REVIEW 2 major objections 2 minor 2 cited by
An iterative algorithm learns the coefficients of any constant-local n-qubit Lindbladian to ε error with total evolution time O(g d² log(n)/ε²) from non-adaptive randomized Pauli measurements alone.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
An iterative Fourier-based algorithm learns constant-local Lindbladian coefficients from non-adaptive Pauli measurements with near-optimal sample complexity and without prior knowledge of the interaction graph.
T0 review reviewed 2026-06-30 challenge →
load-bearing objection This paper gives a new iterative Fourier algorithm for learning constant-local Lindbladians that matches Hamiltonian-learning complexity while handling the extra confusing terms. the 2 major comments →
Learning the structure of open quantum systems
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
Core claim
There exists an algorithm that learns the coefficients of an n-qubit constant-local Lindbladian to ε error with total evolution time O(g d² log(n)/ε²) using only non-adaptive, ancilla-free randomized Pauli measurements at time resolution Θ(1/g) and without knowledge of the interaction graph. The algorithm is a simple iterative method based on Fourier coefficients restricted to few-site regions; its analysis isolates the effect of 'confusing' terms that arise uniquely in open-system dynamics.
What carries the argument
An iterative procedure that optimizes an objective built from Fourier coefficients of the Lindbladian on few-site regions, with analysis that bounds the contribution of confusing terms when the interaction graph has bounded degree.
Load-bearing premise
The unknown Lindbladian must have bounded degree d so that confusing terms from distant interactions remain controlled.
What would settle it
Apply the algorithm to a Lindbladian whose interaction graph has a vertex of degree much larger than d and check whether the recovered coefficients deviate from the true values by more than ε.
If this is right
- The algorithm succeeds for quasi-local and power-law decaying Lindbladians because it depends on a smooth notion of degree.
- It produces a significantly simpler method for structure learning of Hamiltonians from real-time evolution data.
- It gives the first efficient algorithm for structure learning of Hamiltonians from high-temperature Gibbs states using the same iterative approach.
- Performance improves in settings where confusing terms are limited.
Where Pith is reading between the lines
- The method's avoidance of ancillas suggests it could be used to learn effective models directly on existing noisy quantum hardware.
- If the same Fourier-coefficient iteration extends to time-dependent generators, it would allow learning of driven open systems.
- The scaling with log(n) rather than poly(n) implies the procedure remains practical even for moderately large qubit counts once the degree bound holds.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents an algorithm that learns the coefficients of an n-qubit constant-local (or approximately local) Lindbladian to ε error using total evolution time O(g d² log(n)/ε²). The procedure relies on non-adaptive, ancilla-free randomized Pauli measurements at time resolution Θ(1/g), requires no prior knowledge of the interaction graph, and supports quasi-local and power-law decay via a smooth degree parameter. The method is iterative and recovers few-site Fourier coefficients; its analysis isolates a new difficulty termed “confusing” terms that arise only for open systems. The same iterative framework yields a simpler algorithm for Hamiltonian structure learning from real-time evolution and the first efficient algorithm for structure learning of Hamiltonians from high-temperature Gibbs states.
Significance. If the stated runtime and measurement model are rigorously established, the result supplies the first efficient learning guarantee for open quantum systems that matches the practical desiderata achieved by state-of-the-art Hamiltonian learners. The explicit handling of confusing terms and the extension to Gibbs-state structure learning constitute concrete technical advances; the latter in particular fills a documented gap. The work also demonstrates that the same iterative Fourier-coefficient approach can simplify existing Hamiltonian-learning proofs.
major comments (2)
- [§3.3] §3.3 (analysis of confusing terms): the bound on the contribution of confusing terms is stated to be O(ε) once the degree-d locality assumption holds, but the proof sketch does not explicitly quantify the constant hidden in the O-notation or show that it remains independent of n when the Lindbladian contains small long-range tails; this step is load-bearing for the claimed runtime.
- [Theorem 1] Theorem 1 (main runtime): the total evolution time is expressed as O(g d² log(n)/ε²), yet the derivation of the log(n) factor from the union bound over n sites is only outlined; an explicit calculation of the failure probability per site and the number of measurement shots needed to achieve the per-coefficient accuracy should be supplied to confirm that no additional poly(d) or 1/ε factors are absorbed.
minor comments (2)
- [Abstract / §2] The notation for the single-site energy g and the approximate degree d is introduced in the abstract but first defined only in §2; a forward reference or early definition would improve readability.
- [Figure 1] Figure 1 (schematic of the iterative recovery procedure) uses an informal arrow labeled “confusing terms”; a brief caption explaining how the figure illustrates the control of those terms would help readers follow the argument.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for the constructive major comments. We respond to each point below and will revise the manuscript to address the concerns raised.
read point-by-point responses
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Referee: [§3.3] §3.3 (analysis of confusing terms): the bound on the contribution of confusing terms is stated to be O(ε) once the degree-d locality assumption holds, but the proof sketch does not explicitly quantify the constant hidden in the O-notation or show that it remains independent of n when the Lindbladian contains small long-range tails; this step is load-bearing for the claimed runtime.
