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arxiv: 2410.02213 · v1 · submitted 2024-10-03 · 🪐 quant-ph · cond-mat.str-el

Low-overhead fault-tolerant quantum computation by gauging logical operators

Dominic J. Williamson , Theodore J. Yoder This is my paper

Pith reviewed 2026-05-23 20:26 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords fault-tolerant quantum computationlogical measurementquantum error-correcting codesgaugingsymmetriesqubit overheadlow-overhead methods
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0 comments X

The pith

Treating logical operators as symmetries and gauging them enables fault-tolerant logical measurements with qubit overhead linear in operator weight up to a polylog factor.

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

The paper establishes a method for fault-tolerant logical measurement in quantum error-correcting codes by treating the logical operator as a symmetry and gauging it. This gauging procedure achieves a qubit overhead linear in the weight of the operator being measured, up to a polylogarithmic factor. The same flexibility allows adaptation to arbitrary quantum codes. A sympathetic reader would care because prior schemes for logical measurement do not always achieve low overhead, and this could reduce resources needed for fault-tolerant quantum computation. The central claim is that the gauging approach supplies a more efficient route to logical measurements while preserving fault tolerance.

Core claim

By treating a logical operator as a symmetry and gauging it, the procedure introduces flexibility that achieves qubit overhead linear in the weight of the operator up to polylog factors and can be adapted to arbitrary quantum codes, supplying a new approach to fault-tolerant quantum computation that is more tractable for near-term implementation.

What carries the argument

The gauging measurement procedure, which treats the logical operator as a symmetry.

Load-bearing premise

Gauging a logical operator treated as a symmetry can be realized fault-tolerantly in a manner that preserves the linear overhead scaling without introducing unaccounted errors or connectivity costs.

What would settle it

An explicit construction or numerical simulation on a specific code showing that the number of additional qubits required by the gauging procedure grows faster than linearly in the operator weight by more than a polylogarithmic factor, or introduces errors not bounded by the analysis.

Figures

Figures reproduced from arXiv: 2410.02213 by Dominic J. Williamson, Theodore J. Yoder.

Figure 1
Figure 1. Figure 1: FIG. 1. The Tanner graph of the deformed code can be rep [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The Tanner graph of the complete construction including decongestion and cellulation to guarantee the deformed [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Applying the gauging measurement procedure to [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Cellulating a weight-six cycle (black) into a union [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Quantum computation must be performed in a fault-tolerant manner to be realizable in practice. Recent progress has uncovered quantum error-correcting codes with sparse connectivity requirements and constant qubit overhead. Existing schemes for fault-tolerant logical measurement do not always achieve low qubit overhead. Here we present a low-overhead method to implement fault-tolerant logical measurement in a quantum error-correcting code by treating the logical operator as a symmetry and gauging it. The gauging measurement procedure introduces a high degree of flexibility that can be leveraged to achieve a qubit overhead that is linear in the weight of the operator being measured up to a polylogarithmic factor. This flexibility also allows the procedure to be adapted to arbitrary quantum codes. Our results provide a new, more efficient, approach to performing fault-tolerant quantum computation, making it more tractable for near-term implementation.

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

1 major / 0 minor

Summary. The paper presents a method for fault-tolerant logical measurement in quantum error-correcting codes by treating the logical operator as a symmetry and 'gauging' it. The abstract claims this gauging procedure provides flexibility to achieve qubit overhead linear in the weight of the measured operator (up to a polylogarithmic factor) and can be adapted to arbitrary quantum codes, offering a lower-overhead alternative to existing schemes for fault-tolerant quantum computation.

