Pith. sign in

REVIEW 1 major objections 4 minor 75 references

Supersymmetry-inspired circuits host strong zero modes that can be steered ballistically and used to move a logical qubit.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 05:16 UTC pith:FZQBE727

load-bearing objection Clean free-fermion construction of steerable, ballistically propagating SZMs plus an explicit QI-routing protocol; solid within matchgate circuits, not oversold. the 1 major comments →

arxiv 2607.11483 v1 pith:FZQBE727 submitted 2026-07-13 quant-ph

Strong Zero Modes in Supersymmetry-Inspired Quantum Circuits

Alberto Zorzato , Pietro Richelli , Jiri Min\'a\v{r} , Kareljan Schoutens This is my paper
classification quant-ph
keywords strong zero modesquantum circuitssupersymmetryFloquet micromotionDzyaloshinskii-Moriyaquantum information transportfree fermionsmatchgates
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.

The paper builds brick-wall quantum circuits whose two-qubit gates are the elastic S-matrices of a supersymmetric 1+1D field theory. These circuits are free-fermionic and therefore admit exact or approximate strong zero modes—operators that commute with the Floquet evolution and anticommute with parity. In isotropic mass configurations one finds a delocalized mode and a boundary-localized mode. In anisotropic (staggered or single-heavy-mass) configurations the Floquet micromotion generated by a large Dzyaloshinskii–Moriya term enlarges the kernel, producing a second pair of modes. One of these modes stays localized while the other propagates ballistically, following the location of the heavy mass. Encoding a logical qubit in a Dirac fermion built from the pair lets the circuit transfer one qubit of information across an L-site register in 2L−3 layers, with quadratic rather than linear sensitivity to certain single-qubit noise.

Core claim

In the large mass-ratio, small-rapidity regime of the brick-wall circuit the kernel of the Floquet generator becomes four-dimensional and contains a pair of approximately conserved Majorana operators, one of which remains localized while the other propagates ballistically under the micromotion generated by the Dzyaloshinskii–Moriya term; the pair can be used to transfer one logical qubit across an L-site register in 2L−3 layers.

What carries the argument

Floquet micromotion: the nontrivial zeroth-order (in rapidity) partial products of circuit layers that permute Majorana operators according to a strong Dzyaloshinskii–Moriya rotation; this dressing enlarges the kernel of the effective generator and steers the propagating strong zero mode.

Load-bearing premise

The extra strong zero modes become exactly conserved only in the infinite mass-ratio limit; for any finite ratio they are only approximately conserved, and the numerical demonstrations use a large but finite ratio.

What would settle it

Compute the exact commutator of the candidate propagating operator with the full Floquet unitary at finite mass ratio: if the commutator does not fall exponentially with the mass ratio, or if the white magnetization trace fails to track the heavy-mass location under the stated protocol, the claim fails.

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

If this is right

  • A concrete circuit protocol moves one logical qubit from one end of an L-site register to the other in 2L−3 layers by steering a pair of strong zero modes.
  • Encoding the logical qubit in spatially separated Majoranas yields quadratic rather than linear infidelity under certain single-qubit phase errors.
  • The number and location of strong zero modes can be tuned by the mass pattern and the sign of the boundary gates.
  • The same micromotion mechanism may generate additional conserved operators in other free-fermionic brick-wall circuits that contain chiral hopping terms.

Where Pith is reading between the lines

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

  • Because the circuits are matchgate, the entire protocol is efficiently classically simulable via covariance matrices, so any experimental claim can be cross-checked on a classical computer before hardware runs.
  • If the mass-ratio requirement can be relaxed by approximate or dressed zero modes, the same routing idea might transfer to near-term hardware with only moderate anisotropy.
  • The quadratic noise protection suggests a natural comparison with other Majorana-based encodings that also split information across distant sites.

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

1 major / 4 minor

Summary. The manuscript constructs brick-wall quantum circuits whose two-qubit gates are the S-matrix of a supersymmetric 1+1D QFT. Because the gates are matchgates the dynamics is free-fermionic; the associated Floquet operator UF therefore induces an orthogonal rotation OF on the 2L-dimensional Majorana space. In the simultaneous small-rapidity (Hamiltonian) and large mass-ratio limits the kernel of the first-order generator becomes four-dimensional, yielding two exact and two approximate strong zero modes. One pair remains localized at a boundary while the second pair is transported ballistically by the Floquet micromotion generated by a Dzyaloshinskii–Moriya term. The authors give explicit analytic expressions for the modes in both staggered and single-heavy-mass configurations, confirm the ballistic signature with exact Gaussian covariance-matrix numerics, and construct a finite-depth (2L-3-layer) protocol that transfers one logical qubit encoded in a Dirac fermion built from the localized pair. Robustness under selected parity-preserving noise is examined in the Supplemental Material.

