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arxiv: 2606.05664 · v1 · pith:CGESUJTXnew · submitted 2026-06-04 · 🪐 quant-ph

Gauging the Spacetime Code

Gideon Lee This is my paper

Pith reviewed 2026-06-28 01:31 UTC · model grok-4.3

classification 🪐 quant-ph
keywords spacetime codelattice gauge theoryfault tolerancequantum error correctionmeasurement-based quantum computationtopologically ordered mixed statesPauli noise
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The pith

Gauging the spacetime code produces a lattice gauge theory that carries fault tolerance from circuits, with Gauss laws linking error configurations and Wilson loops serving as detectors.

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

The paper establishes that gauging the spacetime code yields a lattice gauge theory inheriting circuit fault tolerance. In this theory Gauss laws represent equivalence relations among spacetime error configurations. Wilson loops function as detectors for those errors. The resulting structure applies to foliated computation, descriptions of classical memory in certain mixed states, and learnable features of circuit noise.

Core claim

Gauging the spacetime code gives rise to a lattice gauge theory that inherits the elements of fault tolerance associated with a circuit, with Gauss laws corresponding to equivalence relations between configurations of spacetime errors and Wilson loops corresponding to detectors.

What carries the argument

The gauging procedure applied to the spacetime code, which transfers circuit fault-tolerance elements into a lattice gauge theory via Gauss laws and Wilson loops.

If this is right

  • The theory contains foliated computation and therefore supplies one version of a gauge theory for measurement-based quantum computation.
  • For certain topologically ordered mixed states the construction supplies a gauge-theoretic description of the associated classical memory.
  • Gauge-invariant observables in the theory coincide with the learnable degrees of freedom under circuit Pauli noise.

Where Pith is reading between the lines

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

  • The same gauging step might extend to other spacetime-based descriptions of dynamical stability in quantum many-body systems.
  • Links could appear between the detector observables here and classical learning tasks outside Pauli noise models.
  • The construction offers a possible route to embed measurement-based protocols inside existing lattice gauge theory toolkits.

Load-bearing premise

The gauging procedure transfers the fault-tolerance structure of the spacetime code to the gauge theory without loss of the correspondences between Gauss laws, Wilson loops, and error detection.

What would settle it

Explicit computation in a small spacetime code example showing whether the Wilson loops of the gauged theory match the detector outcomes of the original circuit under a chosen error model.

Figures

Figures reproduced from arXiv: 2606.05664 by Gideon Lee.

Figure 1
Figure 1. Figure 1: Schematic illustrating the conventions used in this work. In all figures, time goes from right to left. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic depicting our conventions for where we evaluate states and their ISGs. The state [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustrating examples of elementary circuit operators associated with circuit elements. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustrating the graphical representation of the elementary circuit operators depicted [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A simple example of a detector formed by a [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A schematic depiction of the gauge theory obtained by treating the circuit segment in Fig. 5 as a full [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Circuit depicting repeated measurements of a three-qubit repetition code, together with spacetime [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The gauge theory associated with the with the repeated measurement repetition code circuit of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Schematic depicting our conventions for where we evaluate states and errors in the gauged SSC [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The gauge theory associated with the circuit of Fig. 7, with the open boundary fields fixed to [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Schematic depiction of the gauged SSC associated with an entanglement-fidelity experiment. The [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The gauge theory associated with the repeated-measurement circuit of Fig. 7 under alternative [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Local detector structures in the gauged SSC for the data-syndrome code example presented in [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Gauged SSC for the repeated measurement of a surface code with phenomenological errors. [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The gauged SSC for the walking repetition code circuit introduced in Ref. [40]. In particular, a [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The resource state for the foliated computation constructed from the circuit in Fig. 5. The [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
read the original abstract

In recent years, the spacetime code has arisen as a candidate for a unifying view of fault tolerance in space and time. On the other hand, the recent study of dynamical phases has increasingly turned its attention to fault tolerance as a notion of a dynamically stable process. In this work, I explore one pathway between the two, achieved by gauging the spacetime code. This gives rise to a lattice gauge theory that inherits the elements of fault tolerance associated with a circuit, with Gauss laws corresponding to equivalence relations between configurations of spacetime errors and Wilson loops corresponding to detectors. The obtained gauge theory finds a surprisingly wide array of applications, from quantum error correction to condensed matter physics, and even learning theory: (1) It contains in its description foliated computation, and hence gives rise to one version of a gauge theory for measurement-based quantum computation. (2) For a class of topologically ordered mixed states, it gives us a gauge-theoretic language to describe the classical memory associated with the state. (3) The gauge-invariant observables of the theory which describe detectors also coincide with the learnable degrees of freedom of circuit Pauli noise.

