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Ultrahigh threshold nonstabilizer nonlinear quantum error correcting code

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The spinor code encodes a qubit as N identical copies and claims to correct any single-qubit Pauli error, with a code-capacity threshold estimated between 32% and 75%.

desk verdict Genuinely novel spin-based QEC construction, but the ultrahigh threshold and asymptotic claims outrun the band-limited proof and tiny extrapolated numerics. read the letter →

arxiv 2506.10445 v2 pith:HSGONBKJ submitted 2025-06-12 quant-ph

Maga Grafe , Kaixuan Zhou , Zaman Tekin , Zhiyuan Lin , Sen Li , Fengquan Zhang , Valentin Ivannikov , Tim Byrnes This is my paper · ORCID
classification quant-ph PACS 03.67.Pp
keywords spinorcodenonlinearquantumerrorcorrectionnonstabilizertotalspinsectorscoherentstatesdepolarizingchannelthresholdKnill-Laflammeconditions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces the spinor code, a quantum error-correcting code that stores one qubit not as a linear superposition of code words but as a parameterized family of states: the N-fold copy $|\alpha,\beta\rangle\rangle = (\alpha|0\rangle+\beta|1\rangle)^{\otimes N}$, a spin coherent state. The code spaces are labeled by total spin; the error-free space is the maximal-spin sector $s=N/2$, and a single-qubit Pauli error moves the state to $s=N/2-1$, where a syndrome projection and a unitary rotation return it to the maximal-spin sector. For Gaussian-distributed states such as spin coherent states, the distortion introduced by this process becomes small as $N$ grows, and the paper argues the code asymptotically corrects any single-qubit Pauli error. Direct numerical simulation under a depolarizing channel yields an estimated code-capacity threshold in the range 32-75% and, with initialization and measurement errors, a phenomenological threshold in the range 9-75%. These numbers matter because symmetric-depolarizing stabilizer codes typically have thresholds near 1%, so a nonstabilizer nonlinear code with a far higher threshold would be a different route to fault tolerance.

What carries the argument

The load-bearing object is the decomposition of the $N$-qubit Hilbert space into total-spin sectors $|s,l,m\rangle$, with the code words $|C_m\rangle = |N/2,1,m\rangle$. The encoding maps quantum information into spin coherent states $|\alpha,\beta\rangle\rangle$, which live entirely in the maximal-spin sector; the syndrome is the projector $P_{sl}$ onto a sector, and the correction is the unitary $U_{sl}$ that rotates the projected state back to $s=N/2$. The argument is carried by the deformation factors $D_{sl}(m) = \langle s,l,m|\sigma_n^j|N/2,1,m\rangle$, which quantify how much the error-correction cycle distorts the Gaussian distribution of a spin coherent state; the paper shows these are nearly $m$-independent for states away from the poles, so the Knill-Laflamme conditions hold approximately with error $O(N^{-1/2})$.

What would settle it

Run the depolarizing-channel simulation for an equatorial spin coherent state ($\theta=\pi/2$) at $N=20,40,80$ and fixed $p=0.5$, and extrapolate the logical error rate $\gamma_L$ against $1/N$; if $\gamma_L$ does not go to zero in that extrapolation, the claimed 32-75% code-capacity threshold range is falsified.

Watch

Extended reading notes

Core claim

The central claim is that total spin can serve as an error syndrome: every single-qubit Pauli operator changes the total spin of an $N$-qubit ensemble by at most one unit while leaving the magnetic quantum number with respect to the relevant axis effectively intact, so projecting onto the sector $(s,l)$ and rotating back to $s=N/2$ undoes the error. Because the deformation factors that describe the residual distortion are almost flat for spin coherent states far from the Bloch-sphere poles, the correction is only approximate; the paper proves an approximate Knill-Laflamme condition in the band $|m|\le \sqrt{N}$ with error $O(N^{-1/2})$, and numerically finds that the logical error rate extrapolates to zero for physical error probabilities $p\lesssim 0.32$. The same crossover argument puts an upper edge at $p=0.75$, where one round of depolarizing noise already maps every state to the maximally mixed state, so the paper reports a code-capacity threshold of 32-75%; including ancilla initialization and readout errors lowers the conservative estimate to 9-75%.