Significance. The work supplies a concrete, analytically controlled mechanism for ballistically propagating strong zero modes inside free-fermion circuits and converts that mechanism into an explicit quantum-information routing protocol. The derivation of the micromotion-dressed effective Hamiltonian (Eq. 13 and SM E) is careful, the four-dimensional kernel is obtained by direct linear algebra rather than fitting, and the numerics are exact within the Gaussian formalism. These features make the construction a useful addition to the literature on free-fermion Floquet circuits and on protected information transport, even though the setting remains free-fermionic and the approximate modes become exact only as γ o∞.

major comments (1)
  1. The central claim of ballistic propagation and the QI-routing protocol rest on approximate SZM that become exact only for γ o∞ (Eqs. 16–17, 20–21). While the authors correctly state this, the main text and SM F/H present numerical evidence only for a single large value γ=30 and for two parity-preserving noise channels. A short quantitative scan of the residual commutator ||[UF,Ψ̃]|| versus γ (or of the fidelity of the transfer protocol versus γ) would make the practical range of the construction transparent and would strengthen the load-bearing claim that the modes remain useful at finite but large mass ratio.
minor comments (4)
  1. The abstract and introduction use both “Dzyaloshinskii Moriya” and “Dzyaloshinskii–Moriya”; a single hyphenated spelling should be adopted throughout.
  2. Figure 2 caption and SM F refer to “local magnetization” ⟨Zj⟩ as the diagnostic of the SZM; a one-sentence reminder that this is the (2j-1,2j) entry of the covariance matrix would help readers unfamiliar with the Gaussian formalism.
  3. In SM G the three-step transfer protocol is described clearly, but the explicit values of the boundary parameter β and of the mass string after layer L-1 are given only in prose; a short table or circuit diagram annotation would improve reproducibility.
  4. References [39], [41], [42], [44], [45] appear with future arXiv numbers or “sciPost submission” labels; these should be updated or flagged as preprints at the time of publication.

Circularity Check

1 steps flagged

Minor non-load-bearing self-citation of GFS foundation from authors' prior work; new SZM kernel, micromotion dressing, ballistic propagation and QI protocol are derived independently from the free-fermion circuit unitary.

specific steps
  1. self citation load bearing [Introduction / Strong zero modes section, Eqs. (4)–(5) and citation [35]]
    "building on our previous results [35], we define a discrete-time evolution operator UF by choosing the unitary scattering matrix of a (1+1)-dimensional supersymmetric quantum field theory … For PBC, the unitaries UF(α,{mi},θ) admit left and right global fermionic symmetries (GFS) … QL,R({mi},θ)=NL,R ∑i √mi e±θi/2 χA,B i … For OBC … QT=QL+QR survives as GFS … [UF,QT]=0."

    The existence and form of the delocalized SZM ΨQ ≡ QT is taken from the authors' own prior construction rather than re-derived from scratch. This is ordinary background citation and does not force the new four-dimensional kernel, the micromotion-induced approximate modes, or the ballistic propagation; those are obtained independently from the large-γ limit of the circuit rotations and the dressed Heff. Hence the step is minor and non-load-bearing for the paper's strongest claims.

full rationale

The paper's central claims (four-dimensional kernel of A/OF in the large-γ Hamiltonian limit, micromotion-dressed Heff producing one localized and one ballistically propagating approximate SZM, and the explicit finite-depth QI-transfer circuit) follow by direct calculation from the matchgate brick-wall unitary, the SO(2L) Majorana rotation OF, and the zeroth-order partial products P(0)n (Eqs. 7–13 and SM E). These steps do not reduce to a fit, a redefinition, or a uniqueness theorem. The only self-citation is the importation of the global fermionic symmetries QL,R (and their OBC combination QT) from the authors' prior paper [35]; that citation supplies the starting delocalized SZM but is not used to force the existence, localization, or propagation of the additional approximate modes that constitute the novelty. No parameter is fitted to data and then re-presented as a prediction, and the free-fermion algebra plus Gaussian numerics are self-contained. Score 1 reflects a single non-load-bearing self-citation of background structure.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 1 invented entities

