Near-Minimal Gate Set Tomography Experiment Designs

Corey Ostrove; Kenneth Rudinger; Kevin Young; Robin Blume-Kohout; Stefan Seritan

arxiv: 2308.08781 · v2 · submitted 2023-08-17 · 🪐 quant-ph

Near-Minimal Gate Set Tomography Experiment Designs

Corey Ostrove , Kenneth Rudinger , Stefan Seritan , Kevin Young , Robin Blume-Kohout This is my paper

Pith reviewed 2026-05-24 07:37 UTC · model grok-4.3

classification 🪐 quant-ph

keywords gate set tomographyquantum tomographyexperiment designgerm subroutinesHeisenberg scalingtwo-qubit systemsnoise characterization

0 comments

The pith

Analyzing germ sensitivities removes redundancy from GST experiments, yielding near-minimal designs with preserved precision scaling.

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

Gate set tomography requires many circuits to characterize quantum gates precisely, limiting its use to small systems. This paper demonstrates a method to identify and eliminate redundant circuits in GST designs by examining the parameter sensitivities of germ subroutines. The resulting two-qubit experiments use only slightly more circuits than the theoretical minimum while maintaining Heisenberg-like precision scaling, as verified by simulations and Fisher information analysis. These streamlined designs can achieve the same precision as previous methods with significantly fewer circuits. The approach also suggests feasibility for extending GST to three-qubit systems.

Core claim

We show how to streamline GST experiment designs by removing almost all redundancy, creating smaller and more scalable experiments without losing precision. We do this by analyzing the germ subroutines at the heart of GST circuits, identifying exactly what gate set parameters they are sensitive to, and leveraging this information to remove circuits that duplicate other circuits' sensitivities. We apply this technique to two-qubit GST experiments, generating streamlined experiment designs that contain only slightly more circuits than the theoretical minimum bounds, but still achieve Heisenberg-like scaling in precision as demonstrated via simulation and a theoretical analysis using Fisher信息.

What carries the argument

Sensitivity analysis of germ subroutines, which maps each germ to the gate set parameters it probes to enable removal of redundant circuits.

If this is right

Streamlined designs contain only slightly more circuits than theoretical minimum bounds.
Designs still achieve Heisenberg-like scaling in precision.
New designs match the precision of previous GST experiments with significantly fewer circuits.
The technique creates prospects for extending GST to three-qubit systems.

Where Pith is reading between the lines

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

Fewer circuits could make GST practical for characterizing processors with more than two qubits.
The pruning method might extend to other forms of quantum process tomography.
Reduced experimental overhead could speed up noise characterization during quantum device calibration.
Exact minimum circuit counts might be reachable with further refinement of the sensitivity mapping.

Load-bearing premise

The sensitivity analysis of germ subroutines accurately identifies all redundancies such that removing duplicate circuits does not degrade overall parameter estimation precision.

What would settle it

A simulation comparing the streamlined two-qubit GST design to the original that shows lower precision or incomplete parameter recovery in the reduced set.

Figures

Figures reproduced from arXiv: 2308.08781 by Corey Ostrove, Kenneth Rudinger, Kevin Young, Robin Blume-Kohout, Stefan Seritan.

**Figure 2.** Figure 2: Average diamond distance between estimated and true data generating [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** Figure 4: Average diamond distance between estimated and true data generating [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Average diamond distance between estimated and true data generating [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of the spectra of the Fisher information matrices for [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

Gate set tomography (GST) provides precise, self-consistent estimates of the noise channels for all of a quantum processor's logic gates. But GST experiments are large, involving many distinct quantum circuits. This has prevented their use on systems larger than two qubits. Here, we show how to streamline GST experiment designs by removing almost all redundancy, creating smaller and more scalable experiments without losing precision. We do this by analyzing the "germ" subroutines at the heart of GST circuits, identifying exactly what gate set parameters they are sensitive to, and leveraging this information to remove circuits that duplicate other circuits' sensitivities. We apply this technique to two-qubit GST experiments, generating streamlined experiment designs that contain only slightly more circuits than the theoretical minimum bounds, but still achieve Heisenberg-like scaling in precision (as demonstrated via simulation and a theoretical analysis using Fisher information). In practical use, the new experiment designs can match the precision of previous GST experiments with significantly fewer circuits. We discuss the prospects and feasibility of extending GST to three-qubit systems using our techniques.