Load-bearing premise

The load-bearing premise is that the protected state is concentrated away from the poles of the Bloch sphere, because the approximate correction proof and the numerical protection both apply only where the magnetic quantum number is small; a state near a pole would not be correctable, and nonlinear encoding of an unknown qubit requires a no-cloning-free setting.

Editorial extensions

If this is right

  • For physical error probabilities below about 32%, the logical error rate $\gamma_L$ extrapolates to zero as $N$ grows, which the paper takes as the conservative code-capacity threshold.
  • The logical-error curves for different $N$ cross at $p=0.75$, the absolute upper edge of the threshold, because one depolarizing round at that rate maps every state to the maximally mixed state.
  • With ancilla initialization and measurement errors, the conservative phenomenological threshold falls to about 9%, while the crossover point remains at $p=0.75$.
  • In the equatorial band $|m|\le \sqrt{N}$ the code obeys approximate Knill-Laflamme conditions with error $O(N^{-1/2})$, so within that band the logical error vanishes at least as fast as $N^{-1/2}$ as $N\to\infty$.
  • Single-qubit logical gates are implemented by total-spin rotations, which are transversal products of single-qubit gates and therefore include non-Clifford rotations without extra magic-state procedures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the concave dependence of $\gamma_L$ on $1/N$ noted in the paper suggests the true threshold may lie closer to the 75% crossover than to the 32% linear extrapolation; a finite-size scaling study beyond $N=9$ would settle this.
  • Inference: the same sector-projection idea could be applied to non-Gaussian spin states or multipartite spinor states, where the amplitude near the poles is suppressed and the $|m|\le\sqrt{N}$ limitation is less restrictive.
  • Inference: the paper's performance metric, the normalized spin-expectation distance in Eq. (7), is not the fidelity used in most QEC threshold comparisons; Appendix C shows fidelity for a nonlinear encoding has an $N$-dependent factor unrelated to stored information, so threshold comparisons across code families should be made with this metric caveat in mind.
  • Inference: because the nonlinear encoder in Eq. (1) is blocked by no-cloning for a general unknown qubit, the practical reach of the spinor code is limited to settings where the duplicated state is generated by parallel computation or is a known resource state, such as spin squeezed states used in metrology.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces a nonstabilizer, nonlinear quantum error-correcting code called the spinor code. A logical qubit is encoded as an N-fold tensor product of the same qubit state, i.e., as a spin coherent state, and syndrome information is obtained by projecting onto total-spin sectors. The authors argue that single-qubit Pauli errors are approximately correctable for Gaussian-distributed states such as spin coherent states, provide an approximate Knill-Laflamme theorem in Appendix E, and report direct simulations for depolarizing noise with N = 4 to 9. They infer a code-capacity threshold in the range 32-75% and a phenomenological threshold in the range 9-75%.

Significance. If the asymptotic claim were established, the spinor code would be a striking counterpoint to the usual ~1% thresholds for symmetric depolarizing noise, and the proposed nonstabilizer/nonlinear framework would be of considerable conceptual interest. The paper is inventive in treating all Pauli directions symmetrically via total-spin sectors, and it contains self-contained proof machinery in Appendix E plus direct density-matrix simulations. However, the formal result covers only a shrinking band of m-values, the verification for the depolarizing channel relies on an unproved diagonal approximation, and the numerical threshold estimates extrapolate from very small systems. The significance is therefore prospective rather than established.