The central claims rest on the free-fermion character of matchgates, the existence of global fermionic symmetries inherited from the underlying SUSY QFT, and the large-γ expansion that produces the micromotion permutation. Circuit parameters α, γ, θ, β are free choices, not fitted numbers. No new particles or forces are postulated; the SZMs are linear combinations of ordinary Majoranas.

free parameters (4)
  • interaction strength α
    Fixed by hand (usually α=1) for all gates; controls localization length of the edge mode when α≠1.
  • mass ratio γ = log(M/m)
    Taken large (γ=30 in numerics) to make approximate SZMs nearly exact; the infinite-γ limit is an idealization.
  • rapidity difference θ
    Taken small (θ=0.1) to enter the Hamiltonian limit; higher-order terms in θ are neglected.
  • boundary parameter β = ±1
    Chosen by hand to pin the localized mode to the left or right edge and to steer the transfer protocol.
axioms (3)
  • standard math Matchgates generate free-fermion dynamics that can be represented exactly by SO(2L) rotations of Majorana operators.
    Standard result (Valiant, Terhal–DiVincenzo, Jozsa–Miyake) used throughout to reduce the many-body problem to linear algebra on 2L dimensions.
  • domain assumption The left and right global fermionic symmetries of the periodic circuit survive as the single combination QT under open boundaries with β=1.
    Taken from the authors’ prior work and from the SUSY QFT literature; verified by direct commutation for the brick-wall layout.
  • ad hoc to paper In the simultaneous limits θ→0 and γ→∞ the zeroth-order bricks reduce to pure Majorana permutations generated by a Dzyaloshinskii–Moriya term.
    Derived in SM E; the infinite-γ idealization is essential for the four-dimensional kernel and for exact ballistic transport.
invented entities (1)
  • ballistically propagating strong zero mode (tilde-Psi_Q / tilde-Psi_loc) no independent evidence
    purpose: Provides a conserved Majorana that can be steered across the register by the mass configuration, enabling the QI-transfer protocol.
    Constructed explicitly as a linear combination of Majoranas that lies in the kernel of the micromotion-dressed generator A; no independent experimental evidence outside the free-fermion model is claimed.

pith-pipeline@v1.1.0-grok45 · 23686 in / 2697 out tokens · 28937 ms · 2026-07-14T05:16:59.134347+00:00 · methodology

0 comments
read the original abstract

We investigate discrete dynamics in quantum circuits with gates corresponding to the S-matrix of a supersymmetric 1+1D quantum field theory. We show that for a brick-wall configuration such circuits support both localized and delocalized dynamically conserved operators known as strong zero modes (SZM), the number of which depends on the parameter regime. We demonstrate that, while some of the SZM remain localized at boundaries, other SZM propagate ballistically, guided by a choice of circuit parameters. Such propagation can be explained by a strong Dzyaloshinskii Moriya term appearing in the dynamics. We describe how to exploit propagating SZM for quantum information transport and discuss the robustness to various types of noise.

Figures

Figures reproduced from arXiv: 2607.11483 by Alberto Zorzato, Jiri Min\'a\v{r}, Kareljan Schoutens, Pietro Richelli.

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. Evolution of local magnetization under [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. Study of the SZM signature for different parity preserving noises for the evolution of Γ [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

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

75 extracted references · 18 linked inside Pith

  1. [1]

    we show the numerical resilience of this magnetiza- tion signature under parity-preserving perturbations. For the staggered mass configuration we find, again in the largeγlimit, localization of SZM on the right boundary and on the rightmost location where unequal massesmandMcome together. QI routing protocols –Our findings for ballistic prop- agation of S...

