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Cross-code lattice surgery joins two error-correcting codes so that a universal logical gate set can prepare and certify genuine multipartite entanglement among logical qubits.

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2026-07-11 20:47 UTC pith:E4WOBEH5

load-bearing objection Solid first experiment of lattice surgery between complementary universal codes; GME and magic-state claims hold, with the usual d=2 post-selection caveats the authors already flag.

arxiv 2607.04227 v1 pith:E4WOBEH5 submitted 2026-07-05 quant-ph

Genuine Multipartite Entanglement between Logical Qubits via Cross-Code Lattice Surgery

Alex Steiner , Tomasz Andrzejewski , Phila Rembold , Hendrik Poulsen Nautrup , Christian D. Marciniak , Robert Freund , Ivan Pogorelov , Thomas Monz
show 3 more authors
Philipp Schindler Marcel Meyer Nicolai Friis
This is my paper
classification quant-ph
keywords lattice surgerygenuine multipartite entanglementlogical qubitssurface code3D colour codeCCZ gatetrapped-ion quantum processorfault-tolerant quantum computation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

No single quantum error-correcting code can implement a full universal set of logical gates transversally. This experiment joins a four-qubit surface code (which supplies a transversal Hadamard) with an eight-qubit 3D colour code (which supplies a transversal doubly-controlled-Z) by a single joint parity measurement called a smooth merge. On a trapped-ion processor the resulting twelve-physical-qubit merged code is used to prepare both a three-logical-qubit GHZ state and the non-stabiliser resource |CCZ>, each of which is shown by fidelity and stabiliser-norm witnesses to be genuinely multipartite entangled. The same primitives then drive a constant-depth gadget that realises arbitrary single-logical-qubit Z-rotations. The work therefore demonstrates that complementary codes can be surgically combined to generate complex logical states that neither code could produce alone.

Core claim

Lattice surgery that merges a [[4,2,2]] surface code with an [[8,3,2]] 3D colour code realises a transversally implemented universal logical gate set {H, CCZ}. With that set the authors prepare three-logical-qubit GHZ and |CCZ> states whose measured fidelities exceed the respective genuine-multipartite-entanglement thresholds (0.5 and 0.75) and whose stabiliser norm certifies non-stabiliserness of |CCZ>, while the same primitives implement arbitrary logical RZ(θ) rotations.

What carries the argument

Smooth merge: a single weight-four joint parity measurement MZZ = Z_SC ⊗ Z_CC that joins the two codes into a [[12,4,2]] merged code, transferring logical information so that the transversal gates of each code become available on the joint system.

Load-bearing premise

That distance-two, error-detecting codes plus post-selection on flags and merge outcomes already constitute the core building blocks of a scalable fault-tolerant architecture, even though the specific transversal CCZ does not itself belong to a distance-growing family.

What would settle it

A multi-cycle experiment that repeatedly merges and splits the same pair of codes and shows that the accumulated logical error on the resulting GHZ or |CCZ> states falls below the GME thresholds once distance is increased or flags are removed would refute the claim that the present primitives are already the building blocks of a fault-tolerant architecture.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The manuscript reports the first experimental demonstration of lattice surgery between two complementary codes—the [[4,2,2]] surface code and the [[8,3,2]] 3D colour code—on a trapped-ion processor, thereby accessing a transversally implemented universal logical gate set {H, CCZ}. Using the resulting [[12,4,2]] merged code, the authors prepare three-logical-qubit GHZ and non-stabiliser |CCZ angle states, certify genuine multipartite entanglement via fidelity witnesses that exceed the Schmidt-coefficient thresholds (F_GHZ > 0.5 by many σ; F_CCZ > 0.75 by ~1σ with flags), and certify non-stabiliserness of |CCZ angle via the stabiliser norm D = 1.396(15). The same primitives are used to implement heralded arbitrary logical RZ(θ) rotations with average fidelity 83.4(3)%. Circuits, flag analysis, Pauli decompositions, and Pauli-frame handling are documented in detail, with depolarising simulations that largely match experiment.

