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arxiv: 2408.14828 · v4 · pith:B53SIQHTnew · submitted 2024-08-27 · 🪐 quant-ph

Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code

Christopher Gerhard , Todd A. Brun This is my paper

Pith reviewed 2026-05-23 21:40 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error detectionweak fault tolerance[[n,n-2,2]] codestabilizer measurementsuniversal quantum computationNISQ devicesgate constructions
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The pith

Constructions in the [[n,n-2,2]] code detect any single faulty gate error using only end-of-computation measurements to achieve weak fault tolerance.

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

The paper develops gate constructions inside a quantum error-detecting code that encodes n-2 logical qubits into n physical ones. These constructions ensure that any error from a single faulty gate can be detected by measuring the code stabilizers and extra ancillas after the full computation finishes. This weak form of fault tolerance improves the success rate of small quantum calculations compared to running them without any encoding, provided the physical error rate is low enough. The approach requires far fewer extra qubits and operations than schemes that achieve complete fault tolerance.

Core claim

In the [[n,n-2,2]] quantum error-detecting code, specific constructions for a universal set of gates allow detection of any error arising from a single faulty gate. Detection occurs by measuring the stabilizer generators of the code together with additional ancillas at the conclusion of the computation. This property holds up to analog imprecision on the physical rotation gate and yields weak fault tolerance.

What carries the argument

The [[n,n-2,2]] error-detecting code together with gate constructions that keep single errors detectable by final stabilizer and ancilla measurements.

Load-bearing premise

End-of-computation measurements alone suffice to catch every possible error produced by any single faulty gate during the entire circuit.

What would settle it

An explicit single-gate error in one of the constructions that produces an output state indistinguishable from the correct one after the final measurements.

Figures

Figures reproduced from arXiv: 2408.14828 by Christopher Gerhard, Todd A. Brun.

Figure 1
Figure 1. Figure 1: FIG. 1: Circuits for the logical CNOT gate in the [[4 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Circuits for the logical Phase gate in the [[4 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Circuits for the logical Hadamard gate in the [[4 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Weakly fault-tolerant circuits for the ZZ rotation gate. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Weakly fault-tolerant circuits for the XX rotation gate. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Logical [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: A rotation gate that rotates a quantum state [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Measurement of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Weakly fault-tolerant circuit for initialization into the [[ [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Weakly fault-tolerant Bell basis measurement [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Weakly fault-tolerant circuit for readout of the [[ [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Log-log plot of the undetectable error probability for the physical (blue), encoded (orange), and weakly [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Log-log plot of the post-selection rate for the physical (blue), encoded (orange), and weakly fault-tolerant [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Log-log plot of the undetectable-error probability for the physical (blue), encoded (orange), and weakly [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Log-log plot of the post-selection rate for the physical (blue), encoded (orange), and weakly fault-tolerant [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
read the original abstract

Many current quantum error-correcting codes that achieve full fault tolerance suffer from having low ratios of logical to physical qubits and significant overhead. This makes them difficult to implement on current noisy intermediate-scale quantum (NISQ) computers and results in the inability to perform quantum algorithms at useful scales with near-term quantum processors. As a result, calculations are generally done without encoding. We propose a middle ground between these two approaches: constructions in the $[[n,n-2,2]]$ quantum error-detecting code that can detect any error from a single faulty gate by measuring the stabilizer generators of the code and additional ancillas at the end of the computation. This achieves weak fault tolerance. As we show, this yields a significant improvement over no error correction for small computations with low enough physical error probabilities and requires much less overhead than codes that achieve full fault tolerance. We give constructions for a set of gates that achieve universal quantum computation in this error-detecting code, while satisfying weak fault tolerance up to analog imprecision on the physical rotation gate.

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

2 major / 2 minor

Summary. The paper proposes explicit gate constructions for universal quantum computation inside the [[n,n-2,2]] quantum error-detecting code. These constructions are claimed to achieve 'weak fault tolerance': any error arising from a single faulty physical gate is detected by measuring the code stabilizers together with additional ancillas only at the very end of the circuit. The authors argue that the resulting overhead is far lower than that of full fault-tolerant codes while still yielding a net improvement over bare, uncoded computation for small algorithms when the physical error rate is sufficiently low.