  2. [2]

    Lloyd, Science273, 1073 (1996), https://www.science.org/doi/pdf/10.1126/science.273.5278.1073

    S. Lloyd, Science273, 1073 (1996), https://www.science.org/doi/pdf/10.1126/science.273.5278.1073

    doi:10.1126/science.273.5278.1073 1996

  3. [3]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, 5 Phys. Rev. X11, 011020 (2021)

    2021

  4. [4]

    H. Zhao, M. Bukov, M. Heyl, and R. Moessner, PRX Quantum4, 030319 (2023)

    2023

  5. [5]

    M. Heyl, P. Hauke, and P. Zoller, Sci- ence Advances5, eaau8342 (2019), https://www.science.org/doi/pdf/10.1126/sciadv.aau8342

    doi:10.1126/sciadv.aau8342 2019

  6. [6]

    Vernier, B

    E. Vernier, B. Bertini, G. Giudici, and L. Piroli, Phys. Rev. Lett.130, 260401 (2023)

    2023

  7. [7]

    Sels and A

    D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences114, E3909 (2017), https://www.pnas.org/doi/pdf/10.1073/pnas.1619826114

    doi:10.1073/pnas.1619826114 2017

  8. [8]

    P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Phys. Rev. Lett.123, 090602 (2019)

    2019

  9. [9]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Phys. Rev. X7, 031016 (2017)

    2017

  10. [10]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X8, 021014 (2018)

    2018

  11. [11]

    Sierant, M

    P. Sierant, M. Schir` o, M. Lewenstein, and X. Turkeshi, Phys. Rev. Lett.131, 230403 (2023)

    2023

  12. [12]

    Zhou and A

    T. Zhou and A. Nahum, Phys. Rev. B99, 174205 (2019)

    2019

  13. [13]

    M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vijay, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

    2023

  14. [14]

    P. W. Claeys and A. Lamacraft, Phys. Rev. Lett.126, 100603 (2021), arXiv:2009.03791 [quant-ph]

    Pith/arXiv arXiv 2021

  15. [15]

    Bertini and L

    B. Bertini and L. Piroli, Phys. Rev. B102, 064305 (2020)

    2020

  16. [16]

    G. Q. AI and Collaborators, Nature646, 825 (2025)

    2025

  17. [17]

    A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris,et al., Science388, 199 (2025)

    2025

  18. [18]

    Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zale- tel, K. Temme,et al., Nature618, 500 (2023)

    2023

  19. [19]

    Tindall, M

    J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels, PRX Quantum5, 010308 (2024)

    2024

  20. [20]

    Beguˇ si´ c, J

    T. Beguˇ si´ c, J. Gray, and G. K.-L. Chan, Science Ad- vances10, eadk4321 (2024)

    2024

  21. [21]

    M. A. Rampp, P. W. Claeys, A. Chan, and V. Khemani, Phys. Rev. Lett.130, 130402 (2023), arXiv:2208.00329 [quant-ph]

    Pith/arXiv arXiv 2023

  22. [22]

    Krajnik, E

    ˇZ. Krajnik, E. Ilievski, and T. Prosen, SciPost Physics9, 038 (2020), arXiv:2003.05957

    Pith/arXiv arXiv 2020

  23. [23]

    Rosenberg, T

    E. Rosenberg, T. I. Andersen,et al., Science384, 48 (2024), arXiv:2306.09333

    Pith/arXiv arXiv 2024

  24. [24]

    Hudomal, R

    A. Hudomal, R. Smith, A. Hallam, and Z. Papi´ c, PRX Quantum5, 010316 (2024), arXiv:2307.13042

    Pith/arXiv arXiv 2024

  25. [25]

    Paletta and T

    C. Paletta and T. Prosen, SciPost Physics18, 027 (2025), arXiv:2406.12695

    Pith/arXiv arXiv 2025

  26. [26]

    Y. Miao, V. Gritsev, and D. V. Kurlov, SciPost Phys. 16, 078 (2024)

    2024

  27. [27]

    Piroli, B

    L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Phys. Rev. B101, 094304 (2020), arXiv:1911.11175 [cond- mat.stat-mech]

    Pith/arXiv arXiv 2020

  28. [28]

    Miet al., Science378, 785 (2022)

    X. Miet al., Science378, 785 (2022)

    2022

  29. [29]

    Zadnik, M

    L. Zadnik, M. Ljubotina, i. c. v. Krajnik, E. Ilievski, and T. c. v. Prosen, PRX Quantum5, 030356 (2024)

    2024

  30. [30]

    Vanicat, L

    M. Vanicat, L. Zadnik, and T. Prosen, Physical Review Letters121, 030606 (2018), arXiv:1712.00431

    Pith/arXiv arXiv 2018

  31. [31]

    Hutsalyuk, Y

    A. Hutsalyuk, Y. Jiang, B. Pozsgay, H. Xu, and Y. Zhang, SciPost Physics19, 003 (2025), arXiv:2405.16070

    Pith/arXiv arXiv 2025

  32. [32]