Significance. If the experimental claims hold, the work supplies a concrete, hardware-validated primitive for joining codes with complementary transversal gates and for generating both stabiliser and non-stabiliser multipartite entanglement at the logical level. The complete 7- and 29-term Pauli decompositions, flag-based fault-tolerance analysis, and explicit resource counts constitute a reproducible experimental benchmark that goes beyond prior same-code or Clifford-only lattice-surgery demonstrations. The architectural framing is appropriately caveated by the authors themselves (the transversal CCZ is not scalable), so the result remains a solid near-term milestone rather than an overstated architecture claim.

minor comments (4)
  1. Abstract and Sec. V: the phrase “core building blocks of an architecture for fault-tolerant quantum computation” is slightly stronger than the distance-2, post-selected primitives warrant; a single-sentence softening that already appears in A.I.5 would improve precision without changing the claim.
  2. Fig. 2(c) and A.V.2: the ~6–7% discrepancy between experimental and simulated F_CCZ is attributed to the simplified noise model; a brief quantitative remark on which unmodelled channels (idling, coherent T-gate errors) are the leading candidates would help the reader.
  3. A.I.4: the post-selection procedure used for the CCZ protocol (because T/T† prevent Pauli-frame tracking) halves the acceptance rate; stating the raw shot counts before and after this cut would make the statistics fully transparent.
  4. Notation: logical operators are sometimes written with overlines and sometimes with (SC)/(CC) superscripts; a short consistency note early in Sec. II would reduce minor reader friction.

Circularity Check

0 steps flagged

No circularity: experimental fidelities and witnesses are direct measurements against independent theoretical thresholds; noise parameters are used only for comparison simulations.

full rationale

The paper's central claims are experimental: preparation of logical |GHZ> and |CCZ> on a merged [[12,4,2]] code via cross-code lattice surgery, certification of GME by measured fidelities exceeding the known Schmidt-coefficient bounds (F_GHZ > 0.5, F_CCZ > 3/4), non-stabiliserness by stabiliser norm D > 1, and heralded logical RZ( heta) with average fidelity ~83%. Fidelity estimators (Eq. 3 for GHZ; 29-term Pauli decomposition in A.II for |CCZ>) and thresholds follow from standard multipartite-entanglement theory and the explicit form of the target states; they are not fitted or redefined from the data. Depolarising rates p1=0.005, p2=0.015 are taken from prior hardware characterisation of the same trap and appear only in numerical simulations for comparison, never as free parameters that define the reported experimental numbers. Self-citations (prior lattice-surgery and code-switching experiments by overlapping authors) supply experimental context and techniques but are not load-bearing for the new measurements or the GME/non-stabiliserness witnesses. The acknowledged non-scalability of the transversal CCZ is an architectural caveat, not a circular step. The derivation chain is therefore self-contained against external benchmarks and free of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The work is an experimental demonstration resting on standard stabiliser-code theory, the Eastin-Knill theorem, known transversal gates of the two small codes, and established GME/non-stabiliserness witnesses. No new physical entities or free parameters are introduced into the central claims; the only numerical inputs are hardware error rates used solely for simulation comparison.

free parameters (2)
  • single-qubit depolarising probability p1 = 0.005
    Set to 0.005 from prior hardware characterisation; used only in numerical simulations, not in experimental fidelity extraction.
  • two-qubit depolarising probability p2 = 0.015
    Set to 0.015 from prior hardware characterisation; used only in numerical simulations.
axioms (5)
  • domain assumption Eastin-Knill theorem: no single QEC code admits a transversal universal gate set
    Invoked in Introduction and Sec. II to motivate cross-code lattice surgery.
  • standard math Fidelity of any biseparable state with a pure target is at most λ_max^{2} of the target across bipartitions
    Used in Sec. IV to set GME thresholds 0.5 (GHZ) and 0.75 (|CCZ|).
  • domain assumption Stabiliser norm D > 1 witnesses non-stabiliserness (magic)
    Applied in Sec. IV to certify |CCZ> as a magic state; standard resource-theory result.
  • domain assumption [[4,2,2]] admits transversal H⊗H (up to SWAP) and [[8,3,2]] admits transversal CCZ and CZ
    Taken from the hypercube/colour-code literature and verified by explicit operator conjugation in Appendix A.I.
  • domain assumption Smooth merge by measuring Z_SC ⊗ Z_CC realises a fault-tolerant interface that preserves the transversal gates of both blocks
    Core lattice-surgery construction (Sec. II and A.I.3–A.I.5); transversality after merge is shown by commutation arguments.

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Universal quantum computers are expected to generate arbitrary complex quantum states of logical qubits encoded in many physical qubits. This capability hinges on a fault-tolerantly implemented universal gate set, which no single quantum error-correction code admits transversally but which becomes accessible by joining complementary codes via lattice surgery. Here we report on the experimental generation and certification of logical genuine multipartite entanglement in a trapped-ion quantum processor using a transversally implemented universal logical gate set. The gate set is accessed via lattice surgery across two different codes and comprises a Hadamard gate on a four-qubit surface code and a doubly controlled Pauli-$Z$ ($\overline{\mathrm{CCZ}}$) gate on an eight-qubit 3D colour code. To showcase this lattice-surgery toolbox, we generate both stabiliser (Greenberger-Horne-Zeilinger) and non-stabiliser ($|\overline{\mathrm{CCZ}}\rangle$) states of three logical qubits and verify their genuine multipartite entanglement--a form of correlation beyond statistical mixtures of bipartite entanglement across any bipartition. We further use these cross-code primitives to demonstrate arbitrary rotations of single logical qubits via a $\overline{\mathrm{CCZ}}$-based resource gadget accessing the full universal gate set through lattice surgery. Together, these demonstrations showcase the core building blocks of an architecture for fault-tolerant quantum computation and its ability to generate complex logical quantum states.

