pith. sign in

arxiv: 2201.07254 · v1 · submitted 2022-01-18 · 🪐 quant-ph · cond-mat.str-el

Free-Fermion Subsystem Codes

Adrian Chapman , Steven T. Flammia , Alicia J. Koll\'ar This is my paper

Pith reviewed 2026-05-24 12:55 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords free-fermion modelssubsystem codesfrustration graphsBacon-Shor codeJordan-Wigner transformationquantum error correctiontopological qubitsexactly solvable spin models
0
0 comments X

The pith

Subsystem codes realize the first exactly solvable two-dimensional spin model with free-fermion spectrum and exact topological qubits.

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

The paper shows how to turn the gauge generators of certain subsystem codes into the terms of a translation-invariant Hamiltonian whose anticommutation relations form a frustration graph. When that graph permits a free-fermion solution, the entire spectrum and all eigenstates are obtained by a generalized Jordan-Wigner transformation. The authors embed the two-dimensional Bacon-Shor code into such a graph, producing the first known local spin model in two dimensions that is both free-fermion solvable and hosts exact topological qubits. They also supply an efficient graph-theoretic algorithm that decides solvability for any translation-invariant model and extracts the solution when it exists. Finally they relate the energy gaps that suppress errors to two standard graph quantities: the skew energy and the median eigenvalue of an oriented graph.

Core claim

A frustration graph whose vertices are Hamiltonian terms and whose edges mark anticommuting pairs can be embedded in the anticommutation relations of a local spin model; when the embedding succeeds, the model is exactly solvable by a generalized Jordan-Wigner transformation. The first such two-dimensional example is obtained by embedding the gauge generators of the Bacon-Shor code, yielding a free-fermion spectrum together with exact topological qubits protected by the code.

What carries the argument

The frustration graph, whose vertices are the gauge generators and whose edges connect pairs that anticommute; it encodes the solvability condition and supplies the input to the generalized Jordan-Wigner map.

If this is right

  • The spectrum and all eigenstates of the resulting Hamiltonian are known exactly, so error-suppression properties can be computed without approximation.
  • Topological qubits survive inside an exactly solvable free-fermion model, allowing direct study of their protection under the code Hamiltonian.
  • An efficient algorithm decides solvability and constructs the solution for any translation-invariant model by examining only the unit cell of its frustration graph.
  • The relevant gaps above the ground-state configuration are exactly the skew energy difference between symmetry sectors and the median eigenvalue of the oriented frustration graph.
  • Numerical optimization over small unit cells shows that large gaps are favored by low-dimensional lattices with odd coordination numbers.

Where Pith is reading between the lines

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

  • Models with the largest skew-energy gaps may give the strongest thermal suppression of logical errors among all free-fermion subsystem codes.
  • The same embedding technique could be applied to other known subsystem codes to produce additional exactly solvable families in two or higher dimensions.
  • Because the gaps are determined by purely graph-theoretic quantities, one can search for optimal codes by enumerating frustration graphs rather than by simulating Hamiltonians.

Load-bearing premise

Any chosen frustration graph can be realized by the anticommutation relations of a local spin Hamiltonian on a lattice without extra constraints that would break the free-fermion solution or change the topological qubit count.

What would settle it

An explicit frustration graph that cannot be embedded into any local spin model without introducing additional anticommutation relations that invalidate either the free-fermion spectrum or the claimed topological qubit degeneracy.

Figures

Figures reproduced from arXiv: 2201.07254 by Adrian Chapman, Alicia J. Koll\'ar, Steven T. Flammia.