    Ljubotina, L

    M. Ljubotina, L. Zadnik, and T. Prosen, Physical Review Letters122, 150605 (2019), arXiv:1901.05398

    Pith/arXiv arXiv 2019

  33. [33]

    H¨ ubner, E

    F. H¨ ubner, E. Vernier, and L. Piroli, SciPost Physics18, 135 (2025), arXiv:2408.00474

    Pith/arXiv arXiv 2025

  34. [34]

    Suzuki, Physics Letters A146, 319 (1990)

    M. Suzuki, Physics Letters A146, 319 (1990)

    1990

  35. [35]

    Suzuki, Physics Letters A165, 387 (1992)

    M. Suzuki, Physics Letters A165, 387 (1992)

    1992

  36. [36]

    Richelli, K

    P. Richelli, K. Schoutens, and A. Zorzato, SciPost Phys. 17, 087 (2024)

    2024

  37. [37]

    Fendley, Journal of Statistical Mechanics: Theory and Experiment2012, P11020 (2012)

    P. Fendley, Journal of Statistical Mechanics: Theory and Experiment2012, P11020 (2012)

    2012

  38. [38]

    Fendley, Journal of Physics A: Mathematical and The- oretical49, 30LT01 (2016)

    P. Fendley, Journal of Physics A: Mathematical and The- oretical49, 30LT01 (2016)

    2016

  39. [39]

    A. S. Jermyn, R. S. K. Mong, J. Alicea, and P. Fendley, Phys. Rev. B90, 165106 (2014)

    2014

  40. [40]

    F. H. L. Essler, P. Fendley, and E. Vernier (2026), sciPost submission

    2026

  41. [41]

    F. Jin, S. Jiang, X. Zhu, Z. Bao, F. Shen, K. Wang, Z. Zhu, S. Xu, Z. Song, J. Chen, Z. Tan, Y. Wu, C. Zhang, Y. Gao, N. Wang, Y. Zou, A. Zhang, T. Li, J. Zhong, Z. Cui, Y. Han, Y. He, H. Wang, J.-N. Yang, Y. Wang, J. Shen, G. Liu, J. Deng, H. Dong, P. Zhang, W. Li, D. Yuan, Z. Lu, Z.-Z. Sun, H. Li, J. Zhang, C. Song, Z. Wang, Q. Guo, F. Machado, J. Kemp,...

    2025

  42. [42]

    Gehrmann and F

    S. Gehrmann and F. H. L. Essler, arXiv:2511.05490 [cond-mat] (2026)

    Pith/arXiv arXiv 2026

  43. [43]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Strong Zero Modes via Commutant Algebras (2026), 2603.02326 [cond-mat]

    arXiv 2026

  44. [44]

    Vernier, H.-C

    E. Vernier, H.-C. Yeh, L. Piroli, and A. Mitra, Phys. Rev. Lett.133, 050606 (2024)

    2024

  45. [45]

    Gehrmann, Exact strong zero modes are generic in integrable spin systems with large anisotropy (2026), arXiv:2605.26205 [cond-mat.stat-mech]

    S. Gehrmann, Exact strong zero modes are generic in integrable spin systems with large anisotropy (2026), arXiv:2605.26205 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2026

  46. [46]

    Kantha and N

    S. Kantha and N. Laflorencie, Strong zero modes in random Ising-Majorana chains (2026), arXiv:2603.05313 [cond-mat.dis-nn]

    arXiv 2026

  47. [47]

    C. T. Olund, N. Y. Yao, and J. Kemp, Phys. Rev. B111, L201114 (2025), arXiv:2305.16382 [quant-ph]

    Pith/arXiv arXiv 2025

  48. [48]

    Schoutens, Nuclear Physics B344, 665 (1990)

    K. Schoutens, Nuclear Physics B344, 665 (1990)

    1990

  49. [49]

    L. G. Valiant, inProceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC ’01 (Association for Computing Machinery, New York, NY, USA, 2001) pp. 114–123

    2001

  50. [50]

    L. G. Valiant, Theoretical Computer Science289, 457 (2002)

    2002

  51. [51]

    B. M. Terhal and D. P. DiVincenzo, Physical Review A 65, 032325 (2002)

    2002

  52. [52]