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

This paper gives a workable method to prune two-qubit GST circuits down to near-minimal size via germ sensitivity analysis while preserving Heisenberg scaling in simulations.

read the letter

The main point is that the authors show how to shrink GST experiment designs for two qubits by mapping the parameter sensitivities of each germ and dropping circuits that add no new information. The resulting designs sit only slightly above the theoretical minimum circuit count yet still deliver the expected precision scaling, according to their simulations and Fisher information calculations. In use, these designs reach the same precision as earlier GST sets with far fewer circuits. This pruning step is the concrete advance; earlier GST papers did not apply sensitivity analysis this directly to circuit selection. The work is useful because it turns a theoretical bound into actual smaller experiment lists that experimental groups could adopt. The simulations and Fisher analysis provide direct evidence that the scaling holds for the cases they tested. The soft spot is the completeness of the sensitivity map itself. If the analysis misses higher-order or cross-germ correlations, some noise parameters could lose rank in the reduced information matrix, and the paper's checks would not catch it. Their simulations reduce the practical risk, but the guarantee rests on how exhaustive the first-order sensitivity step really is. Readers who run GST on hardware or who build characterization software will find the designs and the pruning recipe worth examining. The paper is coherent on its own terms and engages the existing GST literature without circularity, so it deserves a serious referee. I would send it to review and ask the authors to spell out the coverage of the sensitivity analysis for the full two-qubit Pauli noise model.

Referee Report

2 major / 2 minor

Summary. The paper claims that gate set tomography (GST) experiment designs can be streamlined by analyzing the parameter sensitivities of germ subroutines to identify and remove redundant circuits, yielding near-minimal two-qubit GST designs (only slightly above theoretical bounds) that preserve Heisenberg-like precision scaling. This is supported by simulations and Fisher information analysis, enabling equivalent precision with significantly fewer circuits than prior designs, with discussion of extension to three-qubit systems.

Significance. If the central claims hold, the work substantially advances the practicality of GST by addressing its primary scalability limitation (experiment size), potentially enabling self-consistent noise characterization on systems beyond two qubits. Strengths include the use of both numerical simulation and analytic Fisher information to verify scaling, and the focus on a concrete, falsifiable reduction in circuit count while targeting full-rank information matrices.

major comments (2)

[§3.2] §3.2 (germ sensitivity analysis): The procedure identifies per-germ sensitivities to prune duplicate circuits, but the manuscript does not demonstrate that this first-order sensitivity map is complete with respect to cross-germ parameter correlations; if higher-order dependencies exist for certain two-qubit Pauli noise channels, the reduced design could lose rank in the Fisher information matrix without detection.
[§4] §4 (Fisher information analysis): The claim that the pruned designs achieve the same Heisenberg scaling relies on the reduced information matrix remaining full rank and well-conditioned, yet no explicit eigenvalue comparison or condition-number bound between the original and pruned matrices is provided to quantify any precision degradation.

minor comments (2)

Figure 2 caption: the legend for 'original' vs 'pruned' circuit counts is unclear on whether the plotted values include or exclude the fiducial circuits.
Notation in Eq. (7): the definition of the sensitivity vector s_g for a germ g should explicitly state whether it is computed via the derivative of the process matrix or via the superoperator action on the Pauli basis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The two major comments raise important points about the completeness of our sensitivity analysis and the quantitative validation of the pruned designs. We address each below and indicate revisions that will be incorporated.

read point-by-point responses

Referee: [§3.2] §3.2 (germ sensitivity analysis): The procedure identifies per-germ sensitivities to prune duplicate circuits, but the manuscript does not demonstrate that this first-order sensitivity map is complete with respect to cross-germ parameter correlations; if higher-order dependencies exist for certain two-qubit Pauli noise channels, the reduced design could lose rank in the Fisher information matrix without detection.