Figures

Figures reproduced from arXiv: 2607.04227 by Alex Steiner, Christian D. Marciniak, Hendrik Poulsen Nautrup, Ivan Pogorelov, Marcel Meyer, Nicolai Friis, Phila Rembold, Philipp Schindler, Robert Freund, Thomas Monz, Tomasz Andrzejewski.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Reference graph

Works this paper leans on

88 extracted references · 67 linked inside Pith

  1. [1]

    Richard Jozsa and Noah Linden,On the role of entangle- ment in quantum-computational speed-up,Proc. R. Soc. A459, 2011–2032 (2003), arXiv:quant-ph/0201143

    Pith/arXiv arXiv 2011

  2. [2]

    Lo, Sandu Popescu, and T

    John Preskill,Fault-tolerant quantum computation,in Introduction to Quantum Computation, edited by H.-K. Lo, Sandu Popescu, and T. P. Spiller (World-Scientific,

  3. [3]

    Chap. 8, p. 213–269, arXiv:quant-ph/9712048

    Pith/arXiv arXiv

  4. [4]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang,Quantum Com- putation and Quantum Information, 10th Anniversary Edition ed. (Cambridge University Press, Cambridge, U.K., 2010)

    2010

  5. [5]

    Lidar and Todd A

    Daniel A. Lidar and Todd A. Brun,Quantum Error Cor- rection(Cambridge University Press, Cambridge, 2013)

    2013

  6. [6]

    Bryan Eastin and Emanuel Knill,Restrictions on Transversal Encoded Quantum Gate Sets,Phys. Rev. Lett.102, 110502 (2009), arXiv:0811.4262

    Pith/arXiv arXiv 2009

  7. [7]

    Sergey Bravyi and Alexei Kitaev,Universal Quantum Computation with Ideal Clifford Gates and Noisy An- cillas,Phys. Rev. A71, 022316 (2005), arXiv:quant- ph/0403025

    arXiv 2005

  8. [8]

    Beverland, Aleksander Kubica, and Krysta M

    Michael E. Beverland, Aleksander Kubica, and Krysta M. Svore,Cost of Universality: A Comparative Study of the Overhead of State Distillation and Code Switching with Color Codes,PRX Quantum2, 020341 (2021), arXiv:2101.02211

    Pith/arXiv arXiv 2021

  9. [9]

    Craig Gidney, Noah Shutty, and Cody Jones,Magic State Cultivation: Growing T States as Cheap as CNOT Gates,arXiv:2409.17595 [quant-ph] (2024)

    Pith/arXiv arXiv 2024

  10. [10]

    Phys.18, 043038 (2016), arXiv:1412.5079

    H´ ector Bomb´ ın,Dimensional Jump in Quantum Er- ror Correction,New J. Phys.18, 043038 (2016), arXiv:1412.5079

    Pith/arXiv arXiv 2016

  11. [11]

    Friederike Butt, Sascha Heußen, Manuel Rispler, and Markus M¨ uller,Fault-Tolerant Code Switching Protocols for Near-Term Quantum Processors,PRX Quantum5, 020345 (2024), arXiv:2306.17686

    Pith/arXiv arXiv 2024

  12. [12]

    Marciniak, Philipp Schindler, Markus M¨ uller, and Thomas Monz,Experimental Fault- Tolerant Code Switching,Nat

    Ivan Pogorelov, Friederike Butt, Lukas Postler, Christian D. Marciniak, Philipp Schindler, Markus M¨ uller, and Thomas Monz,Experimental Fault- Tolerant Code Switching,Nat. Phys.21, 298–303 (2025), arXiv:2403.13732

    Pith/arXiv arXiv 2025

  13. [13]

    Brun, and Daniel A

    Shixin Wu, Dawei Zhong, Todd A. Brun, and Daniel A. Lidar,Universal Weakly Fault-Tolerant Quantum Com- putation via Code Switching in the[[8,3,2]]Code, arXiv:2603.15610 [quant-ph] (2026)

    Pith/arXiv arXiv 2026

  14. [14]

    Fowler, Simon Devitt, and Rodney Van Meter,Surface Code Quantum Computing by Lattice Surgery,New J