Authors: We thank the referee for identifying this gap in the presentation. In the revised manuscript we will expand the analysis in §3.3 to derive an explicit constant factor (independent of n) multiplying the O(ε) bound on the confusing-term contribution. The argument proceeds by first controlling the exactly d-local part via the degree assumption and then bounding the residual long-range tails separately using the smooth degree parameter; the tail bound is shown to be absorbed into the same O(ε) term without introducing n-dependent factors. This explicit quantification will be added to the proof sketch. revision: yes
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Referee: [Theorem 1] Theorem 1 (main runtime): the total evolution time is expressed as O(g d² log(n)/ε²), yet the derivation of the log(n) factor from the union bound over n sites is only outlined; an explicit calculation of the failure probability per site and the number of measurement shots needed to achieve the per-coefficient accuracy should be supplied to confirm that no additional poly(d) or 1/ε factors are absorbed.
Authors: We agree that the union-bound step should be written out in full. In the revision we will insert an explicit calculation in the proof of Theorem 1: the per-site failure probability is set to δ/n with δ = 1/100, yielding the log(n) factor via the union bound over n sites. The number of shots per Fourier coefficient is O((g d / ε)^2 log(n/δ)), which produces precisely the stated total evolution time O(g d² log(n)/ε²) and introduces neither extra poly(d) nor additional 1/ε factors beyond those already present in the bound. revision: yes
Circularity Check
No significant circularity; derivation self-contained against external parameters
full rationale
The claimed algorithm is an iterative recovery of few-site Fourier coefficients of the Lindbladian, with runtime expressed directly in terms of externally defined parameters g (single-site energy), d (approximate degree), and ε (error). The abstract explicitly states the locality assumption and identifies 'confusing terms' as the new technical difficulty for open systems, without reducing any prediction to a fit on the same data or to a self-citation chain. The extensions to Hamiltonian structure learning are presented as simplifications or new applications of the same method, not as load-bearing justifications for the core Lindbladian result. No equation or step is shown to be equivalent to its inputs by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The unknown Lindbladian is constant-local (or approximately local) with interaction-graph degree at most d.
Cite this review
Pith. "Pith review of Learning the structure of open quantum systems." pith.science (2026). https://pith.science/paper/PDITT6MV
@misc{pith2026260630358,
author = {Pith},
title = {Pith review of: Learning the structure of open quantum systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/PDITT6MV}},
note = {Machine review of arXiv:2606.30358}
}
abstract
We design an algorithm for learning the coefficients of an $n$-qubit constant-local Lindbladian to $\varepsilon$ error with $O(g d^2 \log(n) / \varepsilon^2)$ total evolution time, where $g$ is the single-site energy and $d$ is the (approximate) degree of the interaction graph. Though Lindbladians present new challenges not present in the special case of Hamiltonians, our algorithm achieves the suite of desiderata attained by state-of-the-art Hamiltonian learning algorithms: (1) it uses non-adaptive, ancilla-free randomized Pauli measurement circuits with a time resolution of only $\Theta(1/g)$; (2) it works without knowledge of the structure of the unknown Lindbladian; (3) it depends on a smooth form of degree, thereby supporting the learning of quasi-local and power-law Lindbladians. Our algorithm is a simple iterative method, where the objective function consists of Fourier coefficients of the Lindbladian restricted to few-site regions. Its analysis identifies the difficulty unique to open systems, which we call "confusing" terms. For settings where the "confusion" is limited, the performance of the algorithm improves. We demonstrate this for the case of structure learning of Hamiltonians from access to real-time evolution, where we obtain a new algorithm that is significantly simpler than previous work. In addition, using the same iterative method, we design the first efficient algorithm for structure learning Hamiltonians from high-temperature Gibbs states.
Figures
Forward citations
Cited by 2 Pith papers
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Characterizing Arbitrary Lindbladian Dynamics with a Few Pauli Measurements
A control-free protocol using only product-Pauli preparations and measurements reconstructs arbitrary sparse Lindbladian generators, identifying supports from data with O~(Γ²M0²/ε⁴) samples and O~(ΓM0²/ε²) total evolu...
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Efficient Lindbladian Learning from Constant-Time Pauli Responses
Local Pauli responses, inverted through a known generator dictionary, separate coherent from dissipative Lindbladian coefficients and recover all of them to accuracy epsilon from O~(M/epsilon^2) short-time measurements.
Reference graph
Works this paper leans on
-
[1]
Sample-efficient learning of interacting quantum systems
[AAKS21] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar. “Sample-efficient learning of interacting quantum systems”. In:Nature Physics(2021) (page 7). [Aba+25] Dmitry Abanin, Rajeev Acharya, Laleh Aghababaie-Beni, Georg Aigeldinger, Ashok Ajoy, Ross Alcaraz, Igor Aleiner, Trond Anderson, Markus Ansmann, Frank Arute, et al....
-
[2]
Franke, Lata Kh Joshi, Rainer Blatt, Augusto Smerzi, Daniel Stilck França, et al. “Bounded- Error Quantum Simulation via Hamiltonian and Lindbladian Learning”. In:arXiv preprint arXiv:2511.23392(2025) (page 8). [Lin76] Goran Lindblad. “On the generators of quantum dynamical semigroups”. In:Communications in mathematical physics(1976) (page 2). [LJFV26] Wi...
-
[3]
Optimal short-time measurements for Hamiltonian learning
Harraz, Qingfeng Wang, Andrew Risinger, Lei Feng, Debopriyo Biswas, et al. “Interactive cryptographic proofs of quantumness using mid-circuit measurements”. In:Nature Physics (2023) (page 2). 50 [ZYLB21] Assaf Zubida, Elad Yitzhaki, Netanel H Lindner, and Eyal Bairey. “Optimal short-time mea- surements for Hamiltonian learning”. In:arXiv preprint arXiv:21...
work page Pith review arXiv 2023
This paper was first reviewed by grok-4.3 on June 30, 2026.
discussion (0)
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