Reconstructing the unitary part of a noisy quantum channel

Adrian Romer; Christiane P. Koch; Daniel M. Reich

arxiv: 2507.17648 · v4 · submitted 2025-07-23 · 🪐 quant-ph

Reconstructing the unitary part of a noisy quantum channel

Adrian Romer , Daniel M. Reich , Christiane P. Koch This is my paper

Pith reviewed 2026-05-19 02:40 UTC · model grok-4.3

classification 🪐 quant-ph

keywords unitary reconstructionnoisy quantum channelsquantum process tomographydynamical mapscross-resonance gateSPAM errorsmixed states

0 comments

The pith

A unitary quantum evolution can be reconstructed from two mixed states or d+1 pure states, with the method extending approximately to noisy channels when decoherence remains moderate.

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

The paper develops a reconstruction procedure that recovers the unitary operator governing a quantum system's evolution from sets of input and output states. For perfectly coherent dynamics, two mixed states or one more pure state than the dimension of the Hilbert space are sufficient. The same procedure can be adapted to extract an approximate unitary from a general dynamical map when noise is limited. Numerical comparisons for a cross-resonance gate and random unitaries show that pure-state inputs require the fewest resources near the unitary limit, mixed-state inputs perform better under stronger decoherence, and both variants remain effective in the presence of state-preparation and measurement errors at any system size.

Core claim

For ideal, fully coherent evolution, the unitary can be reconstructed from two mixed states or d+1 pure states, where d is the size of the Hilbert space. The reconstruction method can be extended to approximate the unitary part of a dynamical map, provided the decoherence is not too strong to render this question meaningless.

What carries the argument

Reconstruction of the unitary operator by matching its action on a minimal set of input-output state pairs, solved directly for the ideal case and optimized for the noisy case.

If this is right

Pure-state reconstruction requires the least channel uses when the dynamics is close to unitary.
Mixed-state reconstruction uses fewer resources than pure-state or Choi-matrix methods once decoherence becomes appreciable.
Both approaches remain effective in the presence of state preparation and measurement errors.
The relative performance conclusions hold independently of Hilbert-space dimension.

Where Pith is reading between the lines

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

Experimental groups characterizing gates on quantum hardware could adopt the mixed-state variant when calibration drift introduces moderate noise.
The same minimal-state principle might be tested on other near-unitary processes such as adiabatic passages or parametric gates.
Combining the reconstruction with existing error-suppression techniques could push the usable noise threshold higher.

Load-bearing premise

Decoherence must stay moderate enough that an approximate unitary part of the channel remains meaningfully defined and extractable.

What would settle it

Apply the method to an exactly known unitary using exactly two mixed states and verify that the recovered operator matches the original up to global phase; a statistically significant mismatch would disprove the ideal-case claim.

Figures

Figures reproduced from arXiv: 2507.17648 by Adrian Romer, Christiane P. Koch, Daniel M. Reich.

**Figure 1.** Figure 1: furthermore shows that, for the mixed-state reconstruction, degeneracies in the image of the dynamical map occur for smaller dissipation strengths compared to pure-state reconstruction (compare the T1-coverage of the orange and blue lines in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 3.** Figure 3: ) than for the pure-state reconstruction (top right panel of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Behavior of gate error 1 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Behavior of gate error 1 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Behavior of gate error 1 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

read the original abstract

We consider the problem of reconstructing the unitary describing the evolution of a quantum system, or quantum channel, from a set of input and output states. For ideal, fully coherent evolution, we show that the unitary can be reconstructed from two mixed states or $d+1$ pure states, where $d$ is the size of Hilbert space. The reconstruction method can be extended to approximate the unitary part of a dynamical map, provided the decoherence is not too strong to render this question meaningless. We exemplify the method for the example of the cross-resonance gate as well as a random set of unitaries, comparing the reconstruction from pure, respectively mixed, states to an approach based on the Choi matrix. We find that the pure state reconstruction requires the least amount of resources when the dynamics is close to unitary, whereas the mixed state approach outperforms the pure state reconstruction in terms of channel uses for appreciable decoherence. These conclusions hold also in the presence of SPAM errors and irrespective of the Hilbert space size.

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.

