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arxiv: 2606.20535 · v1 · pith:Z5JDDINHnew · submitted 2026-06-18 · 🪐 quant-ph

Near-Optimal Learning of Local Lindbladians

Itai Arad , Zhili Chen , Naixu Guo , Patrick Rebentrost , Zhan Yu This is my paper

Pith reviewed 2026-06-26 17:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Lindbladian learningopen quantum systemsquantum channel tomographyclassical shadowsLindblad master equationquantum dynamics learningdissipative coefficient estimation
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0 comments X

The pith

An algorithm learns all coefficients of a local Lindbladian from short black-box evolutions with near-optimal Õ(Λ²/ε²) accesses.

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

The paper gives an algorithm that estimates every Hamiltonian and dissipative coefficient of an unknown local Lindbladian by running the evolution for short times, estimating Pauli transfer matrices with classical shadows, and recovering the coefficients through local Fourier inversion. For fixed locality and bounded dissipation per site the number of evolution uses scales as Õ(Λ²/ε²) and the total evolution time as Õ(Λ/ε²), with only logarithmic dependence on system size. The procedure is non-adaptive, requires no ancillas, and works from random product states and Pauli measurements without knowing the support in advance. Matching lower bounds are proved by constructing a single-qubit dephasing family that forces the same scaling even for adaptive algorithms with arbitrary ancillas and measurements, showing that Heisenberg-limited scaling is impossible once dissipative coefficients must be learned.

Core claim

Local Lindbladians with fixed locality and bounded dissipative site degree can be learned from black-box dynamical evolution accesses using Õ(Λ²/ε²) channel uses and Õ(Λ/ε²) total evolution time, via non-adaptive finite-time probes, classical shadows, and local Fourier inversion, and this is near-optimal as shown by matching lower bounds that rule out Heisenberg scaling.

What carries the argument

Stable local Fourier inversion that converts short-time Pauli transfer matrix estimates into Lindbladian coefficients.

If this is right

  • The algorithm needs no ancillas and uses only random product inputs followed by random Pauli measurements.
  • It recovers coefficients without prior knowledge of the Lindbladian support.
  • The lower bounds apply to any adaptive strategy with arbitrary ancillas and measurements.
  • Estimating dissipative coefficients rules out Heisenberg-limited scaling that is possible for pure Hamiltonian learning.

Where Pith is reading between the lines

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

  • The information-theoretic gap between Hamiltonian and Lindbladian learning may appear in other open-system identification tasks.
  • Practical experiments could test whether the local inversion step remains stable under realistic noise.
  • The non-adaptive nature suggests the method could be parallelized across multiple short-time experiments.

Load-bearing premise

The unknown Lindbladian is local with fixed locality and bounded dissipative site degree per site.

What would settle it

A family of single-qubit dephasing Lindbladians that forces Ω(Λ²/ε²) channel uses and Ω(Λ/ε²) total evolution time even for adaptive algorithms with ancillas.

read the original 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 $\widetilde{O}(\Lambda^2/\varepsilon^2)$ and $\widetilde{O}(\Lambda/\varepsilon^2)$ respectively, in the local dynamical strength bound $\Lambda$ 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 $\Omega(\Lambda^2/\varepsilon^2)$ channel uses and $\Omega(\Lambda/\varepsilon^2)$ 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.

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

0 major / 2 minor

Summary. The manuscript presents an algorithm to learn all Hamiltonian and dissipative coefficients of an unknown local Lindbladian from black-box access to its finite-time evolution. The algorithm uses short-time channel probes, estimates Pauli transfer matrices via classical shadows on random product states, and recovers coefficients via stable local Fourier inversion. For fixed locality and bounded dissipative site degree, it achieves Õ(Λ²/ε²) channel uses and Õ(Λ/ε²) total evolution time with only logarithmic dependence on system size n; the algorithm is non-adaptive and ancilla-free. Matching lower bounds are shown via an explicit single-qubit dephasing Lindbladian family, establishing near-optimality and ruling out Heisenberg-limited scaling for any algorithm (even adaptive with ancillas) once dissipative coefficients are estimated.

Significance. If the central claims hold, the result is significant: it supplies the first near-optimal learning procedure for local open-system dynamics and demonstrates that the presence of dissipation qualitatively changes the sample and time complexity relative to Hamiltonian learning. Strengths include the explicit, parameter-free lower-bound construction, the non-adaptive ancilla-free implementation, and the fact that support knowledge is not required. The bounded-degree hypothesis is used to guarantee stable inversion without hidden n-dependence.

minor comments (2)
  1. The abstract and introduction should explicitly state the precise definition of 'bounded dissipative site degree' (including any implicit constants) so that the assumption's scope is unambiguous to readers.
  2. A short paragraph comparing the obtained Õ(Λ/ε²) total time to the corresponding Hamiltonian-learning bound would help readers immediately see where the extra factor arises.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the provided report, so we have no individual points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The algorithm derives its sample and time complexity bounds from explicit local Fourier inversion of short-time Pauli transfer matrices estimated via classical shadows, under the stated locality and bounded-degree assumptions. The matching lower bounds are constructed independently from an explicit single-qubit dephasing Lindbladian family that applies even to adaptive algorithms with ancillas; this family does not reference or reduce to the upper-bound procedure, any fitted parameters, or self-citations. No self-definitional steps, renamed known results, or load-bearing self-citation chains appear. The derivation is therefore self-contained against external information-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum mechanics (Lindblad form of the master equation) and locality assumptions; no free parameters are fitted inside the algorithm, and no new entities are postulated.

axioms (2)
  • domain assumption Evolution is generated by a time-independent local Lindbladian with fixed locality and bounded dissipative site degree
    Invoked to guarantee that local Fourier inversion recovers all coefficients from short-time probes without support knowledge.
  • standard math Standard quantum channel tomography via classical shadows is possible with random product states and Pauli measurements
    Used to convert finite-time evolution into Pauli transfer matrix estimates.

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discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Robust Structure Learning of $k$-local Lindbladians

    quant-ph 2026-06 unverdicted novelty 8.0

    Protocol learns k-local Lindbladians to ε accuracy with Õ(n^{2k}/ε²) samples and projects to valid generators; improves to log n under sparsity assumptions.

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