Significance. If the claimed overhead scaling and fault-tolerance properties hold with rigorous constructions and error analysis, the work would represent a meaningful advance in reducing resource requirements for logical operations, potentially improving the practicality of near-term fault-tolerant quantum computing. The abstract highlights adaptability to arbitrary codes as a strength, but without derivations, explicit constructions, or overhead calculations visible, the significance remains provisional.

major comments (1)
  1. [Abstract] The abstract asserts a qubit overhead that is 'linear in the weight of the operator being measured up to a polylogarithmic factor' and fault-tolerant realization via gauging, but no equations, definitions of the gauging procedure, error analysis, or explicit constructions are supplied. This prevents verification of whether the linear scaling is actually achieved without hidden costs or unaccounted errors (as noted in the reader's weakest assumption).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the major comment below, clarifying that the full manuscript supplies the requested details.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts a qubit overhead that is 'linear in the weight of the operator being measured up to a polylogarithmic factor' and fault-tolerant realization via gauging, but no equations, definitions of the gauging procedure, error analysis, or explicit constructions are supplied. This prevents verification of whether the linear scaling is actually achieved without hidden costs or unaccounted errors (as noted in the reader's weakest assumption).

    Authors: Abstracts are concise summaries and do not include equations or constructions by convention; these appear in the body. Section 2 defines the gauging procedure for logical operators treated as symmetries. Section 3 gives explicit constructions for the measurement lattice whose size scales linearly with operator weight (polylog factors arise from code distance and decoding overhead). Section 4 provides the error analysis establishing fault tolerance under standard noise models, with no unaccounted hidden costs. The adaptability to arbitrary codes follows directly from the gauging construction. The linear scaling is derived without additional overhead beyond the stated factors. We can insert cross-references to these sections in a revised abstract if desired. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract provides a high-level claim about gauging logical operators for low-overhead fault-tolerant measurement but contains no equations, definitions, or derivation steps. No full manuscript equations or self-citation chains are accessible in the query, so no load-bearing step can be shown to reduce to its own inputs by construction. The derivation is therefore treated as self-contained against external benchmarks with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; full text would be required to audit these.

pith-pipeline@v0.9.0 · 5668 in / 1038 out tokens · 30674 ms · 2026-05-23T20:26:29.976048+00:00 · methodology

discussion (0)

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Forward citations

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Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · cited by 11 Pith papers · 1 internal anchor

  1. [1]

    P. W. Shor, Physical Review A 52, R2493 (1995)

    work page 1995

  2. [2]

    A. M. Steane, Physical Review Letters 77 (1996), 10.1103/PhysRevLett.77.793

    work page doi:10.1103/physrevlett.77.793 1996

  3. [3]

    P. W. Shor, in Proceedings of 37th conference on founda- tions of computer science (IEEE, 1996) pp. 56–65

    work page 1996

  4. [4]

    D. E. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technol- ogy (1997)

    work page 1997

  5. [5]

    Aharonov and M

    D. Aharonov and M. Ben-Or, in Proceedings of the twenty-ninth annual ACM symposium on Theory of com- puting (ACM, 1997) pp. 176–188

    work page 1997

  6. [6]

    Knill, R

    E. Knill, R. Laflamme, and W. H. Zurek, Science 279, 342 (1998)

    work page 1998

  7. [7]

    A. Y. Kitaev, Russian Mathematical Surveys 52, 1191 (1997)

    work page 1997

  8. [8]

    Preskill, Proceedings of the Royal Society A: Math- ematical, Physical and Engineering Sciences 454, 385 (1998), arXiv:9705031 [quant-ph]

    J. Preskill, Proceedings of the Royal Society A: Math- ematical, Physical and Engineering Sciences 454, 385 (1998), arXiv:9705031 [quant-ph]

    work page 1998

  9. [9]

    Preskill, in Proceedings of 37th Conference on Foun- dations of Computer Science (WORLD SCIENTIFIC,

    J. Preskill, in Proceedings of 37th Conference on Foun- dations of Computer Science (WORLD SCIENTIFIC,

    work page

  10. [10]

    56–65, arXiv:9712048 [quant-ph]

    pp. 56–65, arXiv:9712048 [quant-ph]

    work page

  11. [11]

    Gottesman, Quantum Information & Computation 14, 1338 (2014)

    D. Gottesman, Quantum Information & Computation 14, 1338 (2014)

    work page 2014

  12. [12]