Figure 1
Figure 1. Figure 1: Translation-invariant comb graph. We denote the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The terms of the checkerboard-lattice code. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Logical degrees of freedom of the checkerboard [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Building blocks of a triangle model. a Table showing the three effective qubits QP , QS, and QF formed from three physical qubits on the edges of a triangular plaquette. Physical-qubit Pauli operators are indicated in color-coded and labeled circles in the center of the each edge. For ease of view, we indicate the identity by the absence of a label. b, c Stabilizers of a Wen plaquette model constructed fro… view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of our line-graph recognition algo [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic diagram indicating which classes of lat [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Table of known properties for the four main examples in Fig. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: n = 2, m = 0 nanotube. The unoriented version of this nanotube is one graph which achieves the largest gap possible for any 3-regular graph [47, 51]. The elementary orientation is shown in a, with arrows on each edge indicating their orientation. The elementary orientation necessarily has the same DOS (b) as the unoriented graph. Despite the presence of a peak in the DOS at −3 and the absence of states in … view at source ↗
Figure 9
Figure 9. Figure 9: n = 1, m = 1-nanotube. This nanotube, or ladder, is an example of a graph where the exact ground-state orientation is known. Its elementary orientation and the corresponding DOS are shown in c and d and exhibit no single-particle gap. The corresponding plots for the true ground-state orientation are shown in a and b. This orientation has an extremely large single-particle gap: 2λ (τ) 1 = 2. By contrast, th… view at source ↗
read the original abstract

We consider quantum error-correcting subsystem codes whose gauge generators realize a translation-invariant, free-fermion-solvable spin model. In this setting, errors are suppressed by a Hamiltonian whose terms are the gauge generators of the code and whose exact spectrum and eigenstates can be found via a generalized Jordan-Wigner transformation. Such solutions are characterized by the frustration graph of the Hamiltonian: the graph whose vertices are Hamiltonian terms, which are neighboring if the terms anticommute. We provide methods for embedding a given frustration graph in the anticommutation relations of a spin model and present the first known example of an exactly solvable spin model with a two-dimensional free-fermion description and exact topological qubits. This model can be viewed as a free-fermionized version of the two-dimensional Bacon-Shor code. Using graph-theoretic tools to study the unit cell, we give an efficient algorithm for deciding if a given translation-invariant spin model is solvable, and explicitly construct the solution. Further, we examine the energetics of these exactly solvable models from the graph-theoretic perspective and show that the relevant gaps of the spin model correspond to known graph-theoretic quantities: the skew energy and the median eigenvalue of an oriented graph. Finally, we numerically search for models which have large spectral gaps above the ground state spin configuration and thus exhibit particularly robust thermal suppression of errors. These results suggest that optimal models will have low dimensionality and odd coordination numbers, and that the primary limit to energetic error suppression is the skew energy difference between different symmetry sectors rather than single-particle excitations of the free fermions.

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

0 major / 3 minor

Summary. The manuscript develops methods for embedding given frustration graphs into the anticommutation relations of local spin models, yielding translation-invariant free-fermion-solvable subsystem codes whose Hamiltonians are the gauge generators. It supplies an explicit 2D construction (a free-fermionized Bacon-Shor code) claimed to be the first with exact topological qubits, a graph-theoretic algorithm that decides solvability and constructs the solution from the unit cell, a correspondence between relevant spectral gaps and the skew energy plus median eigenvalue of an oriented graph, and numerical searches identifying models with large gaps that favor low dimensionality and odd coordination numbers.

Significance. If the constructions and algorithm are correct, the work supplies the first explicit 2D exactly solvable spin model with both free-fermion solvability and protected topological qubits, together with an efficient decision procedure and a graph-theoretic lens on energetics. These elements directly support the design of Hamiltonians that combine error suppression with analytic tractability.