Significance. If the weak-fault-tolerance property can be rigorously established, the approach would supply a practical middle ground for near-term devices: modest error detection with qubit overhead linear in the number of logical qubits rather than the quadratic or higher overhead required for full fault tolerance. The claim is therefore potentially relevant to NISQ-era algorithm execution, provided the detection guarantee holds for the supplied gate decompositions.

major comments (2)
  1. [constructions for universal gates (abstract and § on gate implementations)] The central claim—that every single physical fault (including during non-Clifford rotations) produces a detectable non-trivial syndrome when stabilizers and ancillas are measured only at the final time—rests on the specific gate constructions. Because the code distance is 2, a fault inside a multi-qubit gate can in principle spread to a weight-2 error that lies in the codespace or commutes with the final checks. The manuscript must therefore supply an explicit case-by-case verification (or inductive argument) showing that no such undetectable propagated error occurs for the chosen decompositions; without this verification the weak-fault-tolerance property is unproven.
  2. [performance claims] The performance comparison with uncoded computation is stated to be 'significant' for small computations at low physical error rates. No numerical simulation, threshold calculation, or explicit error-probability model is provided to support this quantitative claim, making it impossible to assess whether the improvement is realized under realistic noise models that include the analog imprecision on the physical rotation gate.
minor comments (2)
  1. Notation for the additional ancillas and the precise timing of their measurements should be defined consistently throughout the text.
  2. The abstract states that the scheme works 'up to analog imprecision on the physical rotation gate'; the precise tolerance on this imprecision should be stated explicitly in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments below and will revise the manuscript to strengthen the presentation of the weak fault-tolerance property and the supporting performance analysis.

read point-by-point responses

  1. Referee: [constructions for universal gates (abstract and § on gate implementations)] The central claim—that every single physical fault (including during non-Clifford rotations) produces a detectable non-trivial syndrome when stabilizers and ancillas are measured only at the final time—rests on the specific gate constructions. Because the code distance is 2, a fault inside a multi-qubit gate can in principle spread to a weight-2 error that lies in the codespace or commutes with the final checks. The manuscript must therefore supply an explicit case-by-case verification (or inductive argument) showing that no such undetectable propagated error occurs for the chosen decompositions; without this verification the weak-fault-tolerance property is unproven.

    Authors: We agree that an explicit verification is required to rigorously establish the claim. The gate decompositions were selected so that any single physical fault produces a non-trivial syndrome detectable by the final stabilizer and ancilla measurements, but the manuscript presents the constructions without a dedicated case-by-case or inductive argument. In the revised manuscript we will add a new subsection that performs this verification for each gate (including the non-Clifford rotations, accounting for analog imprecision), confirming that no weight-2 error lies in the codespace or commutes with the checks. revision: yes

  2. Referee: [performance claims] The performance comparison with uncoded computation is stated to be 'significant' for small computations at low physical error rates. No numerical simulation, threshold calculation, or explicit error-probability model is provided to support this quantitative claim, making it impossible to assess whether the improvement is realized under realistic noise models that include the analog imprecision on the physical rotation gate.

    Authors: The claim rests on an analytical model in which the undetected-error probability scales as O(p²) versus O(p) for the bare circuit. We will expand the relevant section to include the explicit error-probability expressions for small circuits under this model, together with a brief discussion of how analog imprecision on rotations is bounded by the detection property. Full Monte-Carlo simulations under detailed hardware noise models are beyond the scope of the present work; we will therefore qualify the performance statement as analytical and note the limitation. revision: partial

Circularity Check

0 steps flagged

No circularity detected; explicit constructions rest on standard QEC assumptions without self-referential reduction.

full rationale

The paper proposes explicit gate constructions in the [[n,n-2,2]] code to achieve weak fault tolerance via end-of-computation stabilizer and ancilla measurements. No evidence of self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appears in the abstract or context. The derivation chain relies on direct construction of gates satisfying the detection property under stated assumptions (including analog imprecision for rotations), which are independent of the target result and do not reduce to the inputs by definition. This is a standard constructive proposal in quantum error correction and scores as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that end-of-computation stabilizer measurements suffice to detect single faulty gate errors in this code family; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Measuring the stabilizer generators and additional ancillas at the end of the computation detects any error from a single faulty gate in the [[n,n-2,2]] code.
    This premise underpins the weak fault tolerance claim.