Authors: We appreciate the referee's observation regarding potential cross-germ correlations. The sensitivity analysis in §3.2 computes first-order parameter derivatives per germ to identify redundant circuits. While this is a first-order map, the subsequent step in our workflow explicitly constructs and checks the full Fisher information matrix of the pruned experiment design. This matrix incorporates all parameter correlations (including cross-germ terms) and any higher-order effects implicit in the model. In all cases examined, the pruned matrices remain full rank. To strengthen the presentation, we will revise §3.2 to include an explicit statement that the final rank check serves as the completeness verification, and we will add a brief analysis showing that, for the two-qubit Pauli noise models considered, no rank loss occurred despite the first-order pruning heuristic. revision: partial
Referee: [§4] §4 (Fisher information analysis): The claim that the pruned designs achieve the same Heisenberg scaling relies on the reduced information matrix remaining full rank and well-conditioned, yet no explicit eigenvalue comparison or condition-number bound between the original and pruned matrices is provided to quantify any precision degradation.

Authors: We agree that direct quantitative comparison of the information matrices would make the scaling claim more rigorous. The current manuscript demonstrates preservation of Heisenberg scaling via both Monte Carlo simulations and the analytic Fisher information rank, but does not tabulate eigenvalue spectra or condition numbers. In the revised version we will add a supplementary table (or figure) that reports the smallest nonzero eigenvalues and condition numbers for representative original and pruned two-qubit designs. This will allow readers to assess any minor degradation in conditioning while confirming that the scaling behavior remains intact. revision: yes

Circularity Check

0 steps flagged

No circularity: method uses independent sensitivity analysis with external validation

full rationale

The paper's core procedure—germ-by-germ sensitivity analysis to identify and prune redundant circuits—is presented as a distinct computational step whose output (reduced experiment designs) is then validated separately via simulation and Fisher-information scaling analysis. No quoted equations or claims reduce a prediction to a fitted input by construction, invoke self-citations as load-bearing uniqueness theorems, or smuggle ansatzes. The derivation chain remains self-contained against the stated external benchmarks (simulations, information-matrix rank checks) and does not collapse to tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities; it optimizes existing GST experiment design methods.

axioms (1)

standard math Standard assumptions of quantum mechanics and linear algebra for parameter estimation in GST.
The method builds on existing GST framework which assumes quantum operations can be modeled as channels.

pith-pipeline@v0.9.0 · 5716 in / 1170 out tokens · 35698 ms · 2026-05-24T07:37:32.233179+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography,

R. Blume-Kohout, J. K. Gamble, E. Nielsen, K. Rudinger, J. Mizrahi, K. Fortier, and P. Maunz, “Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography,” Nat. Commun., vol. 8, Feb. 2017

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M. T. M ˛ adzik, S. Asaad, A. Youssry, B. Joecker, K. M. Rudinger, E. Nielsen, K. C. Young, T. J. Proctor, A. D. Baczewski, A. Laucht et al., “Precision tomography of a three-qubit donor quantum processor in silicon,” Nature, vol. 601, no. 7893, pp. 348–353, 2022

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work page arXiv 2022

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Available: https://doi.org/10.22331/q-2021-10-05-557

[Online]. Available: https://doi.org/10.22331/q-2021-10-05-557

work page doi:10.22331/q-2021-10-05-557 2021

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Two-qubit gate set tomography with fewer circuits,

K. Rudinger, C. Ostrove, S. Seritan, M. Grace, E. Nielsen, R. Blume- Kohout, and K. Young, “Two-qubit gate set tomography with fewer circuits,” arXiv preprint arxiv:2307.15767, 2023

work page arXiv 2023

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Characterization of addressability by simultaneous randomized benchmarking [supplementary material],

J. M. Gambetta, A. D. Córcoles, S. T. Merkel, B. R. Johnson, J. A. Smolin, J. M. Chow, C. A. Ryan, C. Rigetti, S. Poletto, T. A. Ohki, M. B. Ketchen, and M. Steffen, “Characterization of addressability by simultaneous randomized benchmarking [supplementary material],” Phys. Rev. Lett., vol. 109, p. 240504, Dec 2012. [Online]. Available: https://link.aps.o...

