology results showing generic noise destroys Heisenberg scaling for phase estimation.

Future Directions:

• Investigate which subsets of Lindbladian functionals allow Heisenberg-limited scaling.
• Optimize scaling constants (currently exponential in locality and degree).
• Extend methods to exact structure learning, stronger metrics (e.g., diamond norm), and beyond Markovian/time-independent generators.

Conclusion

This work rigorously establishes a non-adaptive, ancilla-free algorithm for learning local Lindbladians to near-optimal sample and time complexity. The protocol is support-agnostic and experimentally friendly, and the matching lower bounds confirm the scaling is information-theoretically optimal. This both resolves fundamental questions for open-system identification and provides a practical tool for scalable quantum device characterization (2606.20535).