minor comments (3)
  1. [§3] §3 (embedding procedure): the statement that any frustration graph can be realized locally would be strengthened by an explicit statement of the conditions under which the embedding preserves the exact free-fermion spectrum (i.e., does not introduce additional constraints).
  2. [Numerical search section] The numerical search section would benefit from a table or figure listing the coordination numbers, dimensions, and skew-energy differences for the top-performing models, rather than only the qualitative conclusion.
  3. [Algorithm section] A brief pseudocode block for the unit-cell solvability algorithm would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision for our manuscript on free-fermion subsystem codes. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; constructions are explicit and independent

full rationale

The paper delivers an explicit lattice embedding of the Bacon-Shor frustration graph into local Pauli operators, a graph-theoretic algorithm that decides solvability directly from the unit cell, and a concrete 2D model whose spectrum follows from the standard generalized Jordan-Wigner map. None of these steps reduce by definition or by self-citation to the target claims (topological qubit count or exact solvability); each is supplied as an independent algorithmic or constructive result. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the authors' prior work, and the central example is not obtained by renaming a known pattern. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the applicability of the generalized Jordan-Wigner transformation to frustration graphs of translation-invariant Hamiltonians and on the existence of lattice embeddings that preserve locality and the desired code properties. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Generalized Jordan-Wigner transformation yields the exact spectrum and eigenstates once the frustration graph is realized by a spin model.
    Invoked throughout to obtain exact solutions and to equate gaps with graph quantities.

pith-pipeline@v0.9.0 · 5811 in / 1218 out tokens · 40020 ms · 2026-05-24T12:55:59.294983+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 15 internal anchors

  1. [1]

    Jordan and E

    P. Jordan and E. Wigner, Zeitschrift für Physik47, 631 (1928)

    work page 1928

  2. [2]

    Kitaev, Annals of Physics 321, 2 (2006), january Special Issue

    A. Kitaev, Annals of Physics 321, 2 (2006), january Special Issue

    work page 2006

  3. [3]

    B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A65, 032325 (2002)

    work page 2002

  4. [4]

    Fermionic Linear Optics and Matchgates

    E. Knill, ArXiv e-prints (2001), arXiv:quant-ph/0108033

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  5. [5]

    L. G. Valiant, SIAM Journal on Computing31, 1229 (2002), https://doi.org/10.1137/S0097539700377025

    work page doi:10.1137/s0097539700377025 2002

  6. [6]

    S. B. Bravyi and A. Y. Kitaev, Ann. Phys. (N. Y.)298, 210 (2002)

    work page 2002

  7. [7]

    Bravyi, Phys

    S. Bravyi, Phys. Rev. A73, 042313 (2006)

    work page 2006

  8. [8]

    Jozsa and A

    R. Jozsa and A. Miyake, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences464, 3089 (2008)

    work page 2008

  9. [9]

    de Melo, P

    F. de Melo, P. flwikliński, and B. M. Terhal, New J. Phys. 15, 013015 (2013)

    work page 2013

  10. [10]

    D. J. Brod and A. M. Childs, ArXiv e-prints (2013), arXiv:1308.1463 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  11. [11]

    A. Y. Kitaev, Physics Uspekhi44, 131 (2001), arXiv:cond- mat/0010440 [cond-mat.mes-hall]

    work page arXiv 2001

  12. [12]

    Bravyi, B

    S. Bravyi, B. M. Terhal, and B. Leemhuis,12, 083039 (2010)

    work page 2010

  13. [13]

    M. B. Hastings, arXiv e-prints , arXiv:1703.00612 (2017), arXiv:1703.00612 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Quantum Error Correction for Complex and Majorana Fermion Qubits

    S. Vijay and L. Fu, arXiv e-prints , arXiv:1703.00459 (2017), arXiv:1703.00459 [cond-mat.mes-hall]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    Scalable Fermionic Error Correction in Majorana Surface Codes

    O. Viyuela, S. Vijay, and L. Fu, Phys. Rev. B99, 205114 (2019), arXiv:1812.08477 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  16. [16]

    Chapman and S

    A. Chapman and S. T. Flammia, Quantum4, 278 (2020)

    work page 2020

  17. [17]

    Ogura, Y

    M. Ogura, Y. Imamura, N. Kameyama, K. Minami, and M. Sato, Phys. Rev. B 102, 245118 (2020), arXiv:2003.13264 [cond-mat.stat-mech]

    work page arXiv 2020

  18. [18]