major comments (3)
  1. [Appendix E, Eqs. (E7)-(E8); Section IV, Eq. (4)] Theorem 1 establishes an approximate Knill-Laflamme condition only for the band B_N = {|m| ≤ sqrt(N)}, i.e., for the projector P_N onto that band. The actual encoded states (4) have support on all m ∈ [-N/2, N/2]. For an equatorial spin coherent state, the binomial distribution gives P(|m| > sqrt(N)) → 1 - erf(2) ≈ 4.6% as N → ∞, a constant tail not covered by the theorem. Consequently Eq. (E8) does not imply that the logical error rate vanishes for the states the paper claims to protect, and the abstract's statement that the code is asymptotically capable of protecting spin coherent states is not supported. The authors themselves note in Section III that states near the poles are more susceptible, and Fig. 4(a) shows reduced protection there; the manuscript needs either an explicit bound on the tail contribution or a restriction of the claim to states with negligible pole amplitude.
  2. [Appendix E, Example 1, text after Eq. (E25)] The verification of the theorem's hypotheses for the depolarizing channel is not valid as written. The text says that because σx and σy shift m by at most ±1, one may replace δ_{m,m'} by δ_{m,m±1} up to O(1/N), effectively treating the |m - m'| = 1 matrix elements as diagonal. But for code words |C_m⟩ with m in B_N, the deformation factor D^x_m ≈ 1/2, so the off-diagonal matrix elements ⟨C_{m+1}|σx|C_m⟩ are O(1), not O(1/sqrt(N)). Thus the key assumption (E2) is not satisfied by the original Kraus operators σx and σy. The approximate Knill-Laflamme bound therefore does not follow for the depolarizing channel as stated.
  3. [Section V, Fig. 4(c) and 4(e)] The lower bounds of the thresholds, p ≲ 0.32 and p ≲ 0.09, are obtained by linearly extrapolating γ_L versus 1/N using N = 4 to 9, with no error bars or alternative scaling models. Because the paper explicitly acknowledges that approximate QEC codes can saturate at a nonzero logical error rate, the data cannot distinguish γ_L(∞) = 0 from a small positive floor. Given the constant tail mass identified in Appendix E, such a floor is a real possibility. The central claim that the spinor code asymptotically suppresses logical errors below threshold therefore requires either substantially larger N simulations or a rigorous bound showing γ_L → 0.
minor comments (4)
  1. [Abstract and Section V] The abstract states that two-qubit errors were evaluated, but I could not find any simulation or analysis of two-qubit errors in Section V or the appendices; either the relevant results should be added or the claim should be removed.
  2. [Section III, Eq. (14)] The code distance d = N/2 - m_max is positive only when the m-range is restricted, yet the subsequent analysis drops the restriction and uses the full range m ∈ [-N/2, N/2]. The relation between this distance and the reported threshold behavior should be clarified.
  3. [Section II C, Eq. (7)] The logical error measure (7) uses only the total-spin expectation values. This is a reasonable choice for nonlinear encoding, but it can be insensitive to some state deformations; the discussion in Appendix C explains the motivation, yet the caveat that (7) is not a full state-distance measure should be stated in the main text.
  4. [Appendix E, Lemma 1] The tradeoff lemma is stated informally with exponents such as α and γ without a precise statement; formalizing the lemma would make the bandwidth-versus-convergence discussion more rigorous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the threshold estimates come from direct simulation and a self-contained approximate Knill-Laflamme argument, with self-citations used only as motivating context.

full rationale

The code-capacity and phenomenological threshold estimates are produced by the paper's own simulation protocol in Sec. V (steps 0-4) using explicit density-matrix evolution, syndrome projection, and correction (32), with no fitted parameter inserted to force the central result. The upper bound p < 0.75 is not a fitted claim: the text states that the crossing point is the depolarizing-channel point where one application maps every state to (I/2)^{otimes N}, so it is a property of the channel, not an input to the code. The lower bound p_th ~ 0.32 comes from a stated, deliberately conservative linear extrapolation of the simulated logical-error rates in 1/N (Fig. 4(c)), and the paper itself flags the possibility that approximate QEC rates can saturate. The asymptotic-protection argument rests on the approximate Knill-Laflamme theorem in Appendix E, whose hypotheses (E2)-(E4) are verified in Example 1 for the depolarizing channel; that verification is performed with the paper's own deformation factors and matrix-element bounds. Even if those bounds or the band restriction |m| <= sqrt(N) turn out to be insufficient for the full spin-coherent-state claim, that would be a correctness gap, not a circularity: the conclusion is not assumed in the premises by definition. The self-citations to [39] (nonlinear QEC) and [50] (spinor states) motivate the terminology and the encoding family, but the load-bearing calculations---deformation factors (17), the KL matrix (E25), and the direct numerical evaluation---are carried out within this manuscript rather than imported from those papers. No equation is shown to reduce to another by construction, no fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to forbid alternatives. The paper's own limitation statements, such as caution about the crossover test for approximate QEC, are explicitly weighed and are consistency caveats rather than circular steps.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The code introduces no new physical particles or forces. The main assumptions are the availability of the nonlinear encoding for unknown states and the restriction to states whose amplitudes lie mostly inside the equatorial band. The threshold estimates are obtained by fitting simulated logical error rates to a linear dependence on 1/N.