    Clare Horsman, Austin G. Fowler, Simon Devitt, and Rodney Van Meter,Surface Code Quantum Computing by Lattice Surgery,New J. Phys.14, 123011 (2012), arXiv:1111.4022

    Pith/arXiv arXiv 2012

  15. [15]

    Briegel, Rainer Blatt, Nicolai Friis, and Thomas Monz,Entangling logical qubits with lattice surgery,Na- ture589, 220–224 (2021), arXiv:2006.03071

    Alexander Erhard, Hendrik Poulsen Nautrup, Michael Meth, Lukas Postler, Roman Stricker, Martin Stadler, Vlad Negnevitsky, Martin Ringbauer, Philipp Schindler, Hans J. Briegel, Rainer Blatt, Nicolai Friis, and Thomas Monz,Entangling logical qubits with lattice surgery,Na- ture589, 220–224 (2021), arXiv:2006.03071

    Pith/arXiv arXiv 2021

  16. [16]

    Briegel,Fault-Tolerant Interface Between Quantum Memories and Quantum Processors,Nat

    Hendrik Poulsen Nautrup, Nicolai Friis, and Hans J. Briegel,Fault-Tolerant Interface Between Quantum Memories and Quantum Processors,Nat. Commun.8, 1321 (2017), arXiv:1609.08062

    Pith/arXiv arXiv 2017

  17. [17]

    Daniel Litinski,A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery,Quantum3, 128 (2019), arXiv:1808.02892

    Pith/arXiv arXiv 2019

  18. [18]

    G´ eza T´ oth and Otfried G¨ uhne,Detecting Genuine Multipartite Entanglement with Two Local Measure- ments,Phys. Rev. Lett.94, 060501 (2005), arXiv:quant- ph/0405165

    arXiv 2005

  19. [19]

    Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber,Entanglement Certification From The- ory to Experiment,Nat. Rev. Phys.1, 72–87 (2019), arXiv:1906.10929

    Pith/arXiv arXiv 2019

  20. [20]

    Bertlmann and Nicolai Friis,Modern Quan- tum Theory – From Quantum Mechanics to Entan- glement and Quantum Information(Oxford University Press, Oxford, U.K., 2023) Chapter 18

    Reinhold A. Bertlmann and Nicolai Friis,Modern Quan- tum Theory – From Quantum Mechanics to Entan- glement and Quantum Information(Oxford University Press, Oxford, U.K., 2023) Chapter 18

    2023

  21. [21]

    Phys.19, 093012 (2017), arXiv:1612.05585

    Michael Epping, Hermann Kampermann, Chiara Mac- chiavello, and Dagmar Bruß,Multi-partite entanglement can speed up quantum key distribution in networks,New J. Phys.19, 093012 (2017), arXiv:1612.05585

    Pith/arXiv arXiv 2017

  22. [22]

    Stefan B¨ auml and Koji Azuma,Fundamental limitation on quantum broadcast networks,Quantum Sci. Technol. 2, 024004 (2017), arXiv:1609.03994

    Pith/arXiv arXiv 2017

  23. [23]

    Matej Pivoluska, Marcus Huber, and Mehul Malik,Lay- ered quantum key distribution,Phys. Rev. A97, 032312 (2018), arXiv:1709.00377

    Pith/arXiv arXiv 2018

  24. [24]

    J´ er´ emy Ribeiro, Gl´ aucia Murta, and Stephanie Wehner,Fully device-independent conference key agree- ment,Phys. Rev. A97, 022307 (2018), arXiv:1708.00798

    Pith/arXiv arXiv 2018

  25. [25]

    Hayata Yamasaki, Alexander Pirker, Mio Murao, Wolf- gang D¨ ur, and Barbara Kraus,Multipartite entangle- ment outperforming bipartite entanglement under limited quantum system sizes,Phys. Rev. A98, 052313 (2018), arXiv:1808.00005

    Pith/arXiv arXiv 2018

  26. [26]

    Briegel,A One-Way Quantum Computer,Phys

    Robert Raussendorf and Hans J. Briegel,A One-Way Quantum Computer,Phys. Rev. Lett.86, 5188–5191 (2001), arXiv:quant-ph/0010033

    Pith/arXiv arXiv 2001

  27. [27]

    Briegel and Robert Raussendorf,Persistent En- tanglement in Arrays of Interacting Particles,Phys

    Hans J. Briegel and Robert Raussendorf,Persistent En- tanglement in Arrays of Interacting Particles,Phys. Rev. Lett.86, 910–913 (2001), arXiv:quant-ph/0004051

    Pith/arXiv arXiv 2001

  28. [28]

    Scott,Multipartite entanglement, quantum- error-correcting codes, and entangling power of quantum evolutions,Phys

