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REVIEW 4 major objections 5 minor 2 cited by

Robust Lindbladian Estimation for Quantum Dynamics

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that logarithm-search Lindbladian fitting can be made practical and robust for two-qubit gates on current noisy hardware.

desk verdict A practical, honestly-bounded improvement to logarithm-search Lindbladian fitting with real-hardware demonstrations, but the heuristic c1/c2 safety envelope needs systematic mapping before the robustness claim is fully supported. read the letter →

arxiv 2507.07912 v1 pith:RQWT7EJN submitted 2025-07-10 quant-ph physics.comp-ph

Yinchen Liu , James R. Seddon , Tamara Kohler , Emilio Onorati , Toby S. Cubitt This is my paper · ORCID
classification quant-phphysics.comp-ph MSC 81P6815A1690C25 PACS 03.67.-a
keywords LindbladianestimationquantumprocesstomographygatesetmatrixlogarithmMarkoviannoisetwo-qubitgatesSPAMerrorscharacterization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the long-standing logarithm-search approach to Lindbladian fitting—finding a Lindbladian generator close to a matrix logarithm of a tomographically estimated channel—can be turned into a practical, robust tool for characterising two-qubit gates. It introduces two algorithms: Convex Solve, which handles gates whose ideal transfer matrix has no eigenvalues near the real negative axis, and Alternating Projections, which handles difficult cases such as CNOT and ISWAP by starting from the ideal gate and iteratively projecting between logarithm branches and the Lindbladian cone. It then wraps these in Gate Set Flip-Flop, a protocol that alternates gauge optimisation with Lindbladian fitting to remove state preparation and measurement errors. On simulated data with $10^{4}$ shots, Alternating Projections succeeded on 600/600 cases by the loose success criterion; the whole pipeline was demonstrated on superconducting-qubit data, fitting noise for CNOT, RZX, RZZ, idling, and cross-talk processes. If the claim holds, device-noise characterisation for error mitigation can be done with fewer shots and less classical post-processing than previously thought.

What carries the argument

The machinery is the matrix logarithm set log(E) with its branch structure, together with two projections: projection of a matrix onto the closed convex cone of Lindbladians via a semidefinite program using the Γ-involution characterisation, and projection of a Lindbladian's eigenvectors onto approximate eigenspaces of the tomographic data. Alternating Projections iterates these two projections, seeded by the ideal gate's Lindbladian and by diagonal random perturbations; eigenvalue clustering with precision β merges tomographic eigenvalues into degenerate eigenspaces so that the constructed logarithm Ā belongs to the enlarged set ~log(E). Gate Set Flip-Flop adds a third alternating step: with Lindbladians fixed, it minimises the LogSumExp-smoothed gauge objective over B; with B fixed, it fits Lindbladians to B g̃^{-1} P̃_i $B^{{-1}}$ using Convex Solve or Alternating Projections.

What would settle it

Run Alternating Projections on simulated CNOT data with coherent X, amplitude damping, and dephasing noise at ∥L*−Lideal∥ ≈ 0.5 (average gate fidelity roughly 93%): the paper's own Fig. 6 predicts Success 1 would fail, so a single instance that passes there would falsify the claimed threshold. Alternatively, a hardware dataset with strong idling noise returning no L with ∥e^L−E∥ close to ∥E−E*∥ would falsify the practical-robustness claim.

Watch

Extended reading notes

Core claim

The central discovery is that the non-uniqueness of the matrix logarithm—the obstruction that made earlier logarithm-search algorithms slow or impractical—can be tamed using knowledge of the ideal gate. Convex Solve succeeds on a single principal branch whenever the ideal transfer matrix has no eigenvalues close to the real negative axis; this covers identity, √X⊗I, T⊗I, and more. When real-negative eigenvalues are present, Alternating Projections clusters nearby tomographic eigenvalues into approximate degenerate eigenspaces, assigns the ideal gate's eigenvectors to those subspaces by minimum-cost flow, builds an approximate logarithm, projects it onto the Lindbladian cone, and iterates. The paper reports 100% success on the loose criterion over 600 simulated two-qubit instances, plus successful fits to real hardware data including cross-talk and idling processes. It also proves Theorem 3.1: without statistical error, if the true generator is close enough to the ideal generator in the sense ∥L*−Lideal∥ < (π−ρ(Lideal))/κ(V), the principal-branch Convex Solve already returns a valid Lindbladian logarithm.