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 manuscript proposes gauging the spacetime code to obtain a lattice gauge theory inheriting fault-tolerance properties from the underlying circuit. Gauss laws are identified with equivalence relations among spacetime error configurations, while Wilson loops correspond to detectors. Three applications are outlined: a gauge-theoretic formulation of foliated computation for measurement-based quantum computation, a description of classical memory in topologically ordered mixed states, and the identification of gauge-invariant observables with learnable degrees of freedom under circuit Pauli noise.

Significance. If the central correspondences are rigorously derived, the work supplies a gauge-theoretic language that unifies circuit-level fault tolerance with dynamical phases and error models, potentially enabling new analyses in quantum error correction, condensed-matter dynamics, and noise learning. Explicit machine-checked or parameter-free aspects are not indicated in the provided description.

major comments (2)
  1. [Abstract / §2 (construction)] The central claim that gauging maps circuit-level spacetime error equivalence relations directly onto Gauss-law constraints (and detectors onto Wilson loops) without loss or additional assumptions is load-bearing for all three applications. The abstract asserts this inheritance but supplies no explicit construction, error-model definition, or verification that the promotion of global symmetries to local gauge symmetries preserves the original equivalence classes; this must be shown in the main text (e.g., via a concrete spacetime code example) to establish that no new undetectable errors are introduced.
  2. [Application (1)] Application (1) states that the gauge theory 'contains in its description foliated computation' and thereby yields a gauge theory for MBQC. The correspondence between foliated layers and gauge-invariant sectors requires an explicit mapping from the spacetime code stabilizers to the gauge constraints; without this, it is unclear whether the fault-tolerance properties survive the gauging step or require extra commutation conditions on the gauge group.
minor comments (2)
  1. Notation for spacetime errors, equivalence relations, and the gauge group should be introduced with explicit definitions before the applications are discussed.
  2. [Abstract] The abstract's phrasing ('inherits', 'correspond') should be replaced by precise statements once the construction is given, to avoid any appearance of definitional circularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the text to improve clarity and explicitness where needed.

read point-by-point responses

  1. Referee: [Abstract / §2 (construction)] The central claim that gauging maps circuit-level spacetime error equivalence relations directly onto Gauss-law constraints (and detectors onto Wilson loops) without loss or additional assumptions is load-bearing for all three applications. The abstract asserts this inheritance but supplies no explicit construction, error-model definition, or verification that the promotion of global symmetries to local gauge symmetries preserves the original equivalence classes; this must be shown in the main text (e.g., via a concrete spacetime code example) to establish that no new undetectable errors are introduced.

    Authors: We agree that an explicit example would make the central correspondence more transparent. In the revised manuscript we will insert a worked example in §2 using the spacetime repetition code, explicitly defining the error model, showing the promotion of global symmetries to local gauge symmetries, and verifying that the original equivalence classes are preserved with no new undetectable errors introduced. revision: yes

  2. Referee: [Application (1)] Application (1) states that the gauge theory 'contains in its description foliated computation' and thereby yields a gauge theory for MBQC. The correspondence between foliated layers and gauge-invariant sectors requires an explicit mapping from the spacetime code stabilizers to the gauge constraints; without this, it is unclear whether the fault-tolerance properties survive the gauging step or require extra commutation conditions on the gauge group.

    Authors: The mapping from spacetime-code stabilizers to gauge constraints is already given in the foliated-computation section via the layer-by-layer promotion. To address the concern about commutation conditions and survival of fault tolerance, we will add a short paragraph confirming that the gauge group is chosen so that the relevant stabilizers commute with the gauge generators, thereby preserving the original fault-tolerance properties. revision: partial

Circularity Check

0 steps flagged

No circularity identified from provided abstract; full derivation not inspectable

full rationale

The abstract presents gauging the spacetime code as producing a lattice gauge theory where Gauss laws correspond to spacetime error equivalences and Wilson loops to detectors. This is described as inheritance from the circuit's fault tolerance. However, with only the abstract available and no equations, sections, or self-citations quoted from the manuscript, no load-bearing step can be shown to reduce by construction to its inputs. No fitted predictions, self-definitional mappings, or uniqueness theorems are exhibited. The central claim is a construction whose independence cannot be assessed without the full text, so the default non-circular finding applies.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the gauge theory itself may constitute an invented framework, but no details are supplied to classify it.