free parameters (2)
  • code-capacity threshold lower bound = ~0.32
    Obtained by linear extrapolation in 1/N of simulated logical error rates for N=4 to 9; the upper bound 0.75 is the exact depolarizing crossover where the channel maps all states to the maximally mixed state.
  • phenomenological threshold lower bound = ~0.09
    Obtained by the same linear extrapolation with initialization and measurement errors set to pm=pi=p; not a rigorous threshold.
assumptions (4)
  • domain assumption The nonlinear encoding (1) is physically available for unknown qubits in the parallel-computer scenario.
    Invoked in Sec. II A to circumvent the no-cloning theorem; if this scenario is not accepted, the code cannot encode arbitrary unknown qubits.
  • domain assumption Approximate correction in the band |m|≤√N implies correction for the actual spin coherent states used.
    Appendix E proves approximate Knill-Laflamme conditions only for code words in the band; the paper assumes Gaussian states have negligible weight outside the band, which fails near the poles.
  • domain assumption Total-spin projective syndrome measurements and correction unitaries are implementable.
    Sec. VI states that gate-level implementation is future work; no physical implementation is provided.
  • standard math The degeneracy-label basis |s,l,m> is well defined as in Appendix B.
    Used to define the syndrome projectors (5); follows from standard angular momentum decomposition.

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Cite this review

Pith. "Pith review of Ultrahigh threshold nonstabilizer nonlinear quantum error correcting code." pith.science (2026). https://pith.science/paper/HSGONBKJ

@misc{pith2026250610445,
  author       = {Pith},
  title        = {Pith review of: Ultrahigh threshold nonstabilizer nonlinear quantum error correcting code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSGONBKJ}},
  note         = {Machine review of arXiv:2506.10445}
}
read the original abstract

We introduce a novel type of quantum error correcting code, called the spinor code, based on spaces defined by total spin. The code is a nonstabilizer code, and is also a nonlinear quantum error correcting code, meaning that quantum information is encoded in a parameterized family of quantum states, rather than a linear superposition of code words. Syndrome measurements are performed by projecting on states with differing total spin, with an associated correction to map states back to the maximum total spin space. We show that the code is asymptotically capable of protecting against any single qubit Pauli error for Gaussian distributed states such as spin coherent state. We directly evaluate the performance under the depolarizing channel, considering various cases, with and without initialization and measurement errors, as well as two qubit errors. We estimate the code-capacity threshold to be in the range of 32-75%, while the phenomenological threshold is in the range 9-75%.

Figures

Figures reproduced from arXiv: 2506.10445 by the authors.

Figure 1
Figure 1. FIG. 1. (a) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Deformation factors (16) for various [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Performance of the spinor code. (a) Logical error [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hybrid Quantum Error Correction and Mitigation by Purification

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    A SWAP-test purification protocol that estimates observables of ρ^N from noisy copies without postselection; the claimed mid-circuit quantum error correction is not established because no physical purified state is produced.

Reference graph

Works this paper leans on

82 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    Considering σz-errors first, we have Pslσz n|ψ⟩ = s∑ m=−s ψmD(n) sl (m)|s,l,m ⟩, (15) whereψm = ⟨N 2, 1,m |ψ⟩ are the original amplitudes

    Phase flip errors To understand the way in which errors act on the spinor code, let us evaluate the effect of single qubit er- rors on a general state in the error-free code space after the syndrome measurement. Considering σz-errors first, we have Pslσz n|ψ⟩ = s∑ m=−s ψmD(n) sl (m)|s,l,m ⟩, (15) whereψm = ⟨N 2, 1,m |ψ⟩ are the original amplitudes. The facto...

  2. [2]

    Bit flip errors Now let us consider errors other than phase-flip errors. Considering bit flip errors, we have Pslσx n|ψ⟩ = s∑ m=−s ψ(x) m D(n) sl (m)|s,l,m ⟩(x) (18) where ψ(x) m = ⟨N 2, 1,m |ψ⟩ are the amplitudes in the x- basis and we defined |s,l,m ⟩(x) =e−iSyπ/2|s,l,m ⟩ (19) and the total spin eigenstates in the x-basis. These sat- isfy S2|s,l,m ⟩(x) =s(s...

