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arxiv: 2606.20706 · v1 · pith:OO72JWONnew · submitted 2026-06-15 · 🪐 quant-ph

Efficient and SPAM-Robust Ansatz-Free Lindbladian Learning

Savar D. Sinha This is my paper

Pith reviewed 2026-06-27 03:02 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Lindbladian learningSPAM-robust protocolsBell samplingopen quantum systemsgauge degrees of freedomansatz-free methodsquantum noise characterizationMarkovian dynamics
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The pith

Bell sampling enables an efficient ansatz-free algorithm to learn Lindbladians that stays robust to constant SPAM errors on sparse cases.

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

The paper establishes an efficient method for learning the Lindbladian that governs Markovian open quantum system dynamics without requiring any ansatz on the form of the noise. It shows that Bell sampling allows polynomial-time classical postprocessing to recover the dynamics, and extends this to a SPAM-robust protocol that recovers the gauge-independent components of sparse Lindbladians even when state preparation and measurement errors are constant order. This matters because accurate characterization of dissipative dynamics is necessary for designing fault-tolerant quantum computers, where SPAM noise is a major practical obstacle. The work also gives the first rigorous identification of which parts of the Lindbladian can be learned under such noise.

Core claim

Using Bell sampling, we provide an efficient, ansatz-free Lindbladian learning algorithm with polynomial-time classical postprocessing. We also introduce the first efficient SPAM-robust protocol capable of learning the gauge-independent components of sparse Lindbladians to arbitrary precision in the presence of constant-order SPAM error, while rigorously characterizing the gauge degrees of freedom in noisy Lindbladian learning.

What carries the argument

Bell sampling access model together with the separation of gauge-independent Lindbladian components under SPAM noise

If this is right

  • The algorithm recovers the full Lindbladian using only polynomial classical postprocessing time.
  • Sparse Lindbladians remain learnable to arbitrary precision even with constant-order SPAM error.
  • Gauge degrees of freedom are identified exactly, showing which Lindbladian components cannot be recovered from noisy data.
  • The method requires no ansatz beyond the Markovian assumption and sparsity in the robust variant.

Where Pith is reading between the lines

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

  • The protocol could be used to calibrate real quantum hardware by extracting accurate noise models from Bell samples alone.
  • Gauge characterization implies that global basis choices or overall phases in the Lindbladian will remain undetermined under any amount of SPAM noise.
  • The approach may extend to estimating effective Lindbladians in regimes where full process tomography is impractical due to system size.

Load-bearing premise

The underlying dynamics must be Markovian and thus take Lindblad form, and Bell sampling must be realizable without extra uncontrolled errors.

What would settle it

Apply the protocol to a system with known sparse Lindbladian and constant SPAM error, then check whether the output matches the true gauge-independent components to within the claimed precision.

Figures

Figures reproduced from arXiv: 2606.20706 by Savar D. Sinha.

Figure 3.1
Figure 3.1. Figure 3.1: Lindbladian coefficient learning via Bell sampling. (a) Protocol for learning diagonal components of Lindbladian L = Í 𝑖, 𝑗 𝜒𝑖 𝑗𝑃𝑖𝜌𝑃𝑗 , where 𝑖 = 𝑗, as well as support of Hamiltonian outside dissipator support and its products. By measuring in the Bell basis after evolving a Bell state, we can sample Paulis with probability corresponding to the coefficients of the Lindbladian. (b) By performing either 𝐶 … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Error channels after gauge transformation depicted for a single qubit. The transformation 𝑒 −𝜃Γ acts as a depolarizing channel, compressing the image of E1, while 𝑒 𝜃Γ expands the image of E2. We can only choose 𝜃 such that E2 remains a quantum channel (i.e. the map does not expand outside the Bloch sphere) and E1 does not move too far from the identity channel I [PITH_FULL_IMAGE:figures/full_fig_p037_4… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Lindblad evolution after gauge transformation depicted for a single qubit. The transformation 𝑒 −𝜃Γ · 𝑒 𝜃Γ shrinks the deviation of the center of the ellip￾soid, shifting it towards the origin, effectively making the time evolution operator more unital. Since this ellipsoid always remains within the Bloch sphere for 𝜃 ≥ 0, we have that L′ is always a valid Lindbladian in the single-qubit case [PITH_FULL… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: SPAM-Robust Lindbladian Learning. (a) Protocol for learning diag￾onal and Hamiltonian components of Lindbladian L = Í 𝑖, 𝑗 𝜒𝑖 𝑗𝑃𝑖𝜌𝑃𝑗 using eigen￾value 𝑚𝑘 from Pauli twirled evolution on a 𝑃𝑘 +1-eigenstate and compressed sensing. (b) Modified protocol for estimating off-diagonals by inserting a Clifford 𝐶 after Lindblad evolution. In the Heisenberg picture, we therefore have that L ∗ [𝑄] = ∑︁ 𝑃 𝛾𝑃 [𝑃𝑄𝑃 − … view at source ↗
read the original abstract

Describing the dynamics of open systems is essential for fault-tolerant quantum computation. Under Markovian assumptions, we can characterize dissipative dynamics via the Lindbladian. Using Bell sampling, we provide an efficient, ansatz-free Lindbladian learning algorithm with polynomial-time classical postprocessing. Motivated by the prevalence of state preparation and measurement (SPAM) noise on near-term devices, we also introduce the first efficient SPAM-robust protocol capable of learning the gauge-independent components of sparse Lindbladians to arbitrary precision in the presence of constant-order SPAM error. In doing so, we provide the first rigorous characterization of the gauge degrees of freedom in noisy Lindbladian learning, precisely identifying which components remain learnable under SPAM noise.