Load-bearing premise

The whole method relies on the true noise being close to the ideal gate: the tomographic estimate must be near some Markovian channel, and the true Lindbladian must be near the known ideal Lindbladian, but the paper gives no rigorous bounds on how close is close enough.

Editorial extensions

If this is right

  • For gates like √X⊗I, T⊗I, and identity, a single principal-branch convex solve (one SDP) suffices under weak Markovian noise, dramatically cutting classical post-processing.
  • For CNOT, ISWAP and similar gates, Alternating Projections reduces run-times from weeks to under two minutes for an ISWAP example studied in the prior algorithm.
  • Gate Set Flip-Flop yields Markovian channel estimates that fit real tomographic data better than the ideal gate, and converges within about 8–12 iterations on the hardware data tested.
  • The protocol can flag non-Markovianity: failure to find a Lindbladian that exponentiates close to the data is evidence of non-Markovian or time-dependent noise.
  • SPAM errors no longer have to be assumed absent: the flip-flop procedure estimates gauge and noise self-consistently.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Implicitly, the separate fits of RZX(0.5)^k instances could be unified into a single time-series fit with a time-dependent Lindbladian, which would likely improve shot efficiency and consistency.
  • The 61.4% Success 2 rate suggests the algorithm often overfits statistical error; users wanting the true generator should treat jump-operator details cautiously unless the corresponding γ coefficients are large.
  • The branch-enumeration criterion suggests a cheap pre-screening test: if the ideal transfer matrix has no real-negative eigenvalues, skip random basis search and use Convex Solve on the principal branch.
  • The method's scalability ceiling is set by full tomography; extending it beyond two-qubit gates would likely require combining it with sparse Pauli-Lindblad assumptions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper addresses the problem of fitting time-independent Lindbladian generators to quantum process tomography data. It introduces two Lindbladian fitting algorithms—Convex Solve and Alternating Projections—and augments them with a gate-set-tomography-inspired procedure called Gate Set Flip-Flop to handle SPAM errors. The authors prove a noiseless-case guarantee for Convex Solve (Theorem 3.1), report extensive synthetic benchmarks with simulated shot noise, and demonstrate the methods on data from IBM superconducting-qubit hardware. The central claim is that logarithm-search Lindbladian fitting, previously considered impractical, can be made efficient and robust enough for two-qubit gates on current noisy hardware.

Significance. If the central claim is accepted, the paper would be a useful practical advance for few-qubit noise characterization: it offers a large runtime improvement over the prior logarithm-search algorithm of [55], provides the first hardware demonstrations for this family of methods, and introduces a flexible SPAM-handling protocol that can be paired with other Lindbladian fitters. The synthetic benchmarks are extensive and the main analytic result, Theorem 3.1, is clean and correctly scoped. However, the robustness claim is currently supported mainly by empirical success on a fixed range of noise strengths; the paper explicitly leaves the key closeness parameters c1 and c2 without formal bounds, and the central Alternating Projections algorithm has no convergence guarantee. These gaps are acknowledged in the text but they are load-bearing for the advertised practical reliability, so the contribution is promising rather than fully established.

