pith. sign in

arxiv: 2511.06620 · v2 · submitted 2025-11-10 · 🪐 quant-ph

Fault-Tolerant Encoding of Logical Qudits in Spin Systems

Sumin Lim This is my paper

Pith reviewed 2026-05-18 00:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords fault-tolerant quditslogical encodingquantum error correctionspin systemsqudit codesdistance-3 codesresource-efficient quantum computing
0
0 comments X

The pith

Logical qudits can be encoded fault-tolerantly in spin systems using a single physical qudit for distance-3 and distance-5 protection with smaller Hilbert space than multi-qubit codes.

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

The paper constructs a general framework and explicit examples for encoding logical qudits that correct errors in finite-dimensional spin systems. It gives distance-3 and distance-5 codewords plus a family of 2t+1-distance codes that fit inside one physical qudit or a few coupled qudits. These encodings occupy far fewer dimensions than standard constructions that spread a logical qudit across many qubits. A reader would care because the approach supplies concrete operations and error-correction steps that scale only polynomially and match the multi-level nature of real spin hardware. The constructions are presented as directly compatible with existing experimental platforms.

Core claim

We construct distance-3, distance-5 codewords, and general 2t+1-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations.

What carries the argument

The distance-2t+1 encoding of a logical qudit into the Hilbert space of one or a few physical spin qudits that supplies the stated error-correcting distance at reduced dimension.

If this is right

  • Logical gates and syndrome measurements require only polynomially many elementary operations.
  • Physical layouts exist that are compatible with current spin-qudit devices and their measured gate fidelities.
  • The same framework yields codes of arbitrary odd distance while keeping the physical dimension modest.

Where Pith is reading between the lines

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

  • If the encoding succeeds, direct simulation of multi-level physical Hamiltonians becomes possible without an intermediate qubit mapping.
  • The construction could be tested by preparing the distance-3 codeword in a single trapped-ion qudit and measuring its logical lifetime.
  • Similar encodings might apply to other multi-level systems such as molecular rotors or superconducting circuits with high anharmonicity.

Load-bearing premise

The required logical operations and error-correction protocols can be realized with the stated polynomial scaling and single-gate fidelities in actual finite-dimensional spin systems without introducing uncorrectable errors from the physical implementation.

What would settle it

An experiment that measures the logical error rate on the constructed codewords and finds it does not fall below the physical error rate when the code distance is increased from 3 to 5 would falsify the central claim.

Figures

Figures reproduced from arXiv: 2511.06620 by Sumin Lim.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Brief schematic of qutrit Z-error correction code. (see main text for details) (b) The [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Required Hilbert space for logical qudit encoding in this work (blue) and convensional [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Qutrit fidelity as a function of time without (black) and with (red) error correction. The [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-$3$, distance-$5$ codewords, and general $2t+1$-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.

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 / 1 minor

Summary. The manuscript presents a general framework for encoding fault-tolerant logical qudits in finite-dimensional spin systems. It constructs distance-3 and distance-5 codewords, along with a general construction for 2t+1-distance codes, which can be realized using a single physical qudit or a small number of coupled qudits. These encodings require a significantly smaller Hilbert space dimension compared to conventional constructions based on multiple logical qubits. The manuscript claims that logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. It also discusses schematic designs for physical implementation and required single-gate fidelities that are said to be compatible with current spin qudit platforms.

Significance. If the constructions are correct and the physical implementations feasible, this could offer a resource-efficient path to fault-tolerant qudit-based quantum computing, with advantages for simulating multi-level systems and reduced overhead relative to multi-qubit encodings. The polynomial scaling for logical gates and error correction, if substantiated, would be a notable strength for practical realization in spin platforms.

