Pith. sign in

REVIEW 3 major objections 6 minor 64 references

Towards fault-tolerance with universal phase-error-transparent gates for high-spin cat codes

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A universal set of error-transparent logical gates can be constructed for the spin cat-x code, so phase errors that strike during gate operations stay correctable by a later error-correction step.

desk verdict Sound ET gate construction for spin cat-x codes, but the practical route to the logical X gate has a real physical error and the remaining schemes are underdeveloped. read the letter →

arxiv 2608.05992 v1 pith:YAZHOFOX submitted 2026-08-06 quant-ph

Kelvin Onggadinata , Si Yan Koh , Arghya Maity , Kuan Eng Johnson Goh , Bent Weber , Kay Jin Lim , Hui Khoon Ng , Teck Seng Koh This is my paper · ORCID
classification quant-ph
keywords spincatcodeerror-transparentgatesquantumerrorcorrectionhigh-spindonorsdephasingnoiseuniversalgatesetsiliconcomputingparitymeasurement
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

The paper tries to establish that a single high-spin nucleus in silicon, encoded in the spin cat-x code, can run a complete logical gate set while remaining transparent to the dominant dephasing noise: a phase error that lands during a gate moves the state into a known error subspace instead of scrambling it, and a subsequent error-correction step can still fix it. The authors construct the universal set $\{\bar{Z}(\theta), \bar{CZ}, \bar{X}(\phi)\}$ with generators that satisfy the error-transparency criterion $[\hat{E}_j,\hat{H}(t)]|\psi_L\rangle=0$ on the codespace, and they give the explicit Hamiltonian for the hardest member, the logical $X$ gate, as a sum of odd powers of the nuclear spin operator $\hat{I}_x$. Numerical simulations of these gates followed by noisy error correction show that the error-transparent gates outperform the corresponding non-transparent gates and can cross the break-even line set by an unencoded spin-1/2 qubit. This matters because donor-in-silicon quantum computing is a hardware-efficient route to error correction, but only if the logical operations themselves do not inject errors the code cannot handle; the paper also shows that measurement and a teleportation-based error-correction circuit can be built entirely from error-transparent components, while state preparation intrinsically cannot.

What carries the argument

The central objects are the spin cat-x codewords $|\mu_L\rangle = \frac{1}{\sqrt{2}}\left(|I,I\rangle_x + (-1)^\mu |I,-I\rangle_x\right)$ and their images under repeated dephasing, the errorwords $|E_k^\mu\rangle = \frac{(-1)^k}{\sqrt{2}}\left(|I,I-k\rangle_x + (-1)^\mu |I,-I+k\rangle_x\right)$, which span the logical subspace $k=0$ and the error subspaces $k=1,\dots,r$. The load-bearing identity is the error-transparency criterion $[\hat{E}_j,\hat{H}(t)]|\psi_L\rangle = 0$ for all codespace states $|\psi_L\rangle$, which guarantees that an error during the gate is equivalent to the same error applied after the gate, so a later error-correction round can remove it. For the amplitude-mixing gate this criterion forces $\hat{H}_X$ to contain flips inside every error subspace, which in the $x$-basis becomes a linear combination of odd powers $\hat{I}_x, \hat{I}_x^3, \dots, \hat{I}_x^{2I}$; the coefficients $c_n^{(I)}$ are fixed by solving a system of $\lfloor I\rfloor+1$ linear equations. The $Z$ gate is implemented as a virtual SNAP phase gate, and the $CZ$ gate uses electron-spin-resonance $2\pi$ pulses conditioned on the support states shared by $|1_L,1_L\rangle$ and all $|E_k^1,E_l^1\rangle$, with multi-tone driving collapsing the $(I+1/2)^2$ sequential pulses into one step.