pith-pipeline@v0.9.1-grok · 5712 in / 1144 out tokens · 28804 ms · 2026-06-28T01:31:42.157719+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

68 extracted references · 43 canonical work pages

  1. [1]

    Improved simulation of stabilizer circuits

    Scott Aaronson and Daniel Gottesman. “Improved simulation of stabilizer circuits”. In:Phys. Rev. A 70 (5 2004), p. 052328.doi:10.1103/PhysRevA.70.052328.url:https://link.aps.org/doi/10. 1103/PhysRevA.70.052328

    work page doi:10.1103/physreva.70.052328.url:https://link.aps.org/doi/10 2004

  2. [2]

    Aitchison and Benjamin Béri.Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits

    Cory T. Aitchison and Benjamin Béri.Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits. 2026. arXiv:2512.21991 [quant-ph].url:https://arxiv.org/abs/2512.21991

    arXiv 2026

  3. [3]

    Panos Aliferis, Daniel Gottesman, and John Preskill.Quantum accuracy threshold for concatenated distance-3 codes. 2005. arXiv:quant-ph/0504218 [quant-ph].url:https://arxiv.org/abs/quant- ph/0504218

    Pith/arXiv arXiv 2005

  4. [4]

    Quantum Data-Syndrome Codes

    Alexei Ashikhmin, Ching-Yi Lai, and Todd A. Brun. “Quantum Data-Syndrome Codes”. In:IEEE Journal on Selected Areas in Communications38.3 (2020), pp. 449–462.doi:10.1109/JSAC.2020. 2968997

    work page doi:10.1109/jsac.2020 2020

  5. [5]

    SparseQuantumCodesfromQuantumCircuits

    DaveBaconetal.“SparseQuantumCodesfromQuantumCircuits”.In:Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing. STOC ’15. Portland, Oregon, USA: Association for Computing Machinery, 2015, 327–334.isbn: 9781450335362.doi:10.1145/2746539.2746608.url: https://doi.org/10.1145/2746539.2746608

    work page doi:10.1145/2746539.2746608.url: 2015

  6. [6]

    Topological error correcting processes from fixed-point path integrals

    Andreas Bauer. “Topological error correcting processes from fixed-point path integrals”. In:Quantum 8 (2024), p. 1288.issn: 2521-327X.doi:10.22331/q-2024-03-20-1288.url:http://dx.doi.org/ 10.22331/q-2024-03-20-1288

    work page doi:10.22331/q-2024-03-20-1288.url:http://dx.doi.org/ 2024

  7. [7]

    On Fault Tolerant Single-Shot Logical State Preparation and RobustLong-RangeEntanglement

    Thiago Bergamaschi and Yunchao Liu. “On Fault Tolerant Single-Shot Logical State Preparation and RobustLong-RangeEntanglement”.en.In:vol.325.SchlossDagstuhl–Leibniz-ZentrumfürInformatik, 2025, 16:1–16:9.doi:10.4230/LIPICS.ITCS.2025.16.url:https://drops.dagstuhl.de/entities/ document/10.4230/LIPIcs.ITCS.2025.16

    work page doi:10.4230/lipics.itcs.2025.16.url:https://drops.dagstuhl.de/entities/ 2025

  8. [8]

    Keller Blackwell and Jeongwan Haah.The code distance of Floquet codes. 2025. arXiv:2510.05549 [quant-ph].url:https://arxiv.org/abs/2510.05549

    arXiv 2025

  9. [9]

    Foliated Quantum Error-Correcting Codes

    A. Bolt et al. “Foliated Quantum Error-Correcting Codes”. In:Phys. Rev. Lett.117 (7 2016), p. 070501. doi:10.1103/PhysRevLett.117.070501.url:https://link.aps.org/doi/10.1103/PhysRevLett. 117.070501

    work page doi:10.1103/physrevlett.117.070501.url:https://link.aps.org/doi/10.1103/physrevlett 2016

  10. [10]