    Andrew J. Scott,Multipartite entanglement, quantum- error-correcting codes, and entangling power of quantum evolutions,Phys. Rev. A69, 052330 (2004), arXiv:quant- ph/0310137

    arXiv 2004

  29. [29]

    Dagmar Bruß and Chiara Macchiavello,Multipartite en- tanglement in quantum algorithms,Phys. Rev. A83, 052313 (2011), arXiv:1007.4179

    Pith/arXiv arXiv 2011

  30. [30]

    Plenio, Marcus Huber, Christian Roos, Rainer Blatt, and Ben P

    Nicolai Friis, Oliver Marty, Christine Maier, Cornelius Hempel, Milan Holz¨ apfel, Petar Jurcevic, Martin B. Plenio, Marcus Huber, Christian Roos, Rainer Blatt, and Ben P. Lanyon,Observation of Entangled States of a Fully Controlled 20-Qubit System,Phys. Rev. X8, 021012 (2018), arXiv:1711.11092

    Pith/arXiv arXiv 2018

  31. [31]

    Lanyon,Generation of Multipartite Photonic Entangle- ment Using a Trapped-Ion Quantum Processing Node, arXiv:2510.15693 [quant-ph] (2025)

    Marco Canteri, James Bate, Ida Mishra, Nico- lai Friis, Victor Krutyanskiy, and Benjamin P. Lanyon,Generation of Multipartite Photonic Entangle- ment Using a Trapped-Ion Quantum Processing Node, arXiv:2510.15693 [quant-ph] (2025)

    arXiv 2025

  32. [32]

    Adrian Rodriguez-Blanco, Alejandro Bermudez, Markus M¨ uller, and Farid Shahandeh,Efficient and Robust Cer- tification of Genuine Multipartite Entanglement in Noisy Quantum Error Correction Circuits,PRX Quantum2, 020304 (2021), arXiv:2010.02941

    Pith/arXiv arXiv 2021

  33. [33]

    Victor Veitch, S. A. Hamed Mousavian, Daniel Gottes- man, and Joseph Emerson,The resource theory of sta- 8 bilizer quantum computation,New J. Phys.16, 013009 (2014), arXiv:1307.7171

    Pith/arXiv arXiv 2014

  34. [34]

    Su-un Lee, Ming Yuan, Senrui Chen, Kento Tsub- ouchi, and Liang Jiang,Efficient Benchmarking of Log- ical Magic State,Phys. Rev. Lett.136, 050602 (2026), arXiv:2505.09687

    arXiv 2026

  35. [35]

    Brown, David Hayes, and Isaac H

    Lucas Daguerre, Robin Blume-Kohout, Natalie C. Brown, David Hayes, and Isaac H. Kim,Experimen- tal Demonstration of High-Fidelity Logical Magic States from Code Switching,Phys. Rev. X15, 041008 (2025), arXiv:2506.14169

    arXiv 2025

  36. [36]

    Dorit Aharonov,A Simple Proof that Toffoli and Hadamard are Quantum Universal,arXiv:quant- ph/0301040 [quant-ph] (2003)

    arXiv 2003

  37. [37]

    Fiderer, Hendrik Poulsen Nautrup, Jonas M

    Sofiene Jerbi, Lukas J. Fiderer, Hendrik Poulsen Nautrup, Jonas M. K¨ ubler, Hans J. Briegel, and Vedran Dunjko,Quantum Machine Learning Be- yond Kernel Methods,Nat. Commun.14, 517 (2023), arXiv:2110.13162

    Pith/arXiv arXiv 2023

  38. [38]

    Chuang,Demonstrating the Viability of Universal Quantum Computation Using Teleportation and Single-Qubit Operations,Nature402, 390–393 (1999), arXiv:quant-ph/9908010

    Daniel Gottesman and Isaac L. Chuang,Demonstrating the Viability of Universal Quantum Computation Using Teleportation and Single-Qubit Operations,Nature402, 390–393 (1999), arXiv:quant-ph/9908010

    Pith/arXiv arXiv 1999

  39. [39]

    Leung, and Isaac L

    Xinlan Zhou, Debbie W. Leung, and Isaac L. Chuang,Methodology for Quantum Logic Gate Construc- tion,Phys. Rev. A62, 052316 (2000), arXiv:quant- ph/0002039

    arXiv 2000

  40. [40]

    Yifan Hong, Elijah Durso-Sabina, David Hayes, and Andrew Lucas,Entangling Four Logical Qubits beyond Break-Even in a Nonlocal Code,Phys. Rev. Lett.133, 180601 (2024), arXiv:2406.02666

    Pith/arXiv arXiv 2024

  41. [41]

    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 Vuletic, and Mikhail D. Lukin,...