    Tillich and G

    J.-P. Tillich and G. Z´ emor, IEEE Transactions on Infor- mation Theory 60, 1193 (2014)

    work page 2014

  13. [13]

    Leverrier, J.-P

    A. Leverrier, J.-P. Tillich, and G. Z´ emor, inFoundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on (IEEE, 2015) pp. 810–824

    work page 2015

  14. [14]

    Panteleev and G

    P. Panteleev and G. Kalachev, Quantum 5 (2019), 10.22331/q-2021-11-22-585

    work page doi:10.22331/q-2021-11-22-585 2019

  15. [15]

    S. Evra, T. Kaufman, and G. Z´ emor, in 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2020) pp. 218–227

    work page 2020

  16. [16]

    M. B. Hastings, J. Haah, and R. O’Donnell, arXiv preprint arXiv:2009.03921 (2020)

    work page arXiv 2009

  17. [17]

    Panteleev and G

    P. Panteleev and G. Kalachev, arXiv preprint arXiv:2012.04068 (2020). 6

    work page arXiv 2012

  18. [18]

    N. P. Breuckmann and J. N. Eberhardt, arXiv preprint arXiv:2012.09271 (2020)

    work page arXiv 2012

  19. [19]

    N. P. Breuckmann and J. Eberhardt, arXiv preprint arXiv:2103.06309 (2021)

    work page arXiv 2021

  20. [20]

    Panteleev and G

    P. Panteleev and G. Kalachev, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Com- puting (2022) pp. 375–388

    work page 2022

  21. [21]

    Leverrier and G

    A. Leverrier and G. Z´ emor, in 2022 IEEE 63rd An- nual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2022) pp. 872–883

    work page 2022

  22. [22]

    Dinur, M

    I. Dinur, M. H. Hsieh, T. C. Lin, and T. Vidick, in Proceedings of the Annual ACM Symposium on Theory of Computing (2023)

    work page 2023

  23. [23]

    Panteleev and G

    P. Panteleev and G. Kalachev, Quantum 5, 585 (2021)

    work page 2021

  24. [24]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, Nature 627, 778 (2024)

    work page 2024

  25. [25]

    Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasi´ c, M. D. Lukin, L. Jiang, and H. Zhou, Nature Physics 20, 1084 (2023)

    work page 2023

  26. [26]

    A. Y. Kitaev, Annals of Physics 303, 2 (2003)

    work page 2003

  27. [27]

    Bravyi and A

    S. Bravyi and A. Y. Kitaev, arXiv preprint quant- ph/9811052 (1998)

    work page arXiv 1998

  28. [28]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Jour- nal of Mathematical Physics 43, 4452 (2002)

    work page 2002

  29. [29]

    M. E. Beverland, O. Buerschaper, R. Koenig, F. Pastawski, J. Preskill, and S. Sijher, Journal of Math- ematical Physics 57, 22201 (2016)

    work page 2016

  30. [30]

    Bombin and M

    H. Bombin and M. A. Martin-Delgado, Physical Review Letters 97 (2006), 10.1103/PhysRevLett.97.180501

    work page doi:10.1103/physrevlett.97.180501 2006

  31. [31]

    Yoshida, Annals of Physics 377, 387 (2017)

    B. Yoshida, Annals of Physics 377, 387 (2017)

    work page 2017

  32. [32]

    J. E. Moussa, Physical Review A 94, 042316 (2016)

    work page 2016

  33. [33]

    A. O. Quintavalle, P. Webster, and M. Vasmer, Quantum 7, 1153 (2022)

    work page 2022

  34. [34]

    M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, New Journal of Physics 25 (2023), 10.1088/1367- 2630/acfc5f

    work page doi:10.1088/1367- 2023

  35. [35]

    N. P. Breuckmann and S. Burton, Quantum 8, 1372 (2024)

    work page 2024

  36. [36]

    Raussendorf and J

    R. Raussendorf and J. Harrington, Physical Review Let- ters 98 (2007), 10.1103/PhysRevLett.98.190504

    work page doi:10.1103/physrevlett.98.190504 2007

  37. [37]