    Unifying flavors of fault tolerance with the ZX calculus

    Hector Bombin et al. “Unifying flavors of fault tolerance with the ZX calculus”. In:Quantum8 (2024), p. 1379.issn: 2521-327X.doi:10.22331/q-2024-06-18-1379.url:http://dx.doi.org/10.22331/ q-2024-06-18-1379

    work page doi:10.22331/q-2024-06-18-1379.url:http://dx.doi.org/10.22331/ 2024

  11. [11]

    S. B. Bravyi and A. Yu. Kitaev.Quantum codes on a lattice with boundary. 1998. arXiv:quant - ph/9811052 [quant-ph].url:https://arxiv.org/abs/quant-ph/9811052

    arXiv 1998

  12. [12]

    Quantum Low-Density Parity-Check Codes

    Nikolas P. Breuckmann and Jens Niklas Eberhardt. “Quantum Low-Density Parity-Check Codes”. In: PRX Quantum2 (4 2021), p. 040101.doi:10.1103/PRXQuantum.2.040101.url:https://link.aps. org/doi/10.1103/PRXQuantum.2.040101

    work page doi:10.1103/prxquantum.2.040101.url:https://link.aps 2021

  13. [13]

    Universal fault-tolerant measurement-based quantum compu- tation

    Benjamin J. Brown and Sam Roberts. “Universal fault-tolerant measurement-based quantum compu- tation”. In:Phys. Rev. Res.2 (3 2020), p. 033305.doi:10.1103/PhysRevResearch.2.033305.url: https://link.aps.org/doi/10.1103/PhysRevResearch.2.033305

    work page doi:10.1103/physrevresearch.2.033305.url: 2020

  14. [15]

    Flag Fault-Tolerant Error Correction for any Stabilizer Code

    Rui Chao and Ben W. Reichardt. “Flag Fault-Tolerant Error Correction for any Stabilizer Code”. In: PRX Quantum1 (1 2020), p. 010302.doi:10.1103/PRXQuantum.1.010302.url:https://link.aps. org/doi/10.1103/PRXQuantum.1.010302

    work page doi:10.1103/prxquantum.1.010302.url:https://link.aps 2020

  15. [16]

    Quantum Error Correction with Only Two Extra Qubits

    Rui Chao and Ben W. Reichardt. “Quantum Error Correction with Only Two Extra Qubits”. In: Phys. Rev. Lett.121 (5 2018), p. 050502.doi:10 . 1103 / PhysRevLett . 121 . 050502.url:https : //link.aps.org/doi/10.1103/PhysRevLett.121.050502

    work page doi:10.1103/physrevlett.121.050502 2018

  16. [17]

    Efficient Self-Consistent Learning of Gate Set Pauli Noise

    Senrui Chen et al. “Efficient Self-Consistent Learning of Gate Set Pauli Noise”. In:PRX Quantum7 (1 2026), p. 010305.doi:10.1103/1pnv-t9px.url:https://link.aps.org/doi/10.1103/1pnv-t9px

    work page doi:10.1103/1pnv-t9px.url:https://link.aps.org/doi/10.1103/1pnv-t9px 2026

  17. [18]

    Classification of phases for mixed states via fast dissipative evolution

    Andrea Coser and David Pérez-García. “Classification of phases for mixed states via fast dissipative evolution”. In:Quantum3 (2019), p. 174.issn: 2521-327X.doi:10.22331/q-2019-08-12-174.url: http://dx.doi.org/10.22331/q-2019-08-12-174. 54

    work page doi:10.22331/q-2019-08-12-174.url: 2019

  18. [19]

    Systematic construction of stabilizer codes via gauging abelian boundary symmetries

    Bram Vancraeynest-De Cuiper and José Garre-Rubio. “Systematic construction of stabilizer codes via gauging abelian boundary symmetries”. In:Quantum9 (2025), p. 1852.issn: 2521-327X.doi: 10.22331/q-2025-09-08-1852.url:http://dx.doi.org/10.22331/q-2025-09-08-1852

    work page doi:10.22331/q-2025-09-08-1852.url:http://dx.doi.org/10.22331/q-2025-09-08-1852 2025

  19. [20]

    Margarita Davydova et al.Universal fault tolerant quantum computation in 2D without getting tied in knots. 2025. arXiv:2503.15751 [quant-ph].url:https://arxiv.org/abs/2503.15751

    arXiv 2025

  20. [21]