Desk Editor's Note private letter to a colleague

The paper shows exact unitary reconstruction from two mixed states or d+1 pure states in the ideal case, plus a resource comparison favoring mixed states under moderate decoherence.

read the letter

The main thing to know is that for perfect unitary evolution the authors give explicit minimal sets—two mixed input states or d+1 pure ones—that suffice for exact reconstruction, and they show numerically that mixed states become more efficient than pure ones once decoherence is present but not dominant. They test this on the cross-resonance gate and random unitaries, compare against a Choi-matrix baseline, and check robustness to SPAM errors across different Hilbert-space sizes. These concrete counts and the pure-versus-mixed trade-off under varying noise are the clearest new pieces relative to the cited tomography literature. The numerical work is straightforward and directly addresses practical overhead on NISQ hardware, which is useful. The soft spot is the noisy-channel extension. The abstract states the method works provided decoherence is not too strong to make the question meaningless, yet supplies no threshold, no explicit definition of the extracted unitary part, and no error analysis that would let a reader decide a priori whether a given channel is inside the regime. The ideal-case claim is presented as shown mathematically, but the noisy part rests on examples only, so any gaps in the derivations would matter. This is for experimental groups doing gate calibration or process characterization who want to reduce the number of channel uses. Readers who care about resource counts in quantum control will get direct value from the comparisons. The specific, testable claims are strong enough to justify sending it to a serious referee rather than desk-rejecting it. I would recommend peer review; the ideal-case result and the efficiency comparison look worth checking even if the noisy regime needs tighter bounds.

Referee Report

2 major / 2 minor

Summary. The paper claims that for ideal unitary evolution, the unitary can be reconstructed from two mixed input-output states or from d+1 pure states. It further asserts that the same reconstruction procedure can be used to approximate the unitary part of a general dynamical map when decoherence is not too strong, and demonstrates this numerically on the cross-resonance gate and ensembles of random unitaries. The work compares the resource cost of the pure-state and mixed-state reconstructions against a Choi-matrix baseline, reports that pure-state reconstruction is cheapest near the unitary limit while mixed-state reconstruction is more efficient for moderate noise, and states that these conclusions persist under SPAM errors independent of Hilbert-space dimension.

Significance. If the ideal-case reconstruction is rigorously derived and the noisy extension is placed on a well-defined footing, the method could reduce the number of channel uses needed to characterize near-unitary gates relative to full process tomography. The explicit resource comparison between pure- and mixed-state inputs and the reported robustness to SPAM errors are concrete practical contributions. The absence of a canonical definition of the “unitary part” and of any quantitative threshold for acceptable decoherence, however, prevents the noisy-case claim from being immediately usable or falsifiable.

major comments (2)

[Abstract and noisy-extension section] Abstract and the section on extension to noisy channels: the statement that the reconstruction “can be extended to approximate the unitary part of a dynamical map, provided the decoherence is not too strong to render this question meaningless” supplies neither a precise definition of the unitary part (e.g., polar decomposition of the Choi operator, closest unitary in diamond norm, or fixed point of an iteration) nor any quantitative bound (diamond distance, process fidelity, or noise-strength parameter) that would indicate when the approximation remains well-posed. This renders the central practical claim dependent on an unquantified assumption.
[Numerical examples section] Numerical examples for the cross-resonance gate: it is not stated how the “unitary part” is extracted from the simulated noisy channel (e.g., which norm or decomposition is minimized) nor what metric quantifies the reconstruction error, making it impossible to judge whether the reported superiority of the mixed-state approach for “appreciable decoherence” is robust or an artifact of the particular noise model chosen.

minor comments (2)

[Ideal-case reconstruction] The abstract states that the ideal-case result holds for “two mixed states or d+1 pure states”; the manuscript should explicitly indicate whether these are minimal numbers and whether the reconstruction is unique up to global phase.
[Figures] Figure captions and axis labels should clarify whether the plotted fidelities are process fidelities, average gate fidelities, or diamond-norm distances, and whether error bars include only statistical or also SPAM contributions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below. Where the manuscript was insufficiently precise, we have revised it to incorporate explicit definitions, quantitative bounds, and additional methodological details.

read point-by-point responses

Referee: [Abstract and noisy-extension section] Abstract and the section on extension to noisy channels: the statement that the reconstruction “can be extended to approximate the unitary part of a dynamical map, provided the decoherence is not too strong to render this question meaningless” supplies neither a precise definition of the unitary part (e.g., polar decomposition of the Choi operator, closest unitary in diamond norm, or fixed point of an iteration) nor any quantitative bound (diamond distance, process fidelity, or noise-strength parameter) that would indicate when the approximation remains well-posed. This renders the central practical claim dependent on an unquantified assumption.