major comments (2)
  1. [Abstract] Abstract: The abstract asserts the existence of distance-3, distance-5, and general 2t+1-distance codes with polynomial scaling that can be implemented using a single physical qudit or small number of coupled qudits, but provides no derivations, explicit codewords, or verification steps for the distance properties or the encoding maps.
  2. [Physical implementation discussion] Physical implementation discussion: The schematic designs and claims of compatibility with current single-gate fidelities do not include threshold calculations or explicit noise simulations under realistic spin-system Hamiltonians, where higher levels typically exhibit faster decoherence and control crosstalk that could introduce errors outside the assumed model and violate the distance guarantee.
minor comments (1)
  1. The quantitative comparison of Hilbert space dimensions to conventional multi-qubit constructions should be made explicit with specific examples or tables to support the resource-efficiency claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We address the major comments point by point below and have revised the manuscript accordingly to improve its clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract asserts the existence of distance-3, distance-5, and general 2t+1-distance codes with polynomial scaling that can be implemented using a single physical qudit or small number of coupled qudits, but provides no derivations, explicit codewords, or verification steps for the distance properties or the encoding maps.

    Authors: While the abstract serves as a high-level summary and does not include detailed derivations, the manuscript provides explicit constructions, codewords, and verifications for the distance-3, distance-5, and general 2t+1 codes in the main text. Specifically, the encoding maps and distance properties are derived and verified in Sections 3-5. We have updated the abstract to more clearly indicate that these technical details are elaborated in the body of the paper. revision: partial

  2. Referee: [Physical implementation discussion] Physical implementation discussion: The schematic designs and claims of compatibility with current single-gate fidelities do not include threshold calculations or explicit noise simulations under realistic spin-system Hamiltonians, where higher levels typically exhibit faster decoherence and control crosstalk that could introduce errors outside the assumed model and violate the distance guarantee.

    Authors: The referee correctly notes that our physical implementation section is schematic and does not contain threshold calculations or explicit noise simulations. We agree that incorporating realistic noise models, including faster decoherence for higher levels and control crosstalk, is crucial to fully substantiate the fault-tolerance claims. In the revised manuscript, we have added a more detailed discussion of these potential error sources and their possible effects on the assumed error model. However, performing comprehensive numerical simulations under specific Hamiltonians is a significant extension that we plan to pursue in follow-up work. We believe the current analysis provides a valuable starting point for such investigations. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions are independent mathematical encodings

full rationale

The paper defines explicit distance-3, distance-5, and general 2t+1 codewords for logical qudits in finite-dimensional spin systems, with polynomial-scaling operations. These are presented as new constructions requiring smaller Hilbert space than qubit-based alternatives. No equations reduce by construction to fitted parameters, self-citations, or prior ansatzes from the same authors; the derivations are self-contained algebraic definitions of code subspaces and gates. The physical implementation discussion is schematic and does not rely on self-referential predictions. This is the normal case of an independent encoding result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard quantum error-correction theory applied to finite-dimensional spin systems; no explicit free parameters, new physical entities, or ad-hoc axioms are stated in the abstract.

axioms (1)
  • standard math Standard assumptions of quantum mechanics and quantum error correction (finite-dimensional Hilbert spaces, unitary gates, independent error models)
    The constructions presuppose the usual framework of quantum information theory.

pith-pipeline@v0.9.0 · 5442 in / 1272 out tokens · 37528 ms · 2026-05-18T00:28:21.702141+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM review41, 303 (1999)

    work page 1999

  2. [2]

    L. M. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Experimental realization of shor’s quantum factoring algorithm using nuclear magnetic reso- nance, Nature414, 883 (2001)

    work page 2001

  3. [3]

    L. K. Grover, A fast quantum mechanical algorithm for database search, inProceedings of the twenty-eighth annual ACM symposium on Theory of computing(1996) pp. 212–219

    work page 1996

  4. [4]

    I. L. Chuang, N. Gershenfeld, and M. Kubinec, Experimental implementation of fast quantum searching, Physical review letters80, 3408 (1998)

    work page 1998

  5. [5]

    D. G. Cory, A. F. Fahmy, and T. F. Havel, Ensemble quantum computing by nmr spectroscopy, Proceedings of the National Academy of Sciences94, 1634 (1997)

    work page 1997

  6. [6]

    R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics21, 1572 (1982)

    work page 1982

  7. [7]

    Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    work page 1996

  8. [8]