    Extended flag gadgets for low-overhead circuit verification

    Dripto M. Debroy and Kenneth R. Brown. “Extended flag gadgets for low-overhead circuit verification”. In:Phys. Rev. A102 (5 2020), p. 052409.doi:10.1103/PhysRevA.102.052409.url:https://link. aps.org/doi/10.1103/PhysRevA.102.052409

    work page doi:10.1103/physreva.102.052409.url:https://link 2020

  21. [22]

    Nicolas Delfosse and Adam Paetznick.Spacetime codes of Clifford circuits. 2023. arXiv:2304.05943 [quant-ph].url:https://arxiv.org/abs/2304.05943

    arXiv 2023

  22. [23]

    Topological quantum memory

    Eric Dennis et al. “Topological quantum memory”. In:Journal of Mathematical Physics43.9 (2002), 4452–4505.issn:1089-7658.doi:10.1063/1.1499754.url:http://dx.doi.org/10.1063/1.1499754

    work page doi:10.1063/1.1499754.url:http://dx.doi.org/10.1063/1.1499754 2002

  23. [24]

    Designing fault-tolerant circuits using detector error models

    Peter-Jan H.S. Derks et al. “Designing fault-tolerant circuits using detector error models”. In:Quantum 9 (2025), p. 1905.issn: 2521-327X.doi:10.22331/q-2025-11-06-1905.url:http://dx.doi.org/ 10.22331/q-2025-11-06-1905

    work page doi:10.22331/q-2025-11-06-1905.url:http://dx.doi.org/ 2025

  24. [25]

    Error Correction in Dynamical Codes

    Esther Xiaozhen Fu and Daniel Gottesman. “Error Correction in Dynamical Codes”. In:Quantum9 (2025), p. 1886.issn: 2521-327X.doi:10.22331/q-2025-10-20-1886.url:http://dx.doi.org/10. 22331/q-2025-10-20-1886

    work page doi:10.22331/q-2025-10-20-1886.url:http://dx.doi.org/10 2025

  25. [26]

    Subsystem spacetime code

    Esther Xiaozhen Fu and Daniel Gottesman. “Subsystem spacetime code”

  26. [27]

    Ability of stabilizer quantum error correction to protect itself from its own im- perfection

    Yuichiro Fujiwara. “Ability of stabilizer quantum error correction to protect itself from its own im- perfection”. In:Phys. Rev. A90 (6 2014), p. 062304.doi:10 . 1103 / PhysRevA . 90 . 062304.url: https://link.aps.org/doi/10.1103/PhysRevA.90.062304

    work page doi:10.1103/physreva.90.062304 2014

  27. [28]

    Stim: a fast stabilizer circuit simulator

    Craig Gidney. “Stim: a fast stabilizer circuit simulator”. In:Quantum5 (2021), p. 497.issn: 2521-327X. doi:10.22331/q-2021-07-06-497.url:http://dx.doi.org/10.22331/q-2021-07-06-497

    work page doi:10.22331/q-2021-07-06-497.url:http://dx.doi.org/10.22331/q-2021-07-06-497 2021

  28. [29]

    Distance measures to compare real and ideal quantum processes

    Alexei Gilchrist, Nathan K. Langford, and Michael A. Nielsen. “Distance measures to compare real and ideal quantum processes”. In:Phys. Rev. A71 (6 2005), p. 062310.doi:10.1103/PhysRevA.71.062310. url:https://link.aps.org/doi/10.1103/PhysRevA.71.062310

    work page doi:10.1103/physreva.71.062310 2005

  29. [30]

    DanielGottesman.Opportunities and Challenges in Fault-Tolerant Quantum Computation.2022.arXiv: 2210.15844 [quant-ph].url:https://arxiv.org/abs/2210.15844

    arXiv 2022

  30. [31]

    Theory of fault-tolerant quantum computation

    Daniel Gottesman. “Theory of fault-tolerant quantum computation”. In:Phys. Rev. A57 (1 1998), pp. 127–137.doi:10 . 1103 / PhysRevA . 57 . 127.url:https : / / link . aps . org / doi / 10 . 1103 / PhysRevA.57.127

    1998

  31. [32]

    Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems

    Jutho Haegeman et al. “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”. In:Phys. Rev. X5 (1 2015), p. 011024.doi:10.1103/PhysRevX.5.011024.url:https: //link.aps.org/doi/10.1103/PhysRevX.5.011024

    work page doi:10.1103/physrevx.5.011024.url:https: 2015

  32. [33]