  3. [3]

    The left column of Fig

    Q-functions Finally, to illustrate the effect of various errors acting on spin coherent states, we plot the Q-functions of the state |ψj sln⟩ =Pslσj n|α,β ⟩⟩, (23) which is a spin coherent state with an error in the j- direction, projected to the ( s,l ) subspace. The left column of Fig. 3 shows the Q-functions for errors in the j = x,y,z directions respec...

    work page

  4. [4]

    Initialize the state in ρ = |α,β ⟩⟩⟨⟨α,β |

    work page

  5. [5]

    Apply the depolarizing error channel ρ → 3∑ j=0 E(n) j ρ(E(n) j )† (31) for all n ∈ [1,N ]

    work page

  6. [6]

    Perform the error syndrome measurement and cor- rection ρ → N/2∑ s=0 Ls∑ l=1 UslPslρP † slU † sl. (32)

    work page

  7. [7]

    Measure the logical error (7)

    work page

  8. [8]

    The result of applying many such cycles of errors and QEC is shown in Fig

    Go to step 1. The result of applying many such cycles of errors and QEC is shown in Fig. 4(a). Here we plot the logical error as a function of the number of cycles t of our se- quence for various initial states with and without per- forming the QEC. For the case with no QEC, the pro- cedure is the same as above but we omit step 2 in the above sequence. Al...

    work page

Show all 82 references
  1. [9]

    The same curve is obtained for all ini- tial states for no QEC

    in the sequence. The same curve is obtained for all ini- tial states for no QEC. (b) Logical error rate γL as a func- tion of physical error probability for various code sizes N . The logical error rate is estimated by finding the derivative γL ≈ 2(ǫL(t = 1) −ǫL(t = 0)). The in...

  2. [10]

    P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical review A 52, R2493 (1995)

    1995

  3. [11]

    Gottesman, Stabilizer codes and quantum error cor- rection (California Institute of Technology, 1997)

    D. Gottesman, Stabilizer codes and quantum error cor- rection (California Institute of Technology, 1997)

    1997

  4. [12]

    Peres, Reversible logic and quantum computers, Phys- ical review A 32, 3266 (1985)

    A. Peres, Reversible logic and quantum computers, Phys- ical review A 32, 3266 (1985)

    1985

  5. [13]

    S. J. Devitt, W. J. Munro, and K. Nemoto, Quantum error correction for beginners, Reports on Progress in 10 Physics 76, 076001 (2013)

    2013

  6. [14]

    Roffe, Quantum error correction: an introductory guide, Contemporary Physics 60, 226 (2019)

    J. Roffe, Quantum error correction: an introductory guide, Contemporary Physics 60, 226 (2019)

    2019

  7. [15]

    B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307 (2015)

    2015

  8. [16]

    A. M. Steane, A tutorial on quantum error correction, Quantum Computers, Algorithms and Chaos , 1 (2006)

    2006

  9. [17]

    M. A. Nielsen and I. Chuang, Quantum computation and quantum information (2002)

    2002

  10. [18]

    P. W. Shor, Fault-tolerant quantum computation, in Pro- ceedings of 37th conference on foundations of computer science (IEEE, 1996) pp. 56–65

    1996

  11. [19]

    Aharonov and M

    D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, in Proceedings of the twenty-ninth annual ACM symposium on Theory of com- puting (1997) pp. 176–188

    1997

  12. [20]

    Knill, R

    E. Knill, R. Laflamme, and W. H. Zurek, Resilient quan- tum computation, Science 279, 342 (1998)

    1998

  13. [21]

    A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals of physics 303, 2 (2003)

    2003

  14. [22]

    D. G. Cory, M. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Experi- mental quantum error correction, Physical Review Let- ters 81, 2152 (1998)

    1998

  15. [23]

    Pittman, B

    T. Pittman, B. Jacobs, and J. Franson, Demonstration of quantum error correction using linear optics, Physical Review A—Atomic, Molecular, and Optical Physics 71, 052332 (2005)

    2005

  16. [24]

    Chiaverini, D

    J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Bar- rett, R. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, et al. , Realization of quantum error correction, Nature 432, 602 (2004)

    2004

  17. [25]

    Schindler, J

    P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, and R. Blatt, Ex- perimental repetitive quantum error correction, Science 332, 1059 (2011)