major comments (4)
  1. [Section 3.1 and Appendix D] The two central assumptions, ||E - E*|| ≤ c1 and ||L* - L_ideal|| ≤ c2, are stated as heuristic and no formal bounds on c1 and c2 are provided, as the text itself acknowledges. This is load-bearing because the branch restriction m_j ∈ {-1,0,1} in Algorithm 4 and the initial-guess strategy in Alternating Projections are justified only through bounds such as ||L* - L_ideal|| < 2π/κ(V). For CNOT and ISWAP, Theorem 3.1 cannot apply because ρ(L_ideal) = π, so the safety of the method rests entirely on heuristics and empirical testing. The paper should either supply a formal (even pessimistic) guarantee on admissible c2, or systematically map the failure region over the full set of gate/noise combinations and over a range of c1 values corresponding to realistic shot counts.
  2. [Section 3.1.3 and Appendix D, Algorithms 4-5] The paper states that standard convergence guarantees for alternating projections do not apply to Algorithm 2, yet the Success 1 criterion is reported as 600/600 on the synthetic benchmark. This high success rate is obtained for noise strengths ||L* - L_ideal|| only in [0.089, 0.355] at a fixed 10^4 shot level, and Figure 6 gives a threshold study for only one gate/noise combination (CNOT). The absence of convergence guarantees would be less concerning if the paper provided a systematic failure-threshold map or a quantitative robustness study across the parameter space; without that, the claim that the method is 'robust enough for current hardware' is not fully substantiated.
  3. [Section 4, Figure 5] The success criteria Success 1 and Success 2 measure how well the output Lindbladian fits the input or ground-truth channel, but they do not guarantee that the returned canonical decomposition is physically interpretable. Figure 5 explicitly shows a case that passes Success 2 where the predicted jump operators J1 and J2 are visibly incorrect and the text says they are 'erroneously predicted' even though their rates are small. Since the paper motivates Lindbladian fitting as a way to obtain qualitative and quantitative understanding of noise, the evaluation should include a metric on the canonical decomposition itself, or at least quantify how often the decomposition is statistically compatible with the ground-truth jump operators.
  4. [Section 6.1] The hardware analysis excludes the process RZZ(0.5)^5 because its tomographic eigenvalue structure was found to be 'incompatible with a time-independent Markovian process'. This exclusion weakens the central claim that the method is practical for noisy two-qubit gates on current hardware: one of the tested processes could not be fit at all, and the paper does not discuss whether this indicates a limitation of the method or a genuinely non-Markovian process. The authors should either include this process as a diagnostic example of detecting non-Markovianity, or provide evidence that the exclusion is not a form of cherry-picking.
minor comments (5)
  1. [Section 3.2.2] The paragraph discussing the max formulation states that this formulation 'incorrectly incentivises pulling gate errors in the poorest-fit gate into SPAM errors', then immediately says that this problem 'does not appear in the formulation of eq. (35)'. The logic is confusing and should be rephrased to clarify which formulation is being criticized and which property of eq. (35) avoids the issue.
  2. [Section 3.2.2, eq. (38)] The LSE scaling factor t is said to be fixed based on k and the number of shots, but no concrete formula or sensitivity analysis is given. Since t controls how closely the LSE approximates the max function, the authors should state their chosen values and show that the results are not sensitive to t over a reasonable range.
  3. [Table 1 caption and Section 5] The caption says 'The noise strengths ||L* - L_ideal|| range from 0.089 to 0.355', but Figure 6 separately tests larger values for CNOT. To avoid confusion, the caption should specify that the range refers only to the Table 1 benchmark instances and not to the threshold studies.
  4. [Algorithms 2 and 3] Algorithm 3's line 'B ← minimise h(B, L1, ..., Lk)' is notationally imprecise; it should be written as an optimization step such as 'B ← argmin_B h(B, L1, ..., Lk) subject to the physical constraints'.
  5. [Figures 10-12 and Tables 2-5] The hardware results are reported without uncertainty quantification; at 10^4 shots the statistical noise is non-negligible, as the synthetic tests show. The authors should at least state that no error bars are provided and discuss how shot noise could affect the reported distances and decompositions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: core results are proved in-paper and benchmarks use external ground truth.