What would settle it

A decisive test is to implement the proposed error-transparent $X(\pi/2)$ sequence on a spin-3/2 arsenic donor, apply one round of noisy error correction, and compare the logical entanglement fidelity against the idle-dephasing curve. If the measured infidelity lies above the idle curve or matches the non-ET $\hat{I}_x$ gate instead, the odd-power terms in the effective Hamiltonian are not being generated and the central claim fails; equivalently, a spectroscopy measurement of the effective Hamiltonian that shows only a linear $\hat{I}_x$ term would already rule out the rotating-frame route.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the spin cat-x code has a universal logical gate set whose generating Hamiltonians commute with every rank-$r$ dephasing error on the codespace, so an error occurring at any time during a gate acts as if it occurred at the end of the gate and remains correctable. The $Z(\theta)$ and $CZ$ gates are error-transparent by construction because they act only on the support states of $|1_L\rangle$, which are shared by all the errorwords $|E_k^1\rangle$. The $X(\phi)$ gate is the nontrivial case: the required Hamiltonian is $\hat{H}_X/\Omega = \frac{1}{2}\sum_{k=0}^{r}\left(|E_k^1\rangle\langle E_k^0| + |E_k^0\rangle\langle E_k^1|\right) = \sum_{n\ \mathrm{odd}} c_n^{(I)} \hat{I}_x^n$, with explicit coefficients for $I=3/2,5/2,7/2,9/2$ given in Eq. (20). Simulating the gate followed by a noisy error-correction round, the paper finds that the ET $X(\pi/2)$ gate reproduces the idle dephasing curve and crosses break-even, while the non-ET $\hat{I}_x$ implementation does not; the ET $CZ$ also crosses break-even, with multi-tone simultaneous ESR pulses giving substantially larger gain than sequential pulses. The paper further claims that the ET computational-basis measurement is exactly a spin-parity measurement, and that a teleportation-based error-correction circuit assembled solely from ET operations attains the ideal error-correction fidelity; state preparation, by contrast, cannot be made error-transparent.

Load-bearing premise

The load-bearing assumption is that at least one of the three proposed routes can really be built to generate the higher-order spin rotations that the error-transparent $X(\pi/2)$ gate demands, at usable speed and fidelity; the paper itself calls this gate the most technically demanding component and leaves all three routes open.

Editorial extensions

If this is right

  • A dephasing error that hits during any gate in the set is propagated into a known error subspace and removed by one round of error correction, so gate execution does not have to be faster than the dephasing time to be safe.
  • Under realistic donor parameters, the ET $X(\pi/2)$ and $CZ$ gates followed by noisy error correction cross the unencoded-qubit break-even when $\Gamma_n T_G$ lies in the $10^{-2}$–$10^{-1}$ window, and the ET gate curve coincides with the idle-dephasing curve.
  • The computational-basis logical measurement is realized by the already-demonstrated spin-parity measurement, so no separate readout gadget is needed.
  • A teleportation-based error-correction circuit built entirely from ET gates and ET measurements reaches the same fidelity as ideal error correction; substituting a non-ET dual-basis measurement prevents it from crossing break-even.
  • State preparation is the only non-ET ingredient, but optimized control pulses prepare the required spin-coherent state with fidelity $\gtrsim 99.9\%$, so a fully ET pipeline is blocked only by initialization.

Reading between the lines

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

  • Inference: Because the ET $X$ Hamiltonian is fixed purely by the code's error-subspace geometry, the same construction should transfer to any platform where a high-spin qudit with the spin cat-x encoding can be driven — molecular spins, neutral atoms, or trapped ions — by solving the same linear system for the appropriate $I$.
  • Inference: The paper's three proposed routes to the $X(\pi/2)$ gate have very different implementation costs and uncertainties; a direct experimental benchmark comparing the effective Hamiltonian produced by each route on an error-corrupted cat state would settle which route is viable before a full fault-tolerance demonstration.
  • Inference: Since the ET $CZ$ depends on an electron ancilla, the ancilla's dephasing sets a floor on logical fidelity; the phase diagram in Fig. 4c implies a quantitative prediction that no spin value reaches $G>1$ once the electron dephasing rate exceeds a threshold that grows with spin.
  • Inference: The spin cat-x code's membership in the family of rotationally symmetric spin codes suggests the ET $Z$ gate can be understood as a discrete phase-shift operator, which may let fault-tolerant circuit synthesis reuse constructions already developed for rotation-symmetric bosonic codes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper constructs a universal error-transparent (ET) logical gate set {Z(θ), CZ, X(ϕ)} for the spin cat-x code in high-spin donor systems (I = 3/2 to 9/2), protecting against rank-r dephasing errors E_z^[r] = {1, I_z, ..., I_z^r} via the condition [E_j, H(t)]|ψ_L> = 0. The Z and CZ gates are built from virtual SNAP phases and ESR-based conditional geometric phases; the X gate is specified by a Hamiltonian H_X that acts as logical X in every error subspace and is decomposed into odd powers of I_x with explicit coefficients for each donor species. The paper further constructs an ET parity-based logical Z measurement, adapts Knill's teleportation-based error correction from ET components only, and simulates gate-plus-correction fidelities showing that ET gates can cross the unencoded-qubit break-even. The physical realization of the X gate is identified as the central challenge, with three proposed routes: a generalized rotating-frame transformation, Lie-algebra control, and the matrix-element modification protocol.