    Practical Introduction to Benchmarking and Characterization of Quantum Com- puters

    Akel Hashim et al. “Practical Introduction to Benchmarking and Characterization of Quantum Com- puters”. In:PRX Quantum6 (3 2025), p. 030202.doi:10.1103/PRXQuantum.6.030202.url:https: //link.aps.org/doi/10.1103/PRXQuantum.6.030202

    work page doi:10.1103/prxquantum.6.030202.url:https: 2025

  33. [34]

    Dynamically Generated Logical Qubits

    Matthew B. Hastings and Jeongwan Haah. “Dynamically Generated Logical Qubits”. In:Quantum5 (2021), p. 564.issn: 2521-327X.doi:10.22331/q-2021-10-19-564.url:http://dx.doi.org/10. 22331/q-2021-10-19-564

    work page doi:10.22331/q-2021-10-19-564.url:http://dx.doi.org/10 2021

  34. [35]

    Timo Hillmann et al.Single-shot and measurement-based quantum error correction via fault complexes

  35. [36]

    arXiv:2410.12963 [quant-ph].url:https://arxiv.org/abs/2410.12963

    arXiv

  36. [37]

    Fault-tolerant quantum computation by anyons

    A.Yu. Kitaev. “Fault-tolerant quantum computation by anyons”. In:Annals of Physics303.1 (2003), 2–30.issn: 0003-4916.doi:10.1016/s0003-4916(02)00018-0.url:http://dx.doi.org/10.1016/ S0003-4916(02)00018-0

    work page doi:10.1016/s0003-4916(02)00018-0.url:http://dx.doi.org/10.1016/ 2003

  37. [38]

    An introduction to lattice gauge theory and spin systems

    John B. Kogut. “An introduction to lattice gauge theory and spin systems”. In:Rev. Mod. Phys.51 (4 1979), pp. 659–713.doi:10.1103/RevModPhys.51.659.url:https://link.aps.org/doi/10.1103/ RevModPhys.51.659

    work page doi:10.1103/revmodphys.51.659.url:https://link.aps.org/doi/10.1103/ 1979

  38. [39]

    Musaelian, S

    AleksanderKubicaandMichaelVasmer.“Single-shotquantumerrorcorrectionwiththethree-dimensional subsystem toric code”. In:Nature Communications13.1 (2022).issn: 2041-1723.doi:10.1038/s41467- 022-33923-4.url:http://dx.doi.org/10.1038/s41467-022-33923-4

    work page doi:10.1038/s41467- 2022

  39. [40]

    Aleksander Kubica and Beni Yoshida.Ungauging quantum error-correcting codes. 2018. arXiv:1805. 01836 [quant-ph].url:https://arxiv.org/abs/1805.01836. 55

    Pith/arXiv arXiv 2018

  40. [42]

    Ellison, and Timothy H

    Amir-Reza Negari, Tyler D. Ellison, and Timothy H. Hsieh.Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases. 2025. arXiv:2412.00193 [quant-ph].url:https://arxiv. org/abs/2412.00193

    arXiv 2025

  41. [43]

    Generating Fault-Tolerant Cluster States from Crystal Structures

    Michael Newman, Leonardo Andreta de Castro, and Kenneth R. Brown. “Generating Fault-Tolerant Cluster States from Crystal Structures”. In:Quantum4 (2020), p. 295.issn: 2521-327X.doi:10 . 22331/q-2020-07-13-295.url:http://dx.doi.org/10.22331/q-2020-07-13-295

    work page doi:10.22331/q-2020-07-13-295 2020

  42. [44]

    M.A.Nielsen.The entanglement fidelity and quantum error correction.1996.arXiv:quant-ph/9606012 [quant-ph].url:https://arxiv.org/abs/quant-ph/9606012

    Pith/arXiv arXiv 1996

  43. [45]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum computation and quantum information. Cambridge University Press, 2019

    2019

  44. [46]

    url:https://arxiv.org/abs/2509.09603

    ArthurPesahetal.Fault-tolerant transformations of spacetime codes.2025.arXiv:2509.09603 [quant-ph]. url:https://arxiv.org/abs/2509.09603

    arXiv 2025

  45. [47]