    Pith/arXiv arXiv 2024

  42. [42]

    Phys.22, 189–194 (2026), arXiv:2501.04612

    Ilya Besedin, Michael Kerschbaum, Jonathan Knoll, Ian Hesner, Lukas B¨ odeker, Luis Colmenarez, Luca Hofele, Nathan Lacroix, Christoph Hellings, Fran¸ cois Swiadek, Alexander Flasby, Mohsen Bahrami Panah, Dante Co- lao Zanuz, Markus M¨ uller, and Andreas Wallraff,Lat- tice surgery realized on two distance-three repetition codes with superconducting qubits...

    arXiv 2026

  43. [43]

    Wang, and Ying Li,A superconducting surface-code processor with lattice- surgery logical operations,arXiv:2606.06598 [quant-ph] (2026)

    Yanzhe Wang, Fanhao Shen, Haipeng Xie, Aosai Zhang, Yu Gao, Chuanyu Zhang, Xuhao Zhu, Feitong Jin, Yiren Zou, Ning Wang, Zhengyi Cui, Zehang Bao, Zitian Zhu, Jiarun Zhong, Gongyu Liu, Jia-Nan Yang, Yihang Han, Yiyang He, Jiayuan Shen, Han Wang, Jiahua Huang, Xinrong Zhang, Sailang Zhou, Hang Dong, Jinfeng Deng, Yaozu Wu, Zixuan Song, Hekang Li, Zhen Wang,...

    Pith/arXiv arXiv 2026

  44. [44]

    Wootton,Creating En- tangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes,PRX Quantum5, 040334 (2024), arXiv:2404.15989

    Bence Het´ enyi and James R. Wootton,Creating En- tangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes,PRX Quantum5, 040334 (2024), arXiv:2404.15989

    Pith/arXiv arXiv 2024

  45. [45]

    Ryan-Anderson, N

    C. Ryan-Anderson, N. C. Brown, M. S. Allman, B. Arkin, G. Asa-Attuah, C. Baldwin, J. Berg, J. G. Bohnet, S. Braxton, N. Burdick, J. P. Campora, A. Chernoguzov, J. Esposito, B. Evans, D. Francois, J. P. Gaebler, T. M. Gatterman, J. Gerber, K. Gilmore, D. Gresh, A. Hall, A. Hankin, J. Hostetter, D. Lucchetti, K. Mayer, J. Myers, B. Neyenhuis, J. Santiago, J...

    Pith/arXiv arXiv 2022

  46. [46]

    Nelson, Andrew J

    Jacob S. Nelson, Andrew J. Landahl, and An- drew D. Baczewski,A Small and Interesting Archi- tecture for Early Fault-Tolerant Quantum Computers, arXiv:2507.20387 [quant-ph] (2025)

    Pith/arXiv arXiv 2025

  47. [47]

    Geim, Sophie H

    Dolev Bluvstein, Alexandra A. Geim, Sophie H. Li, Simon J. Evered, J. Pablo Bonilla Ataides, Gefen Baranes, Andi Gu, Tom Manovitz, Muqing Xu, Marcin Kalinowski, Shayan Majidy, Christian Kokail, Nishad Maskara, Elias C. Trapp, Luke M. Stewart, Simon Hol- lerith, Hengyun Zhou, Michael J. Gullans, Susanne F. Yelin, Markus Greiner, Vladan Vuletic, Madelyn Cai...

    arXiv 2026

  48. [48]

    thesis, Caltech (1997), arXiv:quant- ph/9705052

    Daniel Gottesman,Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, Caltech (1997), arXiv:quant- ph/9705052

    arXiv 1997

  49. [49]

    Phys.303, 2–30 (2003), arXiv:quant- ph/9707021

    Alexei Kitaev,Fault-tolerant quantum computation by anyons,Ann. Phys.303, 2–30 (2003), arXiv:quant- ph/9707021

    arXiv 2003

  50. [50]

    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill,Topological quantum memory,J. Math. Phys. 43, 4452–4505 (2002), arXiv:quant-ph/0110143

    Pith/arXiv arXiv 2002

  51. [51]

    Fowler, Matteo Mariantoni, John M

    Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland,Surface codes: Towards practi- cal large-scale quantum computation,Phys. Rev. A86, 032324 (2012), arXiv:1208.0928

    Pith/arXiv arXiv 2012

  52. [52]

    Robert Raussendorf and Jim Harrington,Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions,Phys. Rev. Lett.98, 190504 (2007), arXiv:quant-ph/0610082

    Pith/arXiv arXiv 2007

  53. [53]

    Phys.17, 083026 (2015), arXiv:1503.02065

    Aleksander Kubica, Beni Yoshida, and Fernando Pastawski,Unfolding the color code,New J. Phys.17, 083026 (2015), arXiv:1503.02065