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 / 3 minor

Summary. The paper claims to introduce an efficient, ansatz-free algorithm for learning Lindbladians of open quantum systems via Bell sampling, with polynomial-time classical postprocessing. It further presents the first efficient SPAM-robust protocol that learns the gauge-independent components of sparse Lindbladians to arbitrary precision under constant-order SPAM error, accompanied by the first rigorous characterization of gauge degrees of freedom in noisy Lindbladian learning.

Significance. If the central claims hold, the work would be significant for quantum device characterization on near-term hardware. It supplies the first rigorous treatment of gauge freedom under SPAM noise and an efficient SPAM-robust learning protocol for sparse Lindbladians, directly addressing a practical barrier to fault-tolerant quantum computation. The Bell-sampling access model and polynomial postprocessing are explicit strengths when the derivations are verified.

minor comments (3)
  1. The abstract and introduction should explicitly state the scaling with system dimension n and sparsity parameter s in the runtime and sample complexity bounds, as these are central to the efficiency claim.
  2. Notation for the gauge transformation and the decomposition into gauge-dependent vs. gauge-independent components should be introduced with a dedicated preliminary section or appendix to improve readability for readers outside the immediate subfield.
  3. Figure captions for any runtime or error plots should include the precise parameter settings (e.g., SPAM error magnitude, sparsity level) used in the numerical demonstrations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of our contributions to SPAM-robust Lindbladian learning, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; algorithmic claims rest on external assumptions

full rationale

The paper describes an algorithmic protocol for Lindbladian learning via Bell sampling, with SPAM-robust extensions for sparse cases and a gauge characterization. No equations, derivations, or results in the provided text reduce by construction to fitted parameters, self-definitions, or self-citation chains. The central contributions are efficiency bounds and learnability statements under explicitly stated modeling assumptions (Markovian dynamics, sparsity, Bell sampling realizability), which are independent of the algorithm's outputs. This is a standard non-circular algorithmic result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard assumption that the open-system dynamics admit a Lindblad form (Markovian master equation) and on the existence of a Bell sampling oracle. No free parameters, ad-hoc entities, or non-standard axioms are introduced in the abstract.

axioms (2)
  • domain assumption The system obeys a Markovian master equation in Lindblad form.
    Stated in the opening sentence of the abstract as the setting for the learning task.
  • domain assumption Bell sampling provides direct access to the required measurement statistics.
    The algorithm is built around Bell sampling; this access model is presupposed.

pith-pipeline@v0.9.1-grok · 5644 in / 1410 out tokens · 40490 ms · 2026-06-27T03:02:01.068615+00:00 · methodology

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    Firstly,weneedtosatisfytheSPAM constraint, meaning that 𝑝′ 1 +𝑝 ′ 2 =2−𝑒 −𝜃 (1−𝑝 1) −𝑒 𝜃 (1−𝑝 2) ≤ 𝜀SPAM 2 61 Secondly,while0≤𝑝 ′ 1 ≤1,wehavethat𝑝 ′ 2canbenegativeif𝜃isleftunconstrained. Hence, we must have that 𝑝′ 2 =1−𝑒 𝜃 (1−𝑝 2) ≥0=⇒𝜃≤ −ln(1−𝑝 2) Using the first derivation condition, we find that the𝜃that maximizes the LHS of the first inequality is gi...

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    Consequently, we have that 𝐿 𝑗 𝜌𝐿 † 𝑗 +𝐿 𝑘 𝜌𝐿 † 𝑘 = 1 4𝑛 (𝑃 𝑗 +𝑅𝑃 𝑗 )𝜌(𝑃 𝑗 +𝑃 𝑗 𝑅)= 1 4𝑛 [𝑃 𝑗 𝜌𝑃 𝑗 +𝑅𝑃 𝑗 𝜌𝑃 𝑗 +𝑅𝑃 𝑘 𝜌𝑃 𝑘 +𝑃𝑘 𝜌𝑃 𝑘 ] 64 Hence, we have that these𝐿 𝑗 , 𝐿 𝑘 jump operators collectively capture both the 𝑅𝑃 𝑗 𝜌𝑃 𝑗 and𝑅𝑃 𝑘 𝜌𝑃 𝑘 terms. Hence, if we now iterate over all the Pauli terms with this construction, we would have that ∑︁ 𝑗 𝐿 𝑗 𝜌𝐿 † 𝑗 = 1...