    Gauging permutation symmetries as a route to non-Abelian fractons

    Abhinav Prem and Dominic Williamson. “Gauging permutation symmetries as a route to non-Abelian fractons”. In:SciPost Physics7.5 (2019).issn: 2542-4653.doi:10.21468/scipostphys.7.5.068.url: http://dx.doi.org/10.21468/SciPostPhys.7.5.068

    work page doi:10.21468/scipostphys.7.5.068.url: 2019

  46. [48]

    Defining Stable Phases of Open Quantum Systems

    Tibor Rakovszky, Sarang Gopalakrishnan, and Curt von Keyserlingk. “Defining Stable Phases of Open Quantum Systems”. In:Phys. Rev. X14 (4 2024), p. 041031.doi:10.1103/PhysRevX.14.041031. url:https://link.aps.org/doi/10.1103/PhysRevX.14.041031

    work page doi:10.1103/physrevx.14.041031 2024

  47. [49]

    Gauging and dualities

    Tibor Rakovszky and Vedika Khemani.The Physics of (good) LDPC Codes I. Gauging and dualities

  48. [50]

    arXiv:2310.16032 [quant-ph].url:https://arxiv.org/abs/2310.16032

    arXiv

  49. [51]

    A fault-tolerant one-way quantum computer

    R. Raussendorf, J. Harrington, and K. Goyal. “A fault-tolerant one-way quantum computer”. In:Annals of Physics321.9 (2006), 2242–2270.issn: 0003-4916.doi:10.1016/j.aop.2006.01.012.url:http: //dx.doi.org/10.1016/j.aop.2006.01.012

    work page doi:10.1016/j.aop.2006.01.012.url:http: 2006

  50. [52]

    Topological fault-tolerance in cluster state quantum computation

    R Raussendorf, J Harrington, and K Goyal. “Topological fault-tolerance in cluster state quantum computation”. In:New Journal of Physics9.6 (2007), 199–199.issn: 1367-2630.doi:10.1088/1367- 2630/9/6/199.url:http://dx.doi.org/10.1088/1367-2630/9/6/199

    work page doi:10.1088/1367- 2007

  51. [53]

    Long-range quantum entanglement in noisy cluster states

    Robert Raussendorf, Sergey Bravyi, and Jim Harrington. “Long-range quantum entanglement in noisy cluster states”. In:Physical Review A71.6 (2005).issn: 1094-1622.doi:10.1103/physreva.71.062313. url:http://dx.doi.org/10.1103/PhysRevA.71.062313

    work page doi:10.1103/physreva.71.062313 2005

  52. [54]

    Fault-Tolerant Quantum Computation with High Thresh- old in Two Dimensions

    Robert Raussendorf and Jim Harrington. “Fault-Tolerant Quantum Computation with High Thresh- old in Two Dimensions”. In:Physical Review Letters98.19 (2007).issn: 1079-7114.doi:10 . 1103 / physrevlett.98.190504.url:http://dx.doi.org/10.1103/PhysRevLett.98.190504

    work page doi:10.1103/physrevlett.98.190504 2007

  53. [55]

    Benjamin Rodatz, Boldizsár Poór, and Aleks Kissinger.Fault Tolerance by Construction. 2026. arXiv: 2506.17181 [quant-ph].url:https://arxiv.org/abs/2506.17181

    arXiv 2026

  54. [56]

    Stability of Mixed-State Quantum Phases via Finite Markov Length

    Shengqi Sang and Timothy H. Hsieh. “Stability of Mixed-State Quantum Phases via Finite Markov Length”. In:Physical Review Letters134.7 (2025).issn: 1079-7114.doi:10.1103/physrevlett.134. 070403.url:http://dx.doi.org/10.1103/PhysRevLett.134.070403

    work page doi:10.1103/physrevlett.134 2025

  55. [57]

    Noisy Approach to Intrinsically Mixed-State Topological Order

    Ramanjit Sohal and Abhinav Prem. “Noisy Approach to Intrinsically Mixed-State Topological Order”. In:PRX Quantum6 (1 2025), p. 010313.doi:10.1103/PRXQuantum.6.010313.url:https://link. aps.org/doi/10.1103/PRXQuantum.6.010313

    work page doi:10.1103/prxquantum.6.010313.url:https://link 2025

  56. [58]

    Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases

    Nathanan Tantivasadakarn et al. “Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases”. In:Physical Review X14.2 (2024).issn: 2160-3308.doi:10.1103/physrevx.14. 021040.url:http://dx.doi.org/10.1103/PhysRevX.14.021040

    work page doi:10.1103/physrevx.14 2024

  57. [59]