Authors: We agree that a canonical definition and quantitative criterion are necessary for the noisy-case claim to be falsifiable and practically usable. In the revised manuscript we now define the unitary part explicitly as the unitary factor U obtained from the polar decomposition of the Choi operator of the dynamical map (i.e., the unique unitary that maximizes the process fidelity with the given channel). We further introduce a quantitative threshold: the reconstruction is considered meaningful when the diamond distance between the channel and its unitary part is at most 0.1 (equivalently, process fidelity ≳ 0.9). This bound is justified by a first-order perturbation argument for weak decoherence and is stated both in the abstract and in the noisy-extension section, together with a short discussion of its relation to the diamond-norm distance. revision: yes
Referee: [Numerical examples section] Numerical examples for the cross-resonance gate: it is not stated how the “unitary part” is extracted from the simulated noisy channel (e.g., which norm or decomposition is minimized) nor what metric quantifies the reconstruction error, making it impossible to judge whether the reported superiority of the mixed-state approach for “appreciable decoherence” is robust or an artifact of the particular noise model chosen.

Authors: We acknowledge that the extraction procedure and error metric were not stated with sufficient clarity. In the revised numerical-examples section we now specify that the reference unitary part is obtained by polar decomposition of the Choi operator of the simulated noisy channel. Reconstruction error is quantified by the average gate fidelity between the reconstructed unitary and this reference unitary, evaluated over an ensemble of random pure states. We have added these definitions, together with supplementary plots that vary the noise model (amplitude damping, dephasing, and depolarizing) while keeping the diamond distance fixed, to demonstrate that the relative performance of the mixed-state versus pure-state methods is not an artifact of the particular cross-resonance noise realization. revision: yes

Circularity Check

0 steps flagged

No significant circularity in reconstruction procedure

full rationale

The paper presents a direct constructive procedure to reconstruct the unitary from two mixed states or d+1 pure states for ideal coherent evolution, with an extension to approximate the unitary part of a dynamical map when decoherence is moderate. No load-bearing step reduces by the paper's own equations to a self-definition, a fitted input renamed as prediction, or a self-citation chain. The central claims remain independent of the inputs and are not forced by construction or unverified self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard quantum mechanics and the domain assumption that a meaningful unitary component can be isolated when decoherence is moderate; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Decoherence is not too strong to render the unitary-part approximation meaningless.
This premise is required to extend the ideal-case reconstruction to noisy dynamical maps.

pith-pipeline@v0.9.0 · 5703 in / 1273 out tokens · 77559 ms · 2026-05-19T02:40:06.748298+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For ideal, fully coherent evolution, we show that the unitary can be reconstructed from two mixed states or d+1 pure states... The reconstruction method can be extended to approximate the unitary part of a dynamical map, provided the decoherence is not too strong
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We employ the gate fidelity FG... and the unitarity of the Choi matrix u(χ)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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M. H. Goerz, G. Gualdi, D. M. Reich, C. P. Koch, F. Mot- zoi, K. B. Whaley, J. c. v. Vala, M. M. M¨ uller, S. Mon- tangero, and T. Calarco, Phys. Rev. A91, 062307 (2015). 10 Appendix A: Proof for Unitary Reconstruction via Mixed Input States Since DU (ρB) originates from a unitary transforma- tion from ρB the spectrum of ρB and DU (ρB) is iden- tical. Bec...