    2011

  18. [26]

    M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frun- zio, S. M. Girvin, and R. J. Schoelkopf, Realization of three-qubit quantum error correction with superconduct- ing circuits, Nature 482, 382 (2012)

    2012

  19. [27]

    N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas , B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, et al. , Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature 536, 441 (2016)

    2016

  20. [28]

    Ryan-Anderson, J

    C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, et al., Realization of real-time fault-tolerant quantum error correction, Physical Review X 11, 041058 (2021)

    2021

  21. [29]

    Krinner, N

    S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Her- rmann, et al. , Realizing repeated quantum error correc- tion in a distance-three surface code, Nature 605, 669 (2022)

    2022

  22. [30]

    W. P. Livingston, M. S. Blok, E. Flurin, J. Dressel, A. N. Jordan, and I. Siddiqi, Experimental demonstration of continuous quantum error correction, Nature communi- cations 13, 2307 (2022)

    2022

  23. [31]

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsiout- sios, S. Ganjam, A. Miano, B. Brock, A. Ding, L. Frun- zio, et al. , Real-time quantum error correction beyond break-even, Nature 616, 50 (2023)

    2023

  24. [32]

    Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023)

    2023

  25. [33]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al. , Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024)

    2024

  26. [34]

    R. S. Gupta, N. Sundaresan, T. Alexander, C. J. Wood, S. T. Merkel, M. B. Healy, M. Hillenbrand, T. Jochym- O’Connor, J. R. Wootton, T. J. Yoder, et al. , Encoding a magic state with beyond break-even fidelity, Nature 625, 259 (2024)

    2024

  27. [35]

    Paetznick, M

    A. Paetznick, M. da Silva, C. Ryan-Anderson, J. Bello- Rivas, J. Campora III, A. Chernoguzov, J. Dreiling, C. Foltz, F. Frachon, J. Gaebler, et al. , Demonstra- tion of logical qubits and repeated error correction with better-than-physical error rates, arXiv preprint arXiv:240...

    2024 arXiv

  28. [36]

    Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature 638, 920 (2025)

    2025

  29. [37]

    Raussendorf and J

    R. Raussendorf and J. Harrington, Fault-tolerant quan - tum computation with high threshold in two dimensions, Physical review letters 98, 190504 (2007)

    2007

  30. [38]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Physical Review A—Atomic, Molecular, and Optical Physics 86, 032324 (2012)

    2012

  31. [39]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43, 4452 (2002)

    2002

  32. [40]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low- overhead fault-tolerant quantum memory, Nature 627, 778 (2024)

    2024

  33. [41]

    Bartolucci, P

    S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Daw- son, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, et al. , Fusion-based quantum computation, Nature Communications 14, 912 (2023)

    2023

  34. [42]

    Bravyi, G

    S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, Subsystem surface codes with three-qubit check opera- tors, arXiv preprint arXiv:1207.1443 (2012)

    2012 arXiv

  35. [43]

    Higgott and N

    O. Higgott and N. P. Breuckmann, Subsystem codes with high thresholds by gauge fixing and reduced qubit over- head, Physical Review X 11, 031039 (2021)

    2021

  36. [44]

    J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T . Flammia, and B. J. Brown, The xzzx surface code, Na- ture communications 12, 2172 (2021)

    2021

  37. [45]

    Y. Wu, S. Kolkowitz, S. Puri, and J. D. Thompson, Era- sure conversion for fault-tolerant quantum computing in alkaline earth rydberg atom arrays, Nature communica- tions 13, 4657 (2022)

    2022

  38. [46]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultra- high error threshold for surface codes with biased noise, Physical review letters 120, 050505 (2018)

    2018

  39. [47]

    G. S. B. Smith, Upper and lower bounds on quantum codes (California Institute of Technology, 2006)

    2006

  40. [48]

    Reichert, L

    M. Reichert, L. W. Tessler, M. Bergmann, P. van Loock, and T. Byrnes, Nonlinear quantum error correction, Physical Review A 105, 062438 (2022)

    2022

  41. [49]

    Brion, L

    E. Brion, L. H. Pedersen, M. Saffman, and K. Mølmer, Error correction in ensemble registers for quantum re- peaters and quantum computers, Physical review letters 100, 110506 (2008). 11