    Verstraete and J

    F. Verstraete and J. I. Cirac, Journal of Statistical Me- chanics: Theory and Experiment2005, P09012 (2005)

    work page 2005

  19. [19]

    Setia, S

    K. Setia, S. Bravyi, A. Mezzacapo, and J. D. Whitfield, arXiv e-prints , arXiv:1810.05274 (2018), arXiv:1810.05274 [quant-ph]

    work page arXiv 2018

  20. [20]

    J. T. Seeley, M. J. Richard, and P. J. Love, The Journal of Chemical Physics 137, 224109 (2012), https://doi.org/10.1063/1.4768229

    work page doi:10.1063/1.4768229 2012

  21. [21]

    Tapering off qubits to simulate fermionic Hamiltonians

    S. Bravyi, J. M. Gambetta, A. Mezzacapo, and K. Temme, arXiv e-prints , arXiv:1701.08213 (2017), arXiv:1701.08213 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    Steudtner and S

    M. Steudtner and S. Wehner, New Journal of Physics20, 063010 (2018)

    work page 2018

  23. [23]

    Jiang, A

    Z. Jiang, A. Kalev, W. Mruczkiewicz, and H. Neven, arXiv e-prints , arXiv:1910.10746 (2019), arXiv:1910.10746 [quant-ph]

    work page arXiv 1910

  24. [24]

    Derby, J

    C. Derby, J. Klassen, J. Bausch, and T. Cubitt, Phys. Rev. B 104, 035118 (2021)

    work page 2021

  25. [25]

    Chiew and S

    M. Chiew and S. Strelchuk, arXiv e-prints , arXiv:2110.12792 (2021), arXiv:2110.12792 [quant- ph]

    work page arXiv 2021

  26. [26]

    Jiang, J

    Z. Jiang, J. McClean, R. Babbush, and H. Neven, Physi- cal Review Applied12, 064041 (2019), arXiv:1812.08190 [quant-ph]

    work page arXiv 2019

  27. [27]

    Haah, Communications in Mathematical Physics324, 19 351 (2013)

    J. Haah, Communications in Mathematical Physics324, 19 351 (2013)

    work page 2013

  28. [28]

    Yu, S.-P

    J. Yu, S.-P. Kou, and X.-G. Wen, EPL (Europhysics Letters) 84, 17004 (2008)

    work page 2008

  29. [29]

    Subsystem surface codes with three-qubit check operators

    S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, arXiv e-prints , arXiv:1207.1443 (2012), arXiv:1207.1443 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  30. [30]

    Bacon, Physical Review A73, 012340 (2006), quant- ph/0506023

    D. Bacon, Physical Review A73, 012340 (2006), quant- ph/0506023

    work page arXiv 2006

  31. [31]

    Topological Subsystem Codes

    H. Bombin, Phys. Rev. A 81, 032301 (2010), arXiv:0908.4246 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  32. [32]

    Constructions and Noise Threshold of Topological Subsystem Codes

    M. Suchara, S. Bravyi, and B. Terhal, Journal of Physics A Mathematical General 44, 155301 (2011), arXiv:1012.0425 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  33. [33]

    Adiga, R

    C. Adiga, R. Balakrishnan, and W. So, Linear Algebra and its Applications432, 1825 (2010)

    work page 2010

  34. [34]

    Stabilizer Formalism for Operator Quantum Error Correction

    D. Poulin, Phys. Rev. Lett. 95, 230504 (2005), arXiv:quant-ph/0508131 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  35. [35]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition , 10th ed. (Cambridge University Press, USA, 2011)

    work page 2011

  36. [36]

    Tantivasadakarn, arXiv e-prints , arXiv:2002.11345 (2020), arXiv:2002.11345 [cond-mat.str-el]