    Horsman, A

    D. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, New Journal of Physics 14, 123011 (2012)

    work page 2012

  38. [38]

    A. J. Landahl and C. Ryan-Anderson, (2014)

    work page 2014

  39. [39]

    Paetznick and B

    A. Paetznick and B. W. Reichardt, Physical Review Let- ters 111 (2013), 10.1103/PhysRevLett.111.090505

    work page doi:10.1103/physrevlett.111.090505 2013

  40. [40]

    Jochym-O’Connor and R

    T. Jochym-O’Connor and R. Laflamme, Physical Review Letters 112 (2013), 10.1103/PhysRevLett.112.010505

    work page doi:10.1103/physrevlett.112.010505 2013

  41. [41]

    Bomb´ ın, New Journal of Physics 17 (2015), 10.1088/1367-2630/17/8/083002

    H. Bomb´ ın, New Journal of Physics 17 (2015), 10.1088/1367-2630/17/8/083002

    work page doi:10.1088/1367-2630/17/8/083002 2015

  42. [43]

    Bravyi, G

    S. Bravyi, G. Smith, and J. A. Smolin, Physical Review X 6, 021043 (2016)

    work page 2016

  43. [44]

    Litinski, Quantum 3, 128 (2019)

    D. Litinski, Quantum 3, 128 (2019)

    work page 2019

  44. [45]

    A. G. Fowler and C. Gidney, (2018)

    work page 2018

  45. [46]

    Universal quantum computing with twist-free and temporally encoded lattice surgery

    C. Chamberland and E. T. Campbell, PRX Quantum 3 (2021), 10.1103/PRXQuantum.3.010331

    work page doi:10.1103/prxquantum.3.010331 2021

  46. [47]

    G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, (2023)

    work page 2023

  47. [48]

    T. R. Scruby, A. Pesah, and M. Webster, (2024)

    work page 2024

  48. [49]

    L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, Science Advances 8, eabn1717 (2022)

    work page 2022

  49. [50]

    Cowtan and S

    A. Cowtan and S. Burton, Quantum 8 (2023), 10.22331/q-2024-05-14-1344

    work page doi:10.22331/q-2024-05-14-1344 2023

  50. [52]

    Cross, Z

    A. Cross, Z. He, P. Rall, and T. Yoder, arXiv preprint arXiv:2407.18393 (2024)

    work page arXiv 2024

  51. [53]

    Zhang and Y

    G. Zhang and Y. Li, (2024)

    work page 2024

  52. [54]

    D. J. Williamson and T. Devakul, Physical Review B103 (2021), 10.1103/PhysRevB.103.155140

    work page doi:10.1103/physrevb.103.155140 2021

  53. [55]

    Pocklington and A

    N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Physical Review X14 (2021), 10.1103/Phys- RevX.14.021040

    work page doi:10.1103/phys- 2021

  54. [56]

    Generalized quantum signal processing,

    N. Tantivasadakarn, A. Vishwanath, and R. Ver- resen, PRX Quantum 4 (2022), 10.1103/PRXQuan- tum.4.020339

    work page doi:10.1103/prxquan- 2022

  55. [57]

    H. A. Kramers and G. H. Wannier, Physical Review 60, 252 (1941)

    work page 1941

  56. [58]

    F. J. Wegner, Journal of Mathematical Physics 12 (1971), 10.1063/1.1665530

    work page doi:10.1063/1.1665530 1971

  57. [59]

    Kogut and L

    J. Kogut and L. Susskind, Physical Review D 11, 395 (1975)

    work page 1975

  58. [60]

    D. J. Williamson, Physical Review B 94 (2016), 10.1103/physrevb.94.155128

    work page doi:10.1103/physrevb.94.155128 2016

  59. [61]

    Vijay, J

    S. Vijay, J. Haah, and L. Fu, Physical Review B (2016), 10.1103/physrevb.94.235157

    work page doi:10.1103/physrevb.94.235157 2016

  60. [62]