    2008

  42. [50]

    Mohseni, M

    N. Mohseni, M. Narozniak, A. N. Pyrkov, V. Ivannikov, J. P. Dowling, and T. Byrnes, Error suppression in adi- abatic quantum computing with qubit ensembles, npj Quantum Information 7, 71 (2021)

    2021

  43. [51]

    Omanakuttan, V

    S. Omanakuttan, V. Buchemmavari, J. A. Gross, I. H. Deutsch, and M. Marvian, Fault-tolerant quantum com- putation using large spin-cat codes, PRX Quantum 5, 020355 (2024)

    2024

  44. [52]

    Omanakuttan and T

    S. Omanakuttan and T. Volkoff, Spin squeezed gkp codes for quantum error correction in atomic ensembles, in APS Division of Atomic, Molecular and Optical Physics Meet- ing Abstracts, Vol. 2023 (2023) pp. F01–054

    2023

  45. [53]

    Knill and R

    E. Knill and R. Laflamme, Theory of quantum error- correcting codes, Physical Review A 55, 900 (1997)

    1997

  46. [54]

    W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299, 802 (1982)

    1982

  47. [55]

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

    1996

  48. [56]

    P. W. Shor, Algorithms for quantum computation: dis- crete logarithms and factoring, in Proceedings 35th an- nual symposium on foundations of computer science (Ieee, 1994) pp. 124–134

    1994

  49. [57]

    Byrnes, K

    T. Byrnes, K. Wen, and Y. Yamamoto, Macroscopic quantum computation using Bose-Einstein condensates, Physical Review A 85, 040306(R) (2012)

    2012

  50. [58]

    Byrnes and E

    T. Byrnes and E. O. Ilo-Okeke, Quantum atom optics: Theory and applications to quantum technology (Cam- bridge university press, 2021)

    2021

  51. [59]

    Byrnes, Multipartite spin coherent states and spino r states, Physical Review A 109, 022438 (2024)

    T. Byrnes, Multipartite spin coherent states and spino r states, Physical Review A 109, 022438 (2024)

    2024

  52. [60]

    Gross, Spin squeezing, entanglement and quantum metrology with Bose–Einstein condensates, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 103001 (2012)

    C. Gross, Spin squeezing, entanglement and quantum metrology with Bose–Einstein condensates, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 103001 (2012)

    2012

  53. [61]

    C. C. Gerry and P. L. Knight, Introductory quantum op- tics (Cambridge university press, 2023)

    2023

  54. [62]

    A. M. Stephens, Fault-tolerant thresholds for quantum error correction with the surface code, Physical Review A 89, 022321 (2014)

    2014

  55. [63]

    A. G. Fowler, A. C. Whiteside, and L. C. Hollenberg, To- wards practical classical processing for the surface code, Physical review letters 108, 180501 (2012)

    2012

  56. [64]

    Zhao and D

    Y. Zhao and D. E. Liu, Extracting error thresholds through the framework of approximate quantum error correction condition, Physical Review Research 6 (2024)

    2024

  57. [65]

    Siwach and D

    P. Siwach and D. Lacroix, Filtering states with total sp in on a quantum computer, Physical Review A 104, 062435 (2021)

    2021

  58. [66]

    Jones and S

    T. Jones and S. C. Benjamin, Robust quantum compi- lation and circuit optimisation via energy minimisation, Quantum 6, 628 (2022)

    2022

  59. [67]

    Chen and T

    T. Chen and T. Byrnes, Efficient preparation of the aklt state with measurement-based imaginary time evolution, Quantum 8, 1557 (2024)

    2024

  60. [68]

    Byrnes, D

    T. Byrnes, D. Rosseau, M. Khosla, A. Pyrkov, A. Thomasen, T. Mukai, S. Koyama, A. Abdelrahman, and E. Ilo-Okeke, Macroscopic quantum information pro- cessing using spin coherent states, Optics Communica- tions 337, 102 (2015)

    2015

  61. [69]

    M. F. Riedel, P. B¨ ohi, Y. Li, T. W. H¨ ansch, A. Sina- tra, and P. Treutlein, Atom-chip-based generation of en- tanglement for quantum metrology, Nature 464, 1170 (2010)

    2010

  62. [70]