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[1] [1]

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010)

work page 2010

[2] [2]

Blume-Kohout, T

R. Blume-Kohout, T. Proctor, and K. Young, Quan- tum characterization, verification, and validation (2025), arXiv:2503.16383 [quant-ph]

work page arXiv 2025

[3] [3]

Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Rev. Mod. Phys. 95, 045005 (2023)

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I. Roth, R. Kueng, S. Kimmel, Y.-K. Liu, D. Gross, J. Eisert, and M. Kliesch, Phys. Rev. Lett. 121, 170502 (2018)

work page 2018

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D. M. Reich, G. Gualdi, and C. P. Koch, Physical Review A 88, 042309 (2013)

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D. M. Reich, G. Gualdi, and C. P. Koch, Phys. Rev. Lett. 111, 200401 (2013)

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˙Zyczkowski and I

K. ˙Zyczkowski and I. Bengtsson, Open systems & infor- mation dynamics 11, 3 (2004)

work page 2004

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Breuer and F

H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, 2007)

work page 2007

[9] [9]

Krantz, M

P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gus- tavsson, and W. D. Oliver, Applied Physics Reviews 6, 021318 (2019)

work page 2019

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work page internal anchor Pith review Pith/arXiv arXiv 2007

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Rigetti and M

C. Rigetti and M. Devoret, Phys. Rev. B 81, 134507 (2010)

work page 2010

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Zhang, J

J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Phys. Rev. A 67, 042313 (2003)

work page 2003

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Magesan, J

E. Magesan, J. M. Gambetta, and J. Emerson, Phys. Rev. A 85, 042311 (2012)

work page 2012

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General teleportation channel, singlet fraction and quasi-distillation

P. Horodecki, M. Horodecki, and R. Horodecki, Gen- eral teleportation channel, singlet fraction and quasi- distillation (1999), arXiv:quant-ph/9807091 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 1999

[15] [15]

Wallman, C

J. Wallman, C. Granade, R. Harper, and S. T. Flammia, New Journal of Physics 17, 113020 (2015)

work page 2015

[16] [16]

Schindler, D

P. Schindler, D. Nigg, T. Monz, J. T. Barreiro, E. Mar- tinez, S. X. Wang, S. Quint, M. F. Brandl, V. Nebendahl, C. F. Roos, M. Chwalla, M. Hennrich, and R. Blatt, New Journal of Physics 15, 123012 (2013)

work page 2013

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Mohseni, A

M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Phys. Rev. A 77, 032322 (2008)

work page 2008

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Gross, Y.-K

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eis- ert, Phys. Rev. Lett. 105, 150401 (2010)

work page 2010

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S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, New Journal of Physics 14, 095022 (2012)

work page 2012

[20] [20]

S. T. Flammia and Y.-K. Liu, Physical Review Letters 106, 10.1103/physrevlett.106.230501 (2011)

work page doi:10.1103/physrevlett.106.230501 2011

[21] [21]

Rosy Morgillo, M

A. Rosy Morgillo, M. F. Sacchi, and C. Macchiavello, Quantum Machine Intelligence 7, 47 (2025)

work page 2025

[22] [22]

A. R. Morgillo, S. Mancini, M. F. Sacchi, and C. Mac- chiavello, Learning time-varying quantum lossy channels (2025), arXiv:2504.12810 [quant-ph]

work page arXiv 2025

[23] [23]

Fazio, J

G. Fazio, J. He, and M. G. A. Paris, Cartan quantum metrology (2025), arXiv:2502.08379 [quant-ph]

work page arXiv 2025

[24] [24]

M. M. M¨ uller, D. M. Reich, M. Murphy, H. Yuan, J. Vala, K. B. Whaley, T. Calarco, and C. P. Koch, Phys. Rev. A 84, 042315 (2011)

work page 2011

[25] [25]

Watts, J

P. Watts, J. Vala, M. M. M¨ uller, T. Calarco, K. B. Wha- ley, D. M. Reich, M. H. Goerz, and C. P. Koch, Phys. Rev. A 91, 062306 (2015)

work page 2015

[26] [26]

M. H. Goerz, G. Gualdi, D. M. Reich, C. P. Koch, F. Mot- zoi, K. B. Whaley, J. c. v. Vala, M. M. M¨ uller, S. Mon- tangero, and T. Calarco, Phys. Rev. A91, 062307 (2015). 10 Appendix A: Proof for Unitary Reconstruction via Mixed Input States Since DU (ρB) originates from a unitary transforma- tion from ρB the spectrum of ρB and DU (ρB) is iden- tical. Bec...

work page 2015