full rationale

We find no significant circularity in the claimed derivation chain. The central theoretical guarantee, Theorem 3.1, is proved in Appendix C using Lemma C.1 and the Bauer-Fike theorem, with the weak-noise condition ||L* - L_ideal|| < (pi - rho(L_ideal))/kappa(V) stated as an input assumption rather than derived from the algorithm's output. The Convex Solve and Alternating Projections methods are tested against synthetically generated ground-truth channels E* and SPAM settings B*, with success measured relative to ||E - E*|| and f_max(B*, L*, ...), quantities that are external to the fitted parameters. The paper explicitly acknowledges that the constants c1 and c2 are heuristic and that formal bounds are not provided; this is a stated correctness-robustness limitation, not a circular step. Citations to the authors' prior work [50, 51, 55] supply prior algorithms, the Lindbladian cone characterization, and eigenvalue clustering as building blocks, but the paper's new guarantee is proved in-paper and its practical claims are supported by independent synthetic and hardware benchmarks. Even the Success 1 criterion, which compares the algorithm's objective value to the value attained by the ground-truth L*, is a meaningful sanity check because the heuristic search can fail to find any suitable Lindbladian; the more stringent Success 2 (61.4% on synthetic tests) provides a non-tautological test. Hardware demonstrations report fit residuals such as ||E - exp(L)|| rather than making out-of-sample predictions, so no fitted parameter is renamed as a prediction. The derivation chain therefore does not reduce to its own inputs.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The method rests on standard Lindbladian and GST theory, but relies on several unproven heuristics: weak-noise constants are not quantified, the approximate-logarithm construction is ad hoc, and the projection algorithms have no convergence guarantees. The free parameters are algorithm hyperparameters whose values are chosen by the user or tuned on the test distributions.

free parameters (5)
  • Eigenvalue clustering precision beta = user-specified; beta=0 for Convex Solve, positive values for Alternating Projections
    Controls when near-degenerate eigenvalues are merged into one eigenspace (Section 3.1.3). Choice affects the search space and performance; no automatic selection rule is given.
  • Number of random starts N = 500 for Table 1; 400 for Gate Set Flip-Flop tests
    Alternating Projections perturbs the ideal Lindbladian to escape poor basins (Section 3.1.3). Higher N improves robustness at more computational cost.
  • Maximum search depth T = not stated explicitly in main text
    Number of alternating-projection iterations in the inner loop (Algorithm 2/5). No convergence criterion is proven.
  • LSE scaling factor t = depends on k and number of shots; exact value not given
    Sets the accuracy of the smooth max approximation in the Gate Set Flip-Flop objective (Section 3.2.2).
  • SLSQP slack for physical-constraint violation = 0.001; raised to 0.005 in a few failing cases
    Allowed slack on physicality of B and A in Flip-Flop gauge optimisation (Section 5). Too small a slack makes SLSQP get stuck at the initial gauge.
assumptions (6)
  • domain assumption The estimated transfer matrix E and the ideal gate transfer matrix Eideal are diagonalisable.
    Stated in Section 3.1: 'Throughout this work, we do assume E and Eideal are both diagonalisable.' This underpins the eigenvector-based construction of matrix logarithms.
  • domain assumption Weak-noise assumptions: ||E-E*|| <= c1 and ||L*-Lideal|| <= c2, with c1 and c2 small but unquantified.
    Section 3.1 states these assumptions 'heuristically' and 'we do not have formal bounds on the precise values of c1 and c2 required by our algorithm.' The entire branch-search and initial-guess strategy depends on them.
  • domain assumption The true noise process is approximately time-independent and Markovian, so E* = exp(L*).
    The Lindbladian model (Section 2.1) and the success criteria (Section 3.1.4) assume the noisy gate is generated by a time-independent Lindbladian; Section 6 lists realistic violations.
  • domain assumption No statistical error in Theorem 3.1: E = E*, i.e., c1 = 0.
    Appendix C explicitly assumes c1 = 0 to prove the Convex Solve result. The noisy-case performance is only numerical.
  • ad hoc to paper Approximate logarithm set ~log(E) with precision beta is a valid enlargement of log(E).
    Section 3.1.3 defines ~log(E) by merging eigenvalues within beta; this heuristic is essential for Alternating Projections on near-degenerate spectra and has no formal justification.
  • domain assumption Gauge freedom in gate set tomography is resolved by minimising distance to the ideal gates, and B* is near the chosen gauge.
    Section 2.3 and Section 3.2.1 define the gauge choice; the Flip-Flop objective inherits this convention.