    Aspuru-Guzik, A

    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Simulated quantum compu- tation of molecular energies, Science309, 1704 (2005)

    work page 2005

  9. [9]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Reviews of modern physics 89, 035002 (2017)

    work page 2017

  10. [10]

    Dolde, M

    F. Dolde, M. W. Doherty, J. Michl, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, P. Neu- mann, F. Jelezko, N. B. Manson,et al., Nanoscale detection of a single fundamental charge 12 in ambient conditions using the nv- center in diamond, Physical review letters112, 097603 (2014)

    work page 2014

  11. [11]

    Clarke and F

    J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature453, 1031 (2008)

    work page 2008

  12. [12]

    C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, Theoretical computer science560, 7 (2014)

    work page 2014

  13. [13]

    S. Lim, J. Liu, and A. Ardavan, Fault-tolerant qubit encoding using a spin-7/2 qudit, Physical Review A108, 062403 (2023)

    work page 2023

  14. [14]

    Janković, J.-G

    D. Janković, J.-G. Hartmann, M. Ruben, and P.-A. Hervieux, Noisy qudit vs multiple qubits: conditions on gate efficiency for enhancing fidelity, npj Quantum Information10, 59 (2024)

    work page 2024

  15. [15]

    X. Gao, P. Appel, N. Friis, M. Ringbauer, and M. Huber, On the role of entanglement in qudit-based circuit compression, Quantum7, 1141 (2023)

    work page 2023

  16. [16]

    N. P. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s hamiltonians, npj Quantum Information6, 49 (2020)

    work page 2020

  17. [17]

    Preskill, Quantum computing in the nisq era and beyond, Quantum2, 79 (2018)

    J. Preskill, Quantum computing in the nisq era and beyond, Quantum2, 79 (2018)

    work page 2018

  18. [18]

    R. F. Uy and D. A. Gangloff, Qudit-based quantum error-correcting codes from irreducible representations of su (d), Physical Review A112, 042402 (2025)

    work page 2025

  19. [19]

    B. L. Brock, S. Singh, A. Eickbusch, V. V. Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, Quantum error correction of qudits beyond break-even, Nature641, 612 (2025)

    work page 2025

  20. [20]

    M. H. Michael, M. Silveri, R. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Physical Review X6, 031006 (2016)

    work page 2016

  21. [21]

    Ouyang, Permutation-invariant qudit codes from polynomials, Linear Algebra and its Ap- plications532, 43 (2017)

    Y. Ouyang, Permutation-invariant qudit codes from polynomials, Linear Algebra and its Ap- plications532, 43 (2017)

    work page 2017

  22. [22]

    Mazurek, M

    P. Mazurek, M. Farkas, A. Grudka, M. Horodecki, and M. Studziński, Quantum error- correction codes and absolutely maximally entangled states, Physical Review A101, 042305 (2020)

    work page 2020

  23. [23]

    Schmidt and P

    F. Schmidt and P. van Loock, Quantum error correction with higher gottesman-kitaev-preskill codes: Minimal measurements and linear optics, Physical Review A105, 042427 (2022)

    work page 2022

  24. [24]

    Dutta, D

    S. Dutta, D. Biswas, and P. Mandayam, Noise-adapted qudit codes for amplitude-damping noise, Physical Review A111, 032438 (2025). 13

    work page 2025

  25. [25]

    Knill, R

    E. Knill, R. Laflamme, and W. H. Zurek, Resilient quantum computation, Science279, 342 (1998)

    work page 1998

  26. [26]

    Laflamme, C

    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correcting code, Physical Review Letters77, 198 (1996)

    work page 1996

  27. [27]

    Knill and R

    E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Physical Review A55, 900 (1997)

    work page 1997

  28. [28]

    Suppressing quantum errors by scaling a surface code logical qubit, Nature614, 676 (2023)

    work page 2023

  29. [29]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Physical Review A64, 012310 (2001)

    work page 2001

  30. [30]

    L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. Song, C.-L. Zou,et al., Quantum error correction and universal gate set operation on a binomial bosonic logical qubit, Nature Physics15, 503 (2019)

    work page 2019

  31. [31]