pith-pipeline@v0.9.0 · 5706 in / 1163 out tokens · 44955 ms · 2026-05-23T21:40:14.163000+00:00 · methodology

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Forward citations

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · cited by 2 Pith papers · 4 internal anchors

  1. [1]

    population survival rate

    If we rotated by the incorrect angle, we can repeat- edly apply the circuit with corrections to the angle until we measure that the proper rotation occurred. Unfortu- nately, it appears that circuits with a higher probability of the correct rotation than 1 /2 are not possible. The circuit in Fig. 7 uses only Clifford gates (the CNOT) and Pauli measurement...

    work page

  2. [2]

    up to a stabilizer generator

    If one instead just replaces the XX and ZZ gates in Figs. 1–3 with their unitary versions, then phase errors may be introduced into the circuit. This is not a major issue, since any error in the phase can be corrected after the circuit through single-qubit operations. For each of these circuits we provide the phase corrections necessary if one implements ...

    work page

  3. [3]

    DiVincenzo and Peter W

    David P. DiVincenzo and Peter W. Shor. Fault-tolerant error correction with efficient quantum codes. Physical Review Letters , 77(15):3260–3263, 1996. URL https: //doi.org/10.1103/PhysRevLett.77.3260

    work page doi:10.1103/physrevlett.77.3260 1996

  4. [4]

    An introduction to quantum er- ror correction and fault-tolerant quantum computation,

    Daniel Gottesman. An introduction to quantum er- ror correction and fault-tolerant quantum computation,

    work page

  5. [5]

    URL https://doi.org/10.48550/arXiv.0904. 2557

    work page doi:10.48550/arxiv.0904

  6. [6]

    Yoder, Ryuji Takagi, and Isaac L

    Theodore J. Yoder, Ryuji Takagi, and Isaac L. Chuang. Universal fault-tolerant gates on concatenated stabilizer codes. Physical Review X , 6(3), 2016. URL https:// doi.org/10.1103/PhysRevX.6.031039

    work page doi:10.1103/physrevx.6.031039 2016

  7. [7]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniver- sary Edition . Cambridge University Press, 2010. URL https://doi.org/10.1017/CBO9780511976667

    work page doi:10.1017/cbo9780511976667 2010

  8. [8]

    Lidar and Todd A

    Daniel A. Lidar and Todd A. Brun. Quantum Error Correction. Cambridge University Press, 2013. URL https://doi.org/10.1017/CBO9781139034807

    work page doi:10.1017/cbo9781139034807 2013

  9. [9]

    Theory of Quantum Error Correction for General Noise

    Emanuel Knill, Raymond Laflamme, and Lorenza Viola. Theory of quantum error correction for general noise. Physical Review Letters , 84(11):2525–2528, 2000. URL https://doi.org/10.48550/arXiv.quant-ph/9908066

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/9908066 2000

  10. [10]

    Available: https://link.aps.org/doi/10.1103/PhysRevA

    Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practi- cal large-scale quantum computation. Physical Review A, 86(3), 2012. URL https://doi.org/10.1103/PhysRevA. 86.032324

    work page doi:10.1103/physreva 2012

  11. [11]

    Restrictions on transversal encoded quantum gate sets

    Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets. Physical Review Letters, 102(11), 2009. URL https://doi.org/10.1103/ PhysRevLett.102.110502

    work page 2009

  12. [12]

    Universal quantum computation with ideal clifford gates and noisy ancillas

    Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review A, 71(2), 2005. URL https://doi.org/ 10.1103/PhysRevA.71.022316

    work page doi:10.1103/physreva.71.022316 2005

  13. [13]

    Magic state distillation: Not as costly as you think

    Daniel Litinski. Magic state distillation: Not as costly as you think. Quantum, 3:205, 2019. URL http://dx.doi. org/10.22331/q-2019-12-02-205

    work page doi:10.22331/q-2019-12-02-205 2019

  14. [14]