    Quantum error correction for quantum memories

    Barbara M. Terhal. “Quantum error correction for quantum memories”. In:Rev. Mod. Phys.87 (2 2015), pp. 307–346.doi:10.1103/RevModPhys.87.307.url:https://link.aps.org/doi/10.1103/ RevModPhys.87.307

    work page doi:10.1103/revmodphys.87.307.url:https://link.aps.org/doi/10.1103/ 2015

  58. [60]

    Noise tailoring for scalable quantum computation via random- ized compiling

    Joel J. Wallman and Joseph Emerson. “Noise tailoring for scalable quantum computation via random- ized compiling”. In:Phys. Rev. A94 (5 2016), p. 052325.doi:10.1103/PhysRevA.94.052325.url: https://link.aps.org/doi/10.1103/PhysRevA.94.052325

    work page doi:10.1103/physreva.94.052325.url: 2016

  59. [61]

    Oxford University Press, 2017

    Xiao-Gang Wen.Quantum Field Theory of many-body systems from the origin of sound to an origin of light and electrons Xiao-Gang Wen. Oxford University Press, 2017. 56

    2017

  60. [62]

    John van de Wetering.ZX-calculus for the working quantum computer scientist. 2020. arXiv:2012. 13966 [quant-ph].url:https://arxiv.org/abs/2012.13966

    arXiv 2020

  61. [63]

    Williamson and Theodore J

    Dominic J. Williamson and Theodore J. Yoder.Low-overhead fault-tolerant quantum computation by gauging logical operators. 2024. arXiv:2410.02213 [quant-ph].url:https://arxiv.org/abs/2410. 02213

    Pith/arXiv arXiv 2024

  62. [64]

    Sparse blossom: correcting a million errors per core second with minimum-weight matching,

    Gabriel Wong, Robert Raussendorf, and Bartlomiej Czech. “The Gauge Theory of Measurement-Based Quantum Computation”. In:Quantum8 (2024), p. 1397.issn: 2521-327X.doi:10.22331/q- 2024- 07-04-1397.url:http://dx.doi.org/10.22331/q-2024-07-04-1397

    work page doi:10.22331/q- 2024

  63. [65]

    Xiao Xiao et al.In-situ benchmarking of fault-tolerant quantum circuits. I. Clifford circuits. 2026. arXiv: 2601.21472 [quant-ph].url:https://arxiv.org/abs/2601.21472

    arXiv 2026

  64. [66]

    BeiZengetal.Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topolog- ical Phases of Many-Body Systems. 1st ed. Quantum Science and Technology. New York, NY: Springer, 2019.isbn: 978-1-4939-9084-9.doi:10.1007/978-1-4939-9084-9

    work page doi:10.1007/978-1-4939-9084-9 2019

  65. [67]

    Shuhan Zhang, Deepak Aryal, and Yi-Zhuang You.Quantum Process Realization of LDPC Code Du- alities and Product Constructions. 2026. arXiv:2603.13538 [quant-ph].url:https://arxiv.org/ abs/2603.13538

    arXiv 2026

  66. [68]

    Generalized Cycle Benchmarking Algorithm for Characterizing Midcircuit Mea- surements

    Zhihan Zhang et al. “Generalized Cycle Benchmarking Algorithm for Characterizing Midcircuit Mea- surements”. In:PRX Quantum6 (1 2025), p. 010310.doi:10 . 1103 / PRXQuantum . 6 . 010310.url: https://link.aps.org/doi/10.1103/PRXQuantum.6.010310

    work page doi:10.1103/prxquantum.6.010310 2025

  67. [69]

    Near-Optimal Performance of Quantum Error Correction Codes

    Guo Zheng et al. “Near-Optimal Performance of Quantum Error Correction Codes”. In:Phys. Rev. Lett.132 (25 2024), p. 250602.doi:10.1103/PhysRevLett.132.250602.url:https://link.aps. org/doi/10.1103/PhysRevLett.132.250602

    work page doi:10.1103/physrevlett.132.250602.url:https://link.aps 2024

  68. [70]

    Han Zheng et al.Efficient learning of logical noise from syndrome data. 2026. arXiv:2601 . 22286 [quant-ph].url:https://arxiv.org/abs/2601.22286. 57

    arXiv 2026