    S. Lim, M. V. Vaganov, J. Liu, and A. Ardavan, Demonstrating experimentally the encoding and dynamics of an error-correctable logical qubit on a hyperfine-coupled nuclear spin qudit, Physical Review Letters134, 070603 (2025)

    work page 2025

  32. [32]

    X. Yu, B. Wilhelm, D. Holmes, A. Vaartjes, D. Schwienbacher, M. Nurizzo, A. Kringhøj, M. R. v. Blankenstein, A. M. Jakob, P. Gupta,et al., Schrödinger cat states of a nuclear spin qudit in silicon, Nature Physics , 1 (2025)

    work page 2025

  33. [33]

    Yang, W.-T

    Y. Yang, W.-T. Luo, J.-L. Zhang, S.-Z. Wang, C.-L. Zou, T. Xia, and Z.-T. Lu, Minute-scale schrödinger-cat state of spin-5/2 atoms, Nature Photonics19, 89 (2025)

    work page 2025

  34. [34]

    J. A. Gross, Designing codes around interactions: The case of a spin, Physical Review Letters 127, 010504 (2021)

    work page 2021

  35. [35]

    Omanakuttan and J

    S. Omanakuttan and J. A. Gross, Multispin clifford codes for angular momentum errors in spin systems, Physical Review A108, 022424 (2023)

    work page 2023

  36. [36]

    S.Omanakuttan, V.Buchemmavari, J.A.Gross, I.H.Deutsch,andM.Marvian,Fault-tolerant quantum computation using large spin-cat codes, PRX Quantum5, 020355 (2024)

    work page 2024

  37. [37]

    Omanakuttan, A

    S. Omanakuttan, A. Mitra, E. J. Meier, M. J. Martin, and I. H. Deutsch, Qudit entanglers using quantum optimal control, PRX Quantum4, 040333 (2023)

    work page 2023

  38. [38]

    Chiesa, E

    A. Chiesa, E. Macaluso, F. Petiziol, S. Wimberger, P. Santini, and S. Carretta, Molecular nano- magnets as qubits with embedded quantum-error correction, The journal of physical chemistry letters11, 8610 (2020). 14

    work page 2020

  39. [39]

    S. J. Lockyer, A. Chiesa, G. A. Timco, E. J. McInnes, T. S. Bennett, I. J. Vitorica-Yrezebal, S. Carretta, and R. E. Winpenny, Targeting molecular quantum memory with embedded error correction, Chemical science12, 9104 (2021)

    work page 2021

  40. [40]

    Carretta, D

    S. Carretta, D. Zueco, A. Chiesa, Á. Gómez-León, and F. Luis, A perspective on scaling up quantum computation with molecular spins, Applied Physics Letters118(2021)

    work page 2021

  41. [41]

    Lim and A

    S. Lim and A. Ardavan, Designing quantum error correction codes for practical spin qudit systems, Physical Review A112, 022418 (2025)

    work page 2025

  42. [42]

    Fernández de Fuentes, T

    I. Fernández de Fuentes, T. Botzem, M. A. Johnson, A. Vaartjes, S. Asaad, V. Mourik, F. E. Hudson, K. M. Itoh, B. C. Johnson, A. M. Jakob, J. C. McCallum, D. N. Jamieson, A. S. Dzurak, and A. Morello, Navigating the 16-dimensional hilbert space of a high-spin donor qudit with electric and magnetic fields, Nature communications15, 1380 (2024)

    work page 2024

  43. [43]

    Takahashi, J

    S. Takahashi, J. van Tol, C. C. Beedle, D. N. Hendrickson, . f. L.-C. Brunel, and M. S. Sherwin, Coherent manipulation and decoherence of s= 10 single-molecule magnets, Physical review letters102, 087603 (2009)

    work page 2009

  44. [44]

    Liu, B.-W

    Z. Liu, B.-W. Dong, H.-B. Meng, M.-X. Xu, T.-S. Wang, B.-W. Wang, C.-R. Wang, S.-D. Jiang, and S. Gao, Qubit crossover in the endohedral fullerene sc 3 c 2@ c 80, Chemical Science9, 457 (2018)

    work page 2018

  45. [45]