    Pith/arXiv arXiv 2015

  54. [54]

    Campbell and Mark Howard,Unifying Gate Syn- thesis and Magic State Distillation,Phys

    Earl T. Campbell and Mark Howard,Unifying Gate Syn- thesis and Magic State Distillation,Phys. Rev. Lett.118, 060501 (2017), arXiv:1606.01906

    Pith/arXiv arXiv 2017

  55. [55]

    Campbell and Mark Howard,Unified framework for magic state distillation and multiqubit gate synthe- sis with reduced resource cost,Phys

    Earl T. Campbell and Mark Howard,Unified framework for magic state distillation and multiqubit gate synthe- sis with reduced resource cost,Phys. Rev. A95, 022316 (2017), arXiv:1606.01904

    Pith/arXiv arXiv 2017

  56. [56]

    Daniel Honciuc Menendez, Annie Ray, and Michael Vas- mer,Implementing fault-tolerant non-Clifford gates using the[[8,3,2]]color code,Phys. Rev. A109, 062438 (2024), arXiv:2309.08663 [quant-ph]

    Pith/arXiv arXiv 2024

  57. [57]

    Ivan Pogorelov, Thomas Feldker, Christian D. Marciniak, Lukas Postler, Georg Jacob, Oliver Krieglsteiner, Verena Podlesnic, Michael Meth, Vlad Negnevitsky, Martin Stadler, Bernd H¨ ofer, Christoph W¨ achter, Kirill Lakhmanskiy, Rainer Blatt, Philipp Schindler, and Thomas Monz,Compact Ion-Trap Quantum Computing Demonstrator,PRX Quantum2, 020343 (2021), arX...

    Pith/arXiv arXiv 2021

  58. [58]

    McKay, Christopher J

    David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, and Jay M. Gambetta,EfficientZgates for quantum computing,Phys. Rev. A96, 022330 (2017), arXiv:1612.00858

    Pith/arXiv arXiv 2017

  59. [59]

    Anders Sørensen and Klaus Mølmer,Quantum Compu- tation with Ions in Thermal Motion,Phys. Rev. Lett. 82, 1971–1974 (1999), arXiv:quant-ph/9810039

    Pith/arXiv arXiv 1971

  60. [60]

    Phys.19, 023035 (2017), arXiv:1603.07678

    Dmitri Maslov,Basic circuit compilation techniques for an ion-trap quantum machine,New J. Phys.19, 023035 (2017), arXiv:1603.07678

    Pith/arXiv arXiv 2017

  61. [61]

    Tom Peham, Ludwig Schmid, Lucas Berent, Markus M¨ uller, and Robert Wille,Automated Synthesis of Fault- Tolerant State Preparation Circuits for Quantum Er- ror Correction Codes,PRX Quantum6, 020330 (2025), arXiv:2408.11894

    Pith/arXiv arXiv 2025

  62. [62]

    Gatterman, Justin A

    Yang Wang, Selwyn Simsek, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Nathan Hewitt, Chandler V. Horst, Mitchell Matheny, Tanner Mengle, Brian Neyenhuis, and Ben Criger,Fault-tolerant one-bit addition with the smallest interesting color code, Sci. Adv.10, 29 (2024), arXiv:2309.09893

    Pith/arXiv arXiv 2024

  63. [63]

    Reichardt,Quantum Error Cor- rection with Only Two Extra Qubits,Phys

    Rui Chao and Ben W. Reichardt,Quantum Error Cor- rection with Only Two Extra Qubits,Phys. Rev. Lett. 121, 050502 (2018), arXiv:1705.02329

    Pith/arXiv arXiv 2018

  64. [64]

    Beverland, Flag fault-tolerant error correction with arbitrary dis- tance codes,Quantum2, 53 (2018), arXiv:1708.02246

    Christopher Chamberland and Michael E. Beverland, Flag fault-tolerant error correction with arbitrary dis- tance codes,Quantum2, 53 (2018), arXiv:1708.02246

    Pith/arXiv arXiv 2018

  65. [65]

    Lukas Postler, Sascha Heußen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D. Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus M¨ uller, and Thomas Monz,Demonstration of fault-tolerant universal quan- tum gate operations,Nature605, 675–680 (2022), arXiv:2111.12654

    Pith/arXiv arXiv 2022

  66. [66]

    Matteo Rossi, Marcus Huber, Dagmar Bruß, and Chiara Macchiavello,Quantum hypergraph states,New J. Phys. 15, 113022 (2013), arXiv:1211.5554

    Pith/arXiv arXiv 2013

  67. [67]

    Mark Howard and Earl Campbell,Application of a Re- source Theory for Magic States to Fault-Tolerant Quan- tum Computing,Phys. Rev. Lett.118, 090501 (2017), arXiv:1609.07488