    Julsgaard, A

    B. Julsgaard, A. Kozhekin, and E. S. Polzik, Experimen- tal long-lived entanglement of two macroscopic objects, Nature 413, 400 (2001)

    2001

  63. [71]

    Y. Mao, M. Chaudhary, M. Kondappan, J. Shi, E. O. Ilo- Okeke, V. Ivannikov, and T. Byrnes, Measurement-based deterministic imaginary time evolution, Physical Review Letters 131, 110602 (2023)

    2023

  64. [72]

    Deutsch, F

    C. Deutsch, F. Ramirez-Martinez, C. Lacroˆ ute, F. Rein - hard, T. Schneider, J.-N. Fuchs, F. Pi´ echon, F. Lalo¨ e, J. Reichel, and P. Rosenbusch, Spin self-rephasing and very long coherence times in a trapped atomic ensemble, Physical review letters 105, 020401 (2010)

    2010

  65. [73]

    Eastin and E

    B. Eastin and E. Knill, Restrictions on transversal en- coded quantum gate sets, Physical review letters 102, 110502 (2009)

    2009

  66. [74]

    B. Zeng, A. Cross, and I. L. Chuang, Transversality ver- sus universality for additive quantum codes, IEEE Trans- actions on Information Theory 57, 6272 (2011)

    2011

  67. [75]

    Niset, J

    J. Niset, J. Fiur´ aˇ sek, and N. J. Cerf, No-go theorem fo r gaussian quantum error correction, Physical review let- ters 102, 120501 (2009)

    2009

  68. [76]

    A. G. Fowler, A. M. Stephens, and P. Groszkowski, High- threshold universal quantum computation on the surface code, Physical Review A—Atomic, Molecular, and Opti- cal Physics 80, 052312 (2009). Appendix A: Applications of the spinor code In this section, we discuss the situat...

    2009

  69. [77]

    Second, the amplitude of the states is no different to (4), the only difference is the phase factor eiξ(k−N/2)2

    The key things to note here are that only maximal total spin states are involved, which is a prerequisite for the use of the spinor code. Second, the amplitude of the states is no different to (4), the only difference is the phase factor eiξ(k−N/2)2 . Thus the arguments relating...

  70. [78]

    |s,l,m =s⟩

    Diagonalize S2 − Sz and obtain the eigenstates with maximal Sz eigenvalue, i.e. |s,l,m =s⟩

  71. [79]

    If the |s,l,s ⟩ are not mutually orthogonal with re- spect to the l-label, orthogonalize using a suitable procedure such as Gram-Schmidt orthogonaliza- tion such that |⟨s,l,s |s,l ′,s ⟩|2 =δll′

  72. [80]

    Repeating the above procedure for all spin sectors s ∈ {N/2,N/ 2 − 1,

    For all m ∈ [−s,s ] and l ∈ [1,L s], apply ladder operators and define |s,l,m ⟩ = (S−)s−m|s,l,s ⟩ √ ⟨s,l,s |(S+)s−m(S−)s−m|s,l,s ⟩ (B1) where S± =Sx ±iSy. Repeating the above procedure for all spin sectors s ∈ {N/2,N/ 2 − 1,.... } we obtain the full set of total spin eigenstate...

  73. [81]

    Let E = {E1,...,E r} be a finite set of error operators with local support, where the number of error operators r ≪ N with ‖Ei‖ ≤ 1

    General errors Theorem 1. Let E = {E1,...,E r} be a finite set of error operators with local support, where the number of error operators r ≪ N with ‖Ei‖ ≤ 1. The number of qubits is N ≥ 1. Define the band BN := {m ∈ Z : |m| ≤ ⌊ √ N ⌋} and the codewords |Cm⟩ := |N/2, 1,m ⟩. For ...

  74. [82]

    The depolarizing channel is defined as Dp(ρ) = 3∑ j=0 E(n) j ρ(E(n) j )† (E24) where the Kraus operators are defined according to (30)

    Depolarizing channel Example 1. The depolarizing channel is defined as Dp(ρ) = 3∑ j=0 E(n) j ρ(E(n) j )† (E24) where the Kraus operators are defined according to (30). For these errors we evaluate the Knill-Laflamme matrix as ⟨Cm′ |E† iEj |Cm⟩ ≈ δmm′       1 −p qD x m 0 qDz...

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