    V. V. Albert, J. P. Covey, and J. Preskill, Robust encoding of a qubit in a molecule, Physical Review X10, 031050 (2020)

    work page 2020

  46. [46]

    S. P. Jain, E. R. Hudson, W. C. Campbell, and V. V. Albert, Absorption-emission codes for atomic and molecular quantum information platforms, Physical Review Letters133, 260601 (2024)

    work page 2024

  47. [47]

    M. J. Boguslawski, Z. J. Wall, S. R. Vizvary, I. D. Moore, M. Bareian, D. T. Allcock, D. J. Wineland, E. R. Hudson, and W. C. Campbell, Raman scattering errors in stimulated-raman- induced logic gates in ba+ 133, Physical Review Letters131, 063001 (2023)

    work page 2023

  48. [48]

    X. Yu, J. Mo, T. Lu, T. Y. Tan, and T. L. Nicholson, Magneto-optical trapping of a group-iii atom, Physical Review A105, L061101 (2022)

    work page 2022

  49. [49]

    Leghtas, G

    Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, Hardware-efficient autonomous quantum memory protection, Physical Review Letters111, 120501 (2013)

    work page 2013

  50. [50]

    N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, 15 S. M. Girvin, L. Jiang,et al., Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature536, 441 (2016)

    work page 2016

  51. [51]

    Z. Ni, S. Li, X. Deng, Y. Cai, L. Zhang, W. Wang, Z.-B. Yang, H. Yu, F. Yan, S. Liu,et al., Beating the break-even point with a discrete-variable-encoded logical qubit, Nature616, 56 (2023)

    work page 2023

  52. [52]

    L. Li, D. J. Young, V. V. Albert, K. Noh, C.-L. Zou, and L. Jiang, Phase-engineered bosonic quantum codes, Physical Review A103, 062427 (2021)

    work page 2021

  53. [53]

    M. N. Leuenberger and D. Loss, Quantum computing in molecular magnets, Nature410, 789 (2001)

    work page 2001

  54. [54]

    Murugesu, S

    M. Murugesu, S. Takahashi, A. Wilson, K. A. Abboud, W. Wernsdorfer, S. Hill, and G. Chris- tou, Large mn25 single-molecule magnet with spin s= 51/2: magnetic and high-frequency electron paramagnetic resonance spectroscopic characterization of a giant spin state, Inorganic chemistry47, 9459 (2008)

    work page 2008

  55. [55]

    deMarti iOlius, P

    A. deMarti iOlius, P. Fuentes, R. Orús, P. M. Crespo, and J. E. Martinez, Decoding algorithms for surface codes, Quantum8, 1498 (2024)

    work page 2024

  56. [56]

    E. O. Kiktenko, A. S. Nikolaeva, and A. K. Fedorov, Colloquium: Qudits for decomposing multiqubit gates and realizing quantum algorithms, Reviews of Modern Physics97, 021003 (2025)

    work page 2025

  57. [57]

    L. E. Fischer, A. Chiesa, F. Tacchino, D. J. Egger, S. Carretta, and I. Tavernelli, Universal qudit gate synthesis for transmons, PRX Quantum4, 030327 (2023)

    work page 2023

  58. [58]

    Chizzini, F

    M. Chizzini, F. Tacchino, A. Chiesa, I. Tavernelli, S. Carretta, and P. Santini, Qudit-based quantum simulation of fermionic systems, Physical Review A110, 062602 (2024)

    work page 2024

  59. [59]

    Rambow and M

    P. Rambow and M. Tian, Reduction of circuit depth by mapping qubit-based quantum gates to a qudit basis. arxiv 2021, arXiv preprint arXiv:2109.09902. Appendix A: Pulse sequences for encoding and decoding logical qudits Fig. A1 shows the complete encoding pulse sequence for the qutrit Z-error correction code. Withoutloss of generality, theinitial state can...

    work page arXiv 2021