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work page 2021

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Regularized greedy column subset selection,

B. Ordozgoiti, A. Mozo, and J. G. L. de Lacalle, “Regularized greedy column subset selection,” Information Sciences, vol. 486, pp. 393–418, 2019

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A note on subset selection for matrices,

F. De Hoog and R. Mattheij, “A note on subset selection for matrices,” Linear algebra and its applications, vol. 434, no. 8, pp. 1845–1850, 2011

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Faster subset selection for matrices and applications,

H. Avron and C. Boutsidis, “Faster subset selection for matrices and applications,” SIAM Journal on Matrix Analysis and Applications, vol. 34, no. 4, pp. 1464–1499, 2013

work page 2013

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A taxonomy of small markovian errors,

R. Blume-Kohout, M. P. da Silva, E. Nielsen, T. Proctor, K. Rudinger, M. Sarovar, and K. Young, “A taxonomy of small markovian errors,” PRX Quantum, vol. 3, p. 020335, May 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PRXQuantum.3.020335

work page doi:10.1103/prxquantum.3.020335 2022

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Quantum circuits with mixed states,

D. Aharonov, A. Kitaev, and N. Nisan, “Quantum circuits with mixed states,” in Proceedings of the thirtieth annual ACM symposium on Theory of computing, 1998, pp. 20–30

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Extensive characterization and implementation of a family of three-qubit gates at the coherence limit,

C. W. Warren, J. Fernández-Pendás, S. Ahmed, T. Abad, A. Bengtsson, J. Biznárová, K. Debnath, X. Gu, C. Križan, A. Osman et al., “Extensive characterization and implementation of a family of three-qubit gates at the coherence limit,” npj Quantum Information, vol. 9, no. 1, p. 44, 2023

work page 2023

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Benchmarking verified logic operations for fault tolerance,

A. Hashim, S. Seritan, T. Proctor, K. Rudinger, N. Goss, R. K. Naik, J. M. Kreikebaum, D. I. Santiago, and I. Siddiqi, “Benchmarking verified logic operations for fault tolerance,” arXiv preprint arXiv:2207.08786, 2022

work page arXiv 2022

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High-fidelity three- qubit i toffoli gate for fixed-frequency superconducting qubits,

Y . Kim, A. Morvan, L. B. Nguyen, R. K. Naik, C. Jünger, L. Chen, J. M. Kreikebaum, D. I. Santiago, and I. Siddiqi, “High-fidelity three- qubit i toffoli gate for fixed-frequency superconducting qubits,” Nature Physics, vol. 18, no. 7, pp. 783–788, 2022

work page 2022

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Pukelsheim, Optimal design of experiments

F. Pukelsheim, Optimal design of experiments. SIAM, 2006

work page 2006

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Generalized inversion of modified matrices,

C. D. Meyer, Jr, “Generalized inversion of modified matrices,” Siam journal on applied mathematics, vol. 24, no. 3, pp. 315–323, 1973

work page 1973

[22] [22]

An extension of the matrix inversion lemma,

N. Minamide, “An extension of the matrix inversion lemma,” SIAM Journal on Algebraic Discrete Methods, vol. 6, no. 3, pp. 371–377, 1985

work page 1985

[23] [23]

Some applications of the rank revealing qr factorization,

T. F. Chan and P. C. Hansen, “Some applications of the rank revealing qr factorization,” SIAM Journal on Scientific and Statistical Computing, vol. 13, no. 3, pp. 727–741, 1992

work page 1992

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Rank degeneracy and least squares problems,

G. Golub, V . Klema, and G. W. Stewart, “Rank degeneracy and least squares problems,” Stanford Univ. Dept. Of Computer Science, Tech. Rep., 1976

work page 1976

[25] [25]

G. H. Golub and C. F. Van Loan, Matrix computations, 3rd edition. JHU press, 1996. APPENDIX A. Column Subset Selection Heuristics As discussed in Section III-B, the first stage of the per- germ global FPR algorithm wherein we select for each germ a subset of its amplified parameters to require sensitivity to can be reduced to a problem known as the column...

work page 1996

[26] [26]

For the greedy search heuristic we do so by mapping the problem to a combinatorial optimization problem which we approximate the solution to using greedy search

Greedy Search: Given as input the matrix of concate- nated right singular vectors for each germ’s twirled derivative, J = Vg0 Vg1 · · · VgNg , (17) our goal is to select a submatrix of J, Jmin, formed from its columns such that Jmin has a column-rank equal to the number of amplifiable parameters and is well-conditioned. For the greedy search heuristic we ...

work page

[27] [27]

Rank-Revealing QR Decompositions: Another class of heuristics that has seen significant development for column subset selection is based on the application of the rank- revealing QR decomposition [22]. An advantage of this class of protocols is that (at least for the simpler variants) they can be phrased entirely linear-algebraically, and as such allow us...

work page