    N. Tantivasadakarn, arXiv e-prints , arXiv:2002.11345 (2020), arXiv:2002.11345 [cond-mat.str-el]

    work page arXiv 2002

  37. [37]

    Y.-A. Chen, A. Kapustin, and Ð. Radičević, Annals of Physics 393, 234 (2018)

    work page 2018

  38. [38]

    P. W. Shor, Physical Review A52, R2493 (1995)

    work page 1995

  39. [39]

    Optimal Bacon-Shor codes

    J. Napp and J. Prreskill, Quantum Information and Com- putation 13, 0490 (2013), 1209.0794

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  40. [40]

    A. Y. Kitaev, Annals of Physics303, 1 (2003)

    work page 2003

  41. [41]

    S. J. Elman, A. Chapman, and S. T. Flammia, Commu- nications in Mathematical Physics388, 969 (2021)

    work page 2021

  42. [42]

    E. H. Lieb, Phys. Rev. Lett.73, 2158 (1994)

    work page 1994

  43. [43]

    Whitney, American Journal of Mathematics54, 150 (1932)

    H. Whitney, American Journal of Mathematics54, 150 (1932)

    work page 1932

  44. [44]

    Krausz, Matematikai és Fizikai Lapok50 (1943)

    J. Krausz, Matematikai és Fizikai Lapok50 (1943)

    work page 1943

  45. [45]

    A survey on the skew energy of oriented graphs

    X. Li and H. Lian, arXiv e-prints , arXiv:1304.5707 (2013), arXiv:1304.5707 [math.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  46. [46]

    Denglan and H

    C. Denglan and H. Yaoping, The Electronic Journal of Combinatorics 20 (2013), 10.37236/2864

    work page doi:10.37236/2864 2013

  47. [47]

    A. J. Kollár and P. Sarnak, Communications of the Amer- ican Mathematical Society1, 1 (2021)

    work page 2021

  48. [48]

    Biggs, Algebraic Graph Theory, 2nd ed

    N. Biggs, Algebraic Graph Theory, 2nd ed. (Cambridge University Press, Cambridge, 1993)

    work page 1993

  49. [49]

    D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application (Academic Press, 1980)

    work page 1980

  50. [50]

    Oriented unicyclic graphs with extremal skew energy

    H. Yaoping, S. Xiaoling, and Z. Chongyan, “Oriented unicyclic graphs with extremal skew energy,” (2011), arXiv:1108.6229 [math.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  51. [51]

    Guo and B

    K. Guo and B. Mohar, Linear Alg. Appl.449, 68 (2014)

    work page 2014

  52. [52]

    Median eigenvalues of bipartite subcubic graphs

    B. Mohar, Comb. Prob. Comput. 25, 768 (2016), 1309.7395

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  53. [53]

    S. M. Girvin and K. Yang,Modern Condensed Matter Physics (Cambridge University Press, Inc., 2019)

    work page 2019

  54. [54]

    A. J. Kollár, M. Fitzpatrick, P. Sarnak, and A. A. Houck, Commun. Math. Phys.44, 1601 (2019)

    work page 2019

  55. [55]

    Wildeboer, T

    J. Wildeboer, T. Iadecola, and D. J. Williamson, arXiv e- prints , arXiv:2110.05710 (2021), arXiv:2110.05710 [quant- ph]

    work page arXiv 2021

  56. [56]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Phys. Rev. Lett. 120, 050505 (2018), arXiv:1708.08474 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  57. [57]

    D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, Physical Review X9, 041031 (2019), arXiv:1812.08186 [quant-ph]

    work page arXiv 2019

  58. [58]

    D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, Phys. Rev. Lett. 124, 130501 (2020), arXiv:1907.02554 [quant-ph]. Appendix A: Description of Numerical Simulation Packages The numerical calculations in the work were carried out using a custom suite of graph-theoretic lattice codes defined in Python3. The backbone of this code was devel- oped ...

    work page arXiv 2020