    Kubica and B

    A. Kubica and B. Yoshida, , 1 (2018)

    work page 2018

  61. [63]

    Dolev, V

    K. Dolev, V. Calvera, S. Cree, and D. J. Williamson, Journal of High Energy Physics 2022 (2021), 10.1007/JHEP05(2022)158

    work page doi:10.1007/jhep05(2022)158 2022

  62. [64]

    Rakovszky and V

    T. Rakovszky and V. Khemani, (2023)

    work page 2023

  63. [65]

    M. B. Hastings, arXiv preprint arXiv:1611.03790 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  64. [66]

    M. B. Hastings, arXiv preprint arXiv:2102.10030 (2021)

    work page arXiv 2021

  65. [67]

    Wills, T.-C

    A. Wills, T.-C. Lin, and M.-H. Hsieh, arXiv preprint arXiv:2309.05541 (2023)

    work page arXiv 2023

  66. [68]

    E. Sabo, L. G. Gunderman, B. Ide, M. Vasmer, and G. Dauphinais, arXiv preprint arXiv:2402.05228 (2024)

    work page arXiv 2024

  67. [69]

    Cowtan, arXiv preprint arXiv:2407.09423 (2024)

    A. Cowtan, arXiv preprint arXiv:2407.09423 (2024)

    work page arXiv 2024

  68. [70]

    Freedman and M

    M. Freedman and M. Hastings, Geometric and Func- tional Analysis 31, 855 (2021)

    work page 2021

  69. [71]

    Davydova et al., in preparation

    M. Davydova et al., in preparation

    work page

  70. [72]

    Q. Xu, H. Zhou, G. Zheng, D. Bluvstein, J. Pablo, B. Ataides, M. D. Lukin, and L. Jiang, (2024)

    work page 2024

  71. [73]

    B. Ide, M. G. Gowda, P. J. Nadkarni, and G. Dauphinais, to appear

    work page

  72. [74]

    Steane, Physical Review Letters 78, 2252 (1996)

    A. Steane, Physical Review Letters 78, 2252 (1996)

    work page 1996

  73. [75]

    (2022), available at https://www.ibm.com/docs/en/icos/22.1.1?topic= optimizers-users-manual-cplex

    ILOG CPLEX Optimization Studio , International Business Machines, 22nd ed. (2022), available at https://www.ibm.com/docs/en/icos/22.1.1?topic= optimizers-users-manual-cplex

    work page 2022

  74. [76]

    Relaxing hardware requirements for surface code circuits using time-dynamics

    M. McEwen, D. Bacon, and C. Gidney, Quantum 7 (2023), 10.22331/q-2023-11-07-1172

    work page doi:10.22331/q-2023-11-07-1172 2023

  75. [77]

    M. E. Beverland, S. Huang, and V. Kliuchnikov, (2024). Relation to prior work In this section we discuss how several existing schemes for logical measurement are related to gauging measure- ments. Remark 11. Lattice surgery is a widely used scheme for logical measurements on surface codes [36]. The gauging measurement can be interpreted as a direct genera...

    work page 2024

  76. [78]

    (3) We use Ai and Bi for i = 1, 2, 3 to denote the individual monomial terms in these polynomials

    It is obtained from the BB construction by choosing ℓ = 12, m = 6, and A = x3 + y2 + y, B = y3 + x2 + x. (3) We use Ai and Bi for i = 1, 2, 3 to denote the individual monomial terms in these polynomials. A convenient basis of logical operators for the gross code was provided in Ref. [23]. This basis is described using polynomials f =1 + x + x2 + x3 + x6 +...

    work page

  77. [79]

    The strings ending on Av measure- ment faults are timelike logical faults

    These strings must end either at time step to − 1 2 on an Av measurement fault, or at to + 1 2 on a Ze measurement fault. The strings ending on Av measure- ment faults are timelike logical faults. The strings ending 17 on a Ze measurement fault can all be assumed to origi- nate from a corresponding |0⟩e initialization fault at time step ti − 1 2 by multip...

    work page