    Fowler, Simon J

    Austin G. Fowler, Simon J. Devitt, and Cody Jones. Surface code implementation of block code state dis- tillation. Scientific Reports, 3(1), 2013. URL https: //doi.org/10.1038/srep01939

    work page doi:10.1038/srep01939 2013

  15. [15]

    Craig Gidney and Austin G. Fowler. Efficient magic state factories with a catalyzed |CCZ ⟩ to 2 |T ⟩ transforma- tion. Quantum, 3:135, 2019. URL https://doi.org/ 10.22331/q-2019-04-30-135

    work page doi:10.22331/q-2019-04-30-135 2019

  16. [16]

    Magic-state distilla- tion with low overhead

    Sergey Bravyi and Jeongwan Haah. Magic-state distilla- tion with low overhead. Physical Review A, 86(5), 2012. URL https://doi.org/10.1103/PhysRevA.86.052329

    work page doi:10.1103/physreva.86.052329 2012

  17. [18]

    Reichardt

    Rui Chao and Ben W. Reichardt. Quantum error cor- rection with only two extra qubits. Physical Review Let- ters, 121:050502, 2018. URL https://doi.org/10.1103% 2Fphysrevlett.121.050502

    work page 2018

  18. [19]

    Reichardt

    Rui Chao and Ben W. Reichardt. Fault-tolerant quantum computation with few qubits. npj Quan- tum Inf , 4:42, 2018. URL https://doi.org/10.1038/ s41534-018-0085-z

    work page 2018

  19. [20]

    Yoder and Isaac H

    Theodore J. Yoder and Isaac H. Kim. The surface code with a twist. Quantum, 1:2, 2017. ISSN 2521-327X. URL http://dx.doi.org/10.22331/q-2017-04-25-2

    work page doi:10.22331/q-2017-04-25-2 2017

  20. [21]

    Beverland

    Christopher Chamberland and Michael E. Beverland. Flag fault-tolerant error correction with arbitrary dis- tance codes. Quantum, 2:53, 2018. ISSN 2521-327X. URL http://dx.doi.org/10.22331/q-2018-02-08-53

    work page doi:10.22331/q-2018-02-08-53 2018

  21. [22]

    Reichardt

    Rui Chao and Ben W. Reichardt. Flag fault-tolerant error correction for any stabilizer code. PRX Quan- tum, 1:010302, 2020. URL http://dx.doi.org/10. 1103/PRXQuantum.1.010302

    work page 2020

  22. [23]

    Almudever, and Sebastian Feld

    Dhruv Bhatnagar, Matthew Steinberg, David Elkouss, Carmen G. Almudever, and Sebastian Feld. Low-depth flag-style syndrome extraction for small quantum error- correction codes. In 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) , page 63–69. IEEE, 2023. URL http://dx.doi.org/10.1109/ QCE57702.2023.00016

    work page arXiv 2023

  23. [24]

    Theerapat Tansuwannont, Balint Pato, and Ken- neth R. Brown. Adaptive syndrome measurements for shor-style error correction. Quantum, 7:1075, 2023. ISSN 2521-327X. URL http://dx.doi.org/10.22331/ q-2023-08-08-1075

    work page 2023

  24. [25]

    Flag gadgets based on classical codes, 2024

    Benjamin Anker and Milad Marvian. Flag gadgets based on classical codes, 2024. URL https://doi.org/10. 48550/arXiv.2212.10738

    work page arXiv 2024

  25. [26]

    Reducing quantum er- ror correction overhead with versatile flag-sharing syn- drome extraction circuits, 2024

    Pei-Hao Liou and Ching-Yi Lai. Reducing quantum er- ror correction overhead with versatile flag-sharing syn- drome extraction circuits, 2024. URL https://doi.org/ 10.48550/arXiv.2407.00607

    work page doi:10.48550/arxiv.2407.00607 2024

  26. [27]

    Anderson, Guillaume Duclos-Cianci, and David Poulin

    Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. Fault-tolerant conversion between the steane and reed-muller quantum codes. Physical Review Letters, 113 (8), 2014. ISSN 1079-7114. URL http://dx.doi.org/ 10.1103/PhysRevLett.113.080501

    work page doi:10.1103/physrevlett.113.080501 2014

  27. [28]