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Cite this review

Pith. "Pith review of Robust Lindbladian Estimation for Quantum Dynamics." pith.science (2026). https://pith.science/paper/RQWT7EJN

@misc{pith2026250707912,
  author       = {Pith},
  title        = {Pith review of: Robust Lindbladian Estimation for Quantum Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQWT7EJN}},
  note         = {Machine review of arXiv:2507.07912}
}
read the original abstract

We revisit the problem of fitting Lindbladian models to the outputs of quantum process tomography. A sequence of prior theoretical works approached the problem by considering whether there exists a Lindbladian generator close to a matrix logarithm of the tomographically estimated transfer matrix. This technique must take into account the non-uniqueness of the matrix logarithm, so that in general multiple branches of the logarithm must be checked. In contrast, all practical demonstrations of Lindbladian fitting on real experimental data have to our knowledge eschewed logarithm search, instead adopting direct numerical optimisation or ad-hoc approaches tailored to a particular experimental realisation. In our work, we introduce algorithmic improvements to logarithm search, demonstrating that it can be applied in practice to settings relevant for current quantum computing hardware. We additionally augment the task of Lindbladian fitting with techniques from gate set tomography to improve robustness against state preparation and measurement (SPAM) errors, which can otherwise obfuscate estimates of the model underlying the process of interest. We benchmark our techniques extensively using simulated tomographic data employing a range of realistic error models, before demonstrating their application to tomographic data collected from real superconducting-qubit hardware.

Figures

Figures reproduced from arXiv: 2507.07912 by the authors.