    Pith/arXiv arXiv 2017

  68. [68]

    Tobias Haug and Poetri Sonya Tarabunga,Efficient wit- nessing and testing of magic in mixed quantum states, npj Quantum Inf.12, 40 (2026), arXiv:2504.18098

    Pith/arXiv arXiv 2026

  69. [69]

    Alexei Kitaev,Quantum measurements and the Abelian Stabilizer Problem,arXiv:quant-ph/9511026 (1995)

    Pith/arXiv arXiv 1995

  70. [70]

    Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R

    Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Si- mon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles,Variational quantum algorithms, Nat. Rev. Phys.3, 625–644 (2021), arXiv:2012.09265

    Pith/arXiv arXiv 2021

  71. [71]

    Ross and Peter Selinger,Optimal ancilla-free Clifford+T approximation of z-rotations,Quantum Inf

    Neil J. Ross and Peter Selinger,Optimal ancilla-free Clifford+T approximation of z-rotations,Quantum Inf. Comput.16, 901–953 (2016), arXiv:1403.2975

    Pith/arXiv arXiv 2016

  72. [72]

    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 Quantum5, 010337 (2024), arXiv:2303.13181

    Pith/arXiv arXiv 2024

  73. [73]

    Refaat Ismail, I-Chi Chen, Chen Zhao, Ronen Weiss, Fangli Liu, Hengyun Zhou, Sheng-Tao Wang, An- drew Sornborger, and Milan Kornjaˇ ca,Transversal Architecture for Megaquop-Scale Quantum Simulation with Neutral Atoms,PRX Quantum7, 020343 (2026), arXiv:2509.18294

    Pith/arXiv arXiv 2026

  74. [74]

    Hockings, Nou´ edyn Baspin, Felix Thomsen, Samuel C

    Paul Webster, Lucas Berent, Omprakash Chandra, Evan T. Hockings, Nou´ edyn Baspin, Felix Thomsen, Samuel C. Smith, and Lawrence Z. Cohen,The Pin- nacle Architecture: Reducing the Cost of Breaking RSA- 2048 to 100,000 Physical Qubits Using Quantum LDPC Codes,arXiv:2602.11457 [quant-ph] (2026)

    Pith/arXiv arXiv 2048

  75. [75]

    Yoder, Eddie Schoute, Patrick Rall, Emily Pritchett, Jay M

    Theodore J. Yoder, Eddie Schoute, Patrick Rall, Emily Pritchett, Jay M. Gambetta, Andrew W. Cross, Mal- colm Carroll, and Michael E. Beverland,Tour de gross: A modular quantum computer based on bivariate bicycle codes,arXiv:2506.03094 [quant-ph] (2025)

    Pith/arXiv arXiv 2025

  76. [76]

    Browne,Three-dimensional surface codes: Transversal gates and fault-tolerant architectures,Phys

    Michael Vasmer and Dan E. Browne,Three-dimensional surface codes: Transversal gates and fault-tolerant architectures,Phys. Rev. A100, 012312 (2019), arXiv:1801.04255

    Pith/arXiv arXiv 2019

  77. [77]

    Phys.17, 083002 (2015), arXiv:1311.0879

    H´ ector Bomb´ ın,Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes,New J. Phys.17, 083002 (2015), arXiv:1311.0879

    Pith/arXiv arXiv 2015

  78. [78]

    Cross, Theodore J

    Samuel Stein, Shifan Xu, Andrew W. Cross, Theodore J. Yoder, Ali Javadi-Abhari, Chenxu Liu, Kun Liu, Zeyuan Zhou, Charles Guinn, Yufei Ding, Yongshan Ding, and Ang Li,Architectures for Heterogeneous Quantum Er- ror Correction Codes,inProc. 30th ACM Int. Conf. on Architectural Support for Programming Languages and Operating Systems (ASPLOS ’25), Vol. 2(ACM...

    Pith/arXiv arXiv 2025

  79. [79]

    Landahl and Ciaran Ryan-Anderson, Quantum Computing by Color-Code Lattice Surgery, arXiv:1407.5103 [quant-ph] (2014)

    Andrew J. Landahl and Ciaran Ryan-Anderson, Quantum Computing by Color-Code Lattice Surgery, arXiv:1407.5103 [quant-ph] (2014)

    Pith/arXiv arXiv 2014

  80. [80]

    Lukas B¨ odeker, ´Aron M´ arton, Luis Colmenarez, Ilya Besedin, Andreas Wallraff, and Markus M¨ uller,Lattice surgery for near-term experimental logical qubit entangle- ment creation in planar architectures,arXiv:2606.15190 [quant-ph] (2026)

    arXiv 2026

Showing first 80 references.