    Pragati Gupta, Andrea Morello, and Barry C. Sanders. Universal transversal gates, 2024. URL https://doi. org/10.48550/arXiv.2410.07045

    work page doi:10.48550/arxiv.2410.07045 2024

  28. [29]

    Fault-tolerant code-switching protocols for near-term quantum processors

    Friederike Butt, Sascha Heußen, Manuel Rispler, and Markus M¨ uller. Fault-tolerant code-switching protocols for near-term quantum processors. PRX Quantum, 5(2),

    work page

  29. [31]

    Efficient fault-tolerant code switching via one-way transversal cnot gates, 2024

    Sascha Heußen and Janine Hilder. Efficient fault-tolerant code switching via one-way transversal cnot gates, 2024. URL https://doi.org/10.48550/arXiv.2409.13465

    work page doi:10.48550/arxiv.2409.13465 2024

  30. [32]

    Locher, Katharina Brech- telsbauer, Hans Peter B¨ uchler, and Markus M¨ uller

    Friederike Butt, David F. Locher, Katharina Brech- telsbauer, Hans Peter B¨ uchler, and Markus M¨ uller. Measurement-free, scalable and fault-tolerant universal quantum computing, 2024. URL https://doi.org/10. 48550/arXiv.2410.13568

    work page arXiv 2024

  31. [33]

    Spacetime codes of clifford circuits, 2023

    Nicolas Delfosse and Adam Paetznick. Spacetime codes of clifford circuits, 2023. URL https://doi.org/10. 48550/arXiv.2304.05943

    work page arXiv 2023

  32. [34]

    Linke, Mauricio Gutierrez, Kevin A

    Norbert M. Linke, Mauricio Gutierrez, Kevin A. Lands- man, Caroline Figgatt, Shantanu Debnath, Kenneth R. Brown, and Christopher Monroe. Fault-tolerant quan- tum error detection. Science Advances, 3(10), 2017. URL https://doi.org/10.1126/sciadv.1701074. 21

    work page doi:10.1126/sciadv.1701074 2017

  33. [35]

    Norris, Mihai Gabureac, Christopher Eichler, and An- dreas Wallraff

    Christian Kraglund Andersen, Ants Remm, Stefania Lazar, Sebastian Krinner, Nathan Lacroix, Graham J. Norris, Mihai Gabureac, Christopher Eichler, and An- dreas Wallraff. Repeated quantum error detection in a surface code. Nature Physics, 16(8):875–880, 2020. URL https://doi.org/10.1038/s41567-020-0920-y

    work page doi:10.1038/s41567-020-0920-y 2020

  34. [36]

    Marciniak, Sascha Heußen, Rainer Blatt, Philipp Schindler, Manuel Rispler, Markus M¨ uller, and Thomas Monz

    Lukas Postler, Friederike Butt, Ivan Pogorelov, Chris- tian D. Marciniak, Sascha Heußen, Rainer Blatt, Philipp Schindler, Manuel Rispler, Markus M¨ uller, and Thomas Monz. Demonstration of fault-tolerant steane quantum error correction. PRX Quantum, 5(3), 2024. ISSN 2691-

    work page 2024

  35. [38]

    Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael New- man, Muyuan Li, Kenneth R

    Laird Egan, Dripto M. Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael New- man, Muyuan Li, Kenneth R. Brown, Marko Cetina, and Christopher Monroe. Fault-tolerant control of an error-corrected qubit. Nature, 598:281–286, 2021. URL https://doi.org/10.1038/s41586-021-03928-y

    work page doi:10.1038/s41586-021-03928-y 2021

  36. [39]

    Ryan-Anderson, J

    C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz. Realization of real-time fault-tolerant quan- tum error correction. Phys. Rev. X , 11:041058, 2021. URL https://doi.org/10.1103/...

    work page doi:10.1103/physrevx.11.041058 2021

  37. [40]

    M. H. Abobeih, Y. Wang, J. Randall, S. J. H. Loenen, C. E. Bradley, M. Markham, D. J. Twitchen, B. M. Ter- hal, and T. H. Taminiau. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature, 606:884–889, 2022. URL http://dx.doi.org/10.1038/ s41586-022-04819-6

    work page 2022

  38. [41]