    Jozsa and A

    R. Jozsa and A. Miyake, Proceedings of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 464, 3089 (2008)

    2008

  53. [53]

    Jordan and E

    P. Jordan and E. Wigner, Zeitschrift f¨ ur Physik47, 631 (1928)

    1928

  54. [54]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics16, 407 (1961)

    1961

  55. [55]

    C. D. Batista and G. Ortiz, Physical Review Letters86, 1082 (2001)

    2001

  56. [56]

    strong zero modes in supersymmetry-inspired quantum circuits

    A. Zorzatoet al., Supplemental material for “strong zero modes in supersymmetry-inspired quantum circuits” (2026), supplemental Material to the present manuscript

    2026

  57. [57]

    Moriconi and K

    M. Moriconi and K. Schoutens, Nuclear Physics B487, 756 (1997)

    1997

  58. [58]

    Chepiga and F

    N. Chepiga and F. Mila, Physical Review B107, L081106 (2023)

    2023

  59. [59]

    Chepiga and N

    N. Chepiga and N. Laflorencie, SciPost Physics14, 152 (2023)

    2023

  60. [60]

    Goldman and J

    N. Goldman and J. Dalibard, Physical Review X4, 6 031027 (2014), arXiv:1404.4373 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2014

  61. [61]

    J. K. Asb´ oth, B. Tarasinski, and P. Delplace, Physi- cal Review B90, 125143 (2014), arXiv:1405.1709 [cond- mat.mes-hall]

    Pith/arXiv arXiv 2014

  62. [62]

    M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Physical Review X3, 031005 (2013), arXiv:1212.3324 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2013

  63. [63]

    Bukov, L

    M. Bukov, L. D’Alessio, and A. Polkovnikov, Ad- vances in Physics64, 139 (2015), arXiv:1407.4803 [cond- mat.quant-gas]

    Pith/arXiv arXiv 2015

  64. [64]

    Surace and L

    J. Surace and L. Tagliacozzo, SciPost Phys. Lect. Notes , 54 (2022). 7 Supplementary Materials to the results in the manuscript Strong Zero Modes in Supersymmetry-Inspired Quantum Circuits A. Zorzato, P. Richelli, J. Min´ aˇ r and K. Schoutens T able of contents A- Fermionic Gaussian Formalism B- Jordan-Wigner mapping of the GFS charges C- Explicit expres...

    2022

  65. [65]

    Covariance matrix and efficient simulation of key observables To make the connection to correlators, we introduce the covariance matrix Γ µν for a generalL-qubit state Γµν = i 2 Tr ρ[χ µ, χν] = i 2 ⟨[χµ, χν]⟩=−Im⟨χ µχν⟩,Γ T =−Γ.(A5) Evolving the state with the Gaussian unitaryUamounts to the following update of the covariance matrix, Γ→RΓR T.(A6) For an o...

  66. [66]

    The building blocks are 2-qubit unitaries ˇSi,i+1(α, γ, θ) and 1-qubit boundary unitariesK i(β, θ)

    Evolution operatorO F for brick-wall Floquet operators Object of study in this paper are the evolution operatorsU F for brick-wall circuits with GFS, see SM [55] for detailed expressions. The building blocks are 2-qubit unitaries ˇSi,i+1(α, γ, θ) and 1-qubit boundary unitariesK i(β, θ). Passing to the Majorana basis, we associate Majorana operatorsχ 2j−1 ...

  67. [67]

    Let⃗ v 1, ⃗ v2 ∈Ω⊂R 2L be two orthonormal zero-mode vectors in the Majorana mode space

    Preparation of Gaussian states with zero-modes This routine, used in our numerics, constructs an initial Gaussian state by choosing the occupation of a Dirac fermion built from two Majorana zero-mode operators. Let⃗ v 1, ⃗ v2 ∈Ω⊂R 2L be two orthonormal zero-mode vectors in the Majorana mode space. As stated in the main text, these are coefficient vectors ...