Figure 1
Figure 1. CNOT gate with coherent ZI, IZ, and ZZ errors and dissipative ZI error. Eideal is the transfer matrix of the ideal noiseless CNOT gate, and ∥E − Eideal∥ quantifies the total amount of gate and statistical noises in the input. In this example, there is no statistical error in the input E, so ∥E − E∗∥ = 0. An exhaustive search over the branches of log(E) finds a Lindbladian L satisfying ∥e L − E∥ = ∥e L − E∗∥ = 0 up t… view at source ↗
Figure 2
Figure 2. For the same noisy CNOT gate considered in [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. For the same input E considered in [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: Distribution of the ∥e L − E∥ (top) and ∥e L − E∗∥ (bottom) values for the 20 instances of CNOT with coherent X and dephasing noise tested. The algorithm is overfitting statistical error to some extent. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Canonical decomposition of a CNOT with coherent Z, amplitude damping, and dephasing noise instance. The algorithm succeeded on the input according to both of our success criteria. The Lindbladian found by the algorithm matches the ground truth Hamiltonian accurately an…
Figure 6
Figure 6. Figure 6: CNOT with increasing strengths of (a)(b) coherent X, amplitude damping, and dephasing gate noise and (c)(d) overrotation and dephasing gate noise. (a) The Alternating Projections method passes Success 1 for this test case until ∥L ∗ −L ideal∥ reaches roughly 0.45. (b) …
Figure 7
Figure 7. Figure 7: Each noise model is tested with 10 levels of increasing noise strengths. Each data point corresponds to the average ∥e L − E∥ value of 20 input instances. Across all the data points, the average gate fidelities between Eideal and E∗ range from 99.9% to 44.6% [PITH_FUL…
Figure 8
Figure 8. Figure 8: Synthetic test results for the Gate Set Flip-Flop algorithm on 1000 randomly generated test cases. The blue histograms plot the fmax(B, L1, . . . , Lk) values while the orange histograms plot the fmax(B∗ , L∗ 1 , . . . , L∗ k ) values. A run of the algorithm is deemed …
Figure 9
Figure 9. Figure 9: Gate Set Flip-Flop figures of merit for data collected from ibm_perth. (Top panel) The average gate fidelity between the ideal gate and the estimated Markovian process at each iteration, computed using eq. (44). (Bottom left panel) Distance between raw tomographic data…
Figure 10
Figure 10. Figure 10: Canonical decomposition (see subsection 2.1) of the fitted Lindbladian for the RZX(0.5) process applied to qubit pair (3, 5) on ibm_perth. We compare the ideal unitary process with the initial estimate prior to any gauge optimisation, and the final estimate after conv…
Figure 11
Figure 11. Figure 11: Canonical decomposition (see subsection 2.1) of the fitted Lindbladian for the RZX(0.5)3 = RZX(1.5) process applied to qubit pair (3, 5) on ibm_perth. The ideal process is compared with the initial estimate before gauge optimisation, and the final fitted Lindbladian a…
Figure 12
Figure 12. Figure 12: Canonical decomposition (see subsection 2.1) of the fitted Lindbladian for the RZX(0.5)5 = RZX(2.5) process applied to qubit pair (3, 5) on ibm_perth. We compare the ideal unitary process with fitted estimates before and after running the Gate Set Flip-Flop routine un…
Figure 13
Figure 13. Figure 13: (a) Partial qubit layout on ibm_cairo. Numbered vertices show qubit locations, edges show connections where CNOT gates are available. Dashed lines show connections to qubits not used in the experiment. The qubit pair targeted for gate set tomography is shown shaded gr…
Figure 14
Figure 14. Figure 14: Gate Set Flip-Flop figures of merit for data collected from ibm_cairo, tomographing qubits 13 and 14. (Top panel) The average gate fidelity between the ideal gate and the estimated Markovian process at each iteration, computed using eq. (44). (Bottom left panel) Dista…
Figure 15
Figure 15. Figure 15: Here we compare the canonical decompositions (see subsection 2.1) of the the ideal unitary CNOT gate with the fitted Lindbladians after Gate Set Flip-Flop for the noisy CNOT gate implemented on qubit pair (13, 14) on ibm_cairo. Note that the estimated decomposition fo…
Figure 16
Figure 16. Figure 16: The estimated canonical decomposition (subsection 2.1) of the estimated Lindbladian gen￾erators for the CNOT implemented on on qubit pair (13, 14) on ibm_cairo. Here we compare the ideal unitary process with experimental estimates before and after Gate Set Flip-Flop, …
Figure 17
Figure 17. Figure 17: Estimated canonical decomposition (subsection 2.1) of the estimated Lindbladian generators for the CNOT implemented on on qubit pair (13, 14) on ibm_cairo. Here we compare the ideal unitary process with experimental estimates before and after Gate Set Flip-Flop, for t…
Figure 18
Figure 18. Figure 18: (a) (a) Qubit layout on ibm_perth. Numbered vertices show qubit locations, edges show connections where CNOT gates are available. (b) Shaded green qubits show the position of the pair targeted for process tomography in the (0, 1) run. For the DCNOT process, the CNOT i…
Figure 19
Figure 19. Figure 19: Circuit decomposition for CNOT between qubits 0 and 5 on ibm_perth. Process ∥E − Eideal∥ ∥E − exp(L)∥ CNOT 0.28 0.23 T ⊗ 1 0.25 0.22 1 ⊗ T 0.25 0.23 D 0.57 0.22 D5 2.08 0.18 DCNOT 1.08 0.21 D5 CNOT 4.07 0.22 DX 0.33 0.24 D5 X 0.75 0.21 √ X ⊗ 1 0.27 0.22 (a) Qubit pair…
Figure 20
Figure 20. Figure 20: Gate Set Flip-Flop figures of merit for the idling experiment on qubits (0, 1) from ibm_perth. (Top panel) The average gate fidelity between the ideal gate and the estimated Markovian process at each iteration, computed using eq. (44). (Bottom left panel) Distance bet…
Figure 21
Figure 21. Figure 21: Gate Set Flip-Flop figures of merit for the idling experiment on qubits (0, 5) from ibm_perth. (Top panel) The average gate fidelity between the ideal gate and the estimated Markovian process at each iteration, computed using eq. (44). (Bottom left panel) Distance bet…
Figure 22
Figure 22. Figure 22: Here we plot the canonical decompositions (see subsection 2.1) of the estimated Lindbladian for three variants of the idling process on the adjacent qubit pair (0, 1) (see [PITH_FULL_IMAGE:figures/full_fig_p042_22.png]
Figure 23
Figure 23. Figure 23: Canonical decompositions (see subsection 2.1) of idling processes for the separated qubit pair (0, 5) on ibm_perth (see [PITH_FULL_IMAGE:figures/full_fig_p043_23.png]

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Pith tools

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