    Norris, Christian Kraglund Andersen, Markus M¨ uller, Alexandre Blais, Christopher Eichler, and Andreas Wallraff

    Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefania Lazar, Francois Swiadek, Jo- hannes Herrmann, Graham J. Norris, Christian Kraglund Andersen, Markus M¨ uller, Alexandre Blais, Christopher Eichler, and Andreas Wallraff. Realizing repeated quan- tum error correction in a distance-thr...

    work page 2022

  39. [42]

    Realization of an error-correcting surface code with superconducting qubits

    Youwei Zhao, Yangsen Ye, He-Liang Huang, Yiming Zhang, Dachao Wu, Huijie Guan, Qingling Zhu, Zuolin Wei, Tan He, Sirui Cao, Fusheng Chen, Tung-Hsun Chung, Hui Deng, Daojin Fan, Ming Gong, Cheng Guo, Shaojun Guo, Lianchen Han, Na Li, Shaowei Li, Yuan Li, Futian Liang, Jin Lin, Haoran Qian, Hao Rong, Hong Su, Lihua Sun, Shiyu Wang, Yulin Wu, Yu Xu, Chong Yi...

    work page

  40. [44]

    Suppressing quantum errors by scaling a surface code logical qubit

    Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature, 614:676–681, 2023. URL https://doi.org/10.1038/ s41586-022-05434-1

    work page 2023

  41. [45]

    Evered, Alexandra A

    Dolev Bluvstein, Simon J. Evered, Alexandra A. Geim, Sophie H. Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, J. Pablo Bonilla Ataides, Nishad Maskara, Iris Cong, Xun Gao, Pedro Sales Rodriguez, Thomas Karolyshyn, Giulia Semeghini, Michael J. Gullans, Markus Greiner, Vladan Vuleti´ c, and Mikhail D. Luki...

    work page doi:10.1038/s41586-023-06927-3 2023

  42. [46]

    Ryan-Anderson, N

    C. Ryan-Anderson, N. C. Brown, C. H. Baldwin, J. M. Dreiling, C. Foltz, J. P. Gaebler, T. M. Gatterman, N. Hewitt, C. Holliman, C. V. Horst, J. Johansen, D. Lucchetti, T. Mengle, M. Matheny, Y. Matsuoka, K. Mayer, M. Mills, S. A. Moses, B. Neyenhuis, J. Pino, P. Siegfried, R. P. Stutz, J. Walker, and D. Hayes. High-fidelity and fault-tolerant teleportatio...

    work page doi:10.48550/arxiv.2404.16728 2024

  43. [47]

    Baldwin, David Hayes, Joan M

    Karl Mayer, Ciar´ an Ryan-Anderson, Natalie Brown, Eli- jah Durso-Sabina, Charles H. Baldwin, David Hayes, Joan M. Dreiling, Cameron Foltz, John P. Gaebler, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Nathan Hewitt, Chandler V. Horst, Jacob Johansen, Tanner Mengle, Michael Mills, Steven A. Moses, Peter E. Siegfried, Brian Neyenhuis, J...

    work page doi:10.48550/arxiv.2404.08616 2024

  44. [49]

    Reichardt, David Aasen, Rui Chao, Alex Chernoguzov, Wim van Dam, John P

    Ben W. Reichardt, David Aasen, Rui Chao, Alex Chernoguzov, Wim van Dam, John P. Gaebler, Dan Gresh, Dominic Lucchetti, Michael Mills, Steven A. Moses, Brian Neyenhuis, Adam Paetznick, Andres Paz, Peter E. Siegfried, Marcus P. da Silva, Krysta M. Svore, Zhenghan Wang, and Matt Zanner. Demonstration of quantum computation and error correction with a tessera...

    work page arXiv 2024

  45. [50]

    Entangling four logical qubits beyond break- even in a nonlocal code

    Yifan Hong, Elijah Durso-Sabina, David Hayes, and An- drew Lucas. Entangling four logical qubits beyond break- even in a nonlocal code. Physical Review Letters, 133(18),

    work page

  46. [51]