  68. [68]

    PBC In the PBC case for the{M, m, M, m, . . .}configuration, the Floquet operator is given by the standard brick-wall construction U P F (α, γ, θ) =U odd Ueven = L/2Y i=1 ˇS2i,2i+1(α, γ, θ) L/2Y i=1 ˇS2i−1,2i(α, γ, θ) ,(C1) whereL+ 1≡1. By construction, the evolution preserves the fermionic charges for all parameter values: [U P F (α, γ, θ), QL/R(γ, θ)] = 0.(C2)

  69. [69]

    In all OBC realizations the left and right boundary matrices are evaluated at∓θ/2 respectively

    OBC,γ= 0 U O F (α,0, θ) =K 1(−θ/2) L/2−1Y i=1 ˇS2i,2i+1(α,0, θ) KL(θ/2) L/2Y i=1 ˇS2i−1,2i(α,0, θ) ,(C3) satisfies the commutation relation [U O F (α,0, θ), Q T (0, θ)] = 0, withQ T =Q L +Q R. In all OBC realizations the left and right boundary matrices are evaluated at∓θ/2 respectively

  70. [70]

    OBC,γ̸= 0 In this case satisfying the commutation requires 2Llayers applied in a layout like Fig. 1(c). An initial string{m i}, sets the values of{γ i}={γ 1, γ2, . . . , γT }in theT=L(L−1) two-qubit gates throughout the 2Llayers according to (M, M) or (m, m) :γ i = 0, (M, m) :γ i = +γ, (m, M) :γ i =−γ.(C4) For evenL, the number of two-qubit gates in layer...

  71. [71]

    Pauli Basis ˇS(α, γ, θ) =N(α, γ, θ) −1 M(α, γ, θ) (D1) M(α, γ, θ) =   2αcosh γ 2 csch(θ) +i0 0αsech θ 2 0αcsch θ 2 2αsinh γ 2 csch(θ) +i0 0−2αsinh γ 2 csch(θ) +i αcsch θ 2 0 −αsech θ 2 0 0 2αcosh γ 2 csch(θ)−i   (D2) N(α, γ, θ) = q 2α2 cosh(γ) csch2(θ) + 2α2 coth(θ) csch(θ) + 1 (D3) K(β, θ) =   √ 2 cos π 4 + iβθ 2 p sech(βθ) 0 0 √ 2 cos π 4 − iβ...

  72. [72]

    (D6) Then ˇSM(α, γ, θ) =   A− −E− C++ F− E+ A+ −F− C+− C−− F+ A− −E+ −F+ C−+ E− A+   .(D7) K M(β, θ) = sech(βθ) tanh(βθ) −tanh(βθ) sech(βθ) .(D8) 11 E

    Majorana Basis D= 2α 2 coshγ+ coshθ + sinh2 θ, eD= 2α 2 csch2 θ coshγ+ coshθ + 1 = D sinh2 θ , A± = 4α2 cosh γ±θ 2 D , E ± = α csch θ 2 ±sech θ 2 eD , F ± = 2αe±γ/2 cschθ eD , C++ = 2α2 sinhγ+ sinhθ 2α2 + sinhθ D , C +− = 2α2 sinhγ+ sinhθ sinhθ−2α 2 D , C−+ = −2α2 sinhγ+ sinhθ 2α2 + sinhθ D , C −− = −2α2 sinhγ+ sinhθ sinhθ−2α 2 D . (D6) Then ˇSM(α, γ, θ) ...

  73. [73]

    , M, m}, (L−2) circuit layers guide the heavy mass from lineL−1 to line 1, so thatηevolves toη ′ = 1√ 8(χA 1 −χ B 1 +i(χ A L −χ B L ))

    Starting from mass configuration{m, m, . . . , M, m}, (L−2) circuit layers guide the heavy mass from lineL−1 to line 1, so thatηevolves toη ′ = 1√ 8(χA 1 −χ B 1 +i(χ A L −χ B L ))

  74. [74]

    Circuit layerL−1 has 1-qubit gatesF 1 andF L at lines 1 andL, which are such thatF 1F4η′ =ζ ′F1F4, with ζ ′ = 1√ 8(χA 1 +χ B 1 +i(χ A L +χ B L ))

  75. [75]

    To achieve this, we need to change the sign in the boundary gates, settingβ=−1, and re-initialize the mass configuration in circuit lineLas{m, m,

    The layersLthrough 2L−3 are such that one SZM is stuck at line 1, while the other evolves from lineLto line 2. To achieve this, we need to change the sign in the boundary gates, settingβ=−1, and re-initialize the mass configuration in circuit lineLas{m, m, . . . , m, M}. This fixes the SZM 1√ 2(χA 1 +χ B 1 ) at line 1, while it propagates the SZM 1√ 2(χA ...