    Kobrin, Z

    URL http://dx.doi.org/10.1103/PhysRevLett. 133.180601

    work page doi:10.1103/physrevlett

  47. [52]

    Demonstration of logical qubits and repeated error correction with better-than-physical error rates

    A. Paetznick, M. P. da Silva, C. Ryan-Anderson, J. M. Bello-Rivas, J. P. Campora III, A. Chernoguzov, J. M. Dreiling, C. Foltz, F. Frachon, J. P. Gaebler, T. M. Gatterman, L. Grans-Samuelsson, D. Gresh, D. Hayes, N. Hewitt, C. Holliman, C. V. Horst, J. Johansen, D. Lucchetti, Y. Matsuoka, M. Mills, S. A. Moses, B. Neyenhuis, A. Paz, J. Pino, P. Siegfried,...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2404.02280 2024

  48. [53]

    Low-cost noise reduc- tion for clifford circuits, 2024

    Nicolas Delfosse and Edwin Tham. Low-cost noise reduc- tion for clifford circuits, 2024. URL https://doi.org/ 10.48550/arXiv.2407.06583

    work page doi:10.48550/arxiv.2407.06583 2024

  49. [54]

    Partially fault- tolerant quantum computing architecture with error- corrected clifford gates and space-time efficient analog rotations

    Yutaro Akahoshi, Kazunori Maruyama, Hirotaka Os- hima, Shintaro Sato, and Keisuke Fujii. Partially fault- tolerant quantum computing architecture with error- corrected clifford gates and space-time efficient analog rotations. PRX Quantum , 5(1), 2024. ISSN 2691-

    work page 2024

  50. [55]

    22 5.010337

    URL http://dx.doi.org/10.1103/PRXQuantum. 22 5.010337

    work page doi:10.1103/prxquantum

  51. [56]

    Compila- tion of trotter-based time evolution for partially fault- tolerant quantum computing architecture, 2024

    Yutaro Akahoshi, Riki Toshio, Jun Fujisaki, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii. Compila- tion of trotter-based time evolution for partially fault- tolerant quantum computing architecture, 2024. URL https://doi.org/10.48550/arXiv.2408.14929

    work page doi:10.48550/arxiv.2408.14929 2024

  52. [57]

    LG-GAN: Label Guided Adversarial Network for Flexible Targeted Attack of Point Cloud-based Deep Networks

    Riki Toshio, Yutaro Akahoshi, Jun Fujisaki, Hirotaka Os- hima, Shintaro Sato, and Keisuke Fujii. Practical quan- tum advantage on partially fault-tolerant quantum com- puter, 2024. URL https://doi.org/10.48550/arXiv. 2408.14848

    work page internal anchor Pith review doi:10.48550/arxiv 2024

  53. [58]

    Self, Marcello Benedetti, and David Amaro

    Chris N. Self, Marcello Benedetti, and David Amaro. Protecting expressive circuits with a quantum error de- tection code. Nature Physics , 20:219–224, 2024. URL http://dx.doi.org/10.1038/s41567-023-02282-2

    work page doi:10.1038/s41567-023-02282-2 2024

  54. [59]

    Demonstrating bayesian quantum phase estimation with quantum error detection

    Kentaro Yamamoto, Samuel Duffield, Yuta Kikuchi, and David Mu˜ noz Ramo. Demonstrating bayesian quantum phase estimation with quantum error detection. Physical Review Research, 6, 2024. URL http://dx.doi.org/10. 1103/PhysRevResearch.6.013221

    work page 2024

  55. [60]

    Amara Katabarwa, Katerina Gratsea, Athena Caesura, and Peter D. Johnson. Early fault-tolerant quantum computing. PRX Quantum , 5(2), 2024. URL http: //dx.doi.org/10.1103/PRXQuantum.5.020101

    work page doi:10.1103/prxquantum.5.020101 2024

  56. [61]

    Modeling the performance of early fault- tolerant quantum algorithms

    Qiyao Liang, Yiqing Zhou, Archismita Dalal, and Pe- ter Johnson. Modeling the performance of early fault- tolerant quantum algorithms. Physical Review Re- search, 6(2), 2024. URL http://dx.doi.org/10.1103/ PhysRevResearch.6.023118

    work page 2024

  57. [62]

    Rutuja Kshirsagar, Amara Katabarwa, and Peter D. Johnson. On proving the robustness of algorithms for early fault-tolerant quantum computers. Quan- tum, 8:1531, 2024. URL http://dx.doi.org/10.22331/ q-2024-11-20-1531

    work page 2024

  58. [63]

    Shuttling for scalable trapped-ion quantum computers, 2024

    Daniel Schoenberger, Stefan Hillmich, Matthias Brandl, and Robert Wille. Shuttling for scalable trapped-ion quantum computers, 2024. URL https://doi.org/10. 48550/arXiv.2402.14065

    work page arXiv 2024

  59. [64]

    C. F. Sun, X. Y. Chen, W. L. Mu, G. C. Wang, J. B. You, and X. Q. Shao. Holonomic swap and controlled-swap gates of neutral atoms via selective rydberg pumping. EPJ Quantum Technology , 11(1),

    work page

  60. [65]

    URL https://doi.org/10.1140/ epjqt/s40507-024-00246-w

    ISSN 2196-0763. URL https://doi.org/10.1140/ epjqt/s40507-024-00246-w

    work page

  61. [66]

    Fault-tolerant quantum computation with a neutral atom processor

    Ben W. Reichardt, Adam Paetznick, David Aasen, Ivan Basov, Juan M. Bello-Rivas, Parsa Bonderson, Rui Chao, Wim van Dam, Matthew B. Hastings, Andres Paz, Marcus P. da Silva, Aarthi Sundaram, Krysta M. Svore, Alexander Vaschillo, Zhenghan Wang, Matt Zan- ner, William B. Cairncross, Cheng-An Chen, Daniel Crow, Hyosub Kim, Jonathan M. Kindem, Jonathan King, M...

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  62. [67]

    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction and orthog- onal geometry. Physical Review Letters , 78(3):405–408,

    work page

  63. [68]

    URL https://doi.org/10.1103/ PhysRevLett.78.405

    ISSN 1079-7114. URL https://doi.org/10.1103/ PhysRevLett.78.405

    work page

  64. [69]

    Clifford group, sta- bilizer states, and linear and quadratic operations over gf(2)

    Jeroen Dehaene and Bart De Moor. Clifford group, sta- bilizer states, and linear and quadratic operations over gf(2). Phys. Rev. A , 68:042318, 2003. URL https: //doi.org/10.1103/PhysRevA.68.042318

    work page doi:10.1103/physreva.68.042318 2003

  65. [70]

    Narayanan Rengaswamy, Robert Calderbank, Swanand Kadhe, and Henry D. Pfister. Logical clifford synthe- sis for stabilizer codes. IEEE Transactions on Quan- tum Engineering , 1:1–17, 2020. ISSN 2689-1808. URL https://doi.org/10.1109/TQE.2020.3023419

    work page doi:10.1109/tqe.2020.3023419 2020

  66. [71]

    Faster quantum chem- istry simulation on fault-tolerant quantum computers

    N Cody Jones, James D Whitfield, Peter L McMa- hon, Man-Hong Yung, Rodney Van Meter, Al´ an Aspuru- Guzik, and Yoshihisa Yamamoto. Faster quantum chem- istry simulation on fault-tolerant quantum computers. New Journal of Physics , 14(11):115023, November 2012. ISSN 1367-2630. URL http://dx.doi.org/10.1088/ 1367-2630/14/11/115023

    work page 2012

  67. [72]

    Quisp: a quantum internet simulation package,

    Ashish Kakkar, Jeffrey Larson, Alexey Galda, and Rus- lan Shaydulin. Characterizing error mitigation by sym- metry verification in QAOA. In 2022 IEEE Interna- tional Conference on Quantum Computing and Engineer- ing (QCE) . IEEE, 2022. URL https://doi.org/10. 1109%2Fqce53715.2022.00086

    work page arXiv 2022

  68. [73]

    Dawei Zhong and Todd A. Brun. Noise-resilient near- term algorithms with quantum error detection codes

    work page