Significance. The mathematical core of the paper is solid and deserves credit: the ET X-gate Hamiltonian of Eq. (10) is constructed so that [I_z^k, H_X]|ψ_L> = 0 holds by design on every error subspace; the coefficients c_n in Eq. (20) are solved from a linear system with no fitting to the performance targets; the ET Z gate, the ESR-based CZ gate, the parity realization of ET logical measurement, and the Knill-EC adaptation are all explicit; and the simulations benchmark against an unencoded spin-1/2 qubit with realistic donor parameters. The candid limitation statements in the Discussion — that X(π/2) implementation is open, that Knill-EC assumes noiseless gadgets, and that electron noise breaks full fault tolerance — are appropriately scoped and do not, by themselves, undermine the theoretical construction. The load-bearing weakness is practical universality: the only concrete X-gate implementation route (Eq. 12) is invalid, and the two alternative routes are sketches that do not establish error transparency; the X-gate simulations are consequently conditional on an unproven control Hamiltonian.

major comments (3)
  1. [Results, 'Error-transparent X gate' (Eqs. 11–12)] This implementation route is not valid as a derivation of a physical gate. Under the claimed frame transformation U_rf(t) = exp[-it(b_3 I_x^3 + b_5 I_x^5 + ···)], the rotating-frame Hamiltonian is H_R = U_rf† H_GRF U_rf − i U_rf† ∂_t U_rf = (γ_n B_1/2) I_x − (b_3 I_x^3 + b_5 I_x^5 + ···), and the physical evolution is U_S(T) = U_rf(T) U_R(T). Since U_rf is a function of I_x alone, it commutes with H_GRF, so U_S(T) = exp(−iT(γ_n B_1/2) I_x): the b_n terms cancel exactly and the implemented operation is the standard linear rotation, not the H_X of Eq. (10). A frame transformation cannot generate the I_x^n (n ≥ 3) terms required by Eq. (11) in the lab frame; those terms must be present in the physical drive Hamiltonian. Consequently, the matching of coefficients and the gate time T_{π/2} = π c_1^(I)/(γ_n B_1), including the numerical comparison (0.741 ms for the ET rotation), rest on an incorrect premise and should be removed or replaced by a genuine nonlinear-coupling scheme. The Discussion's candid statement that X(π/2) implementation remains open is appropriate, but it does not repair the incorrect derivation in the main text.
  2. [Results, 'Error-transparent X gate' (second and third approaches)] The Lie-algebra and MEM routes establish, at most, universal control of the qudit, not an error-transparent X(π/2). For the Lie-algebra route, Theorem 1 of Ref. [38] guarantees reachability of the unitary from the generators {I_x, I_y, I_x^2}, but the ET criterion (Eq. 7) requires [E_j, H(t)]|ψ_L> = 0 at every instant; [I_z, I_x] = i I_y maps P_0 into P_1, so none of the proposed elementary generators is ET and a concatenation of them cannot satisfy the instantaneous condition. For the MEM route, the generator M ∝ Σ_n c_n [I_-, I_z^n] is not of the H_X form of Eq. (10) and is not shown to commute with the error set on the codespace; the two open questions listed (adiabatic speed, coupling strength) do not address error transparency. The paper should either provide an explicit ET decomposition for one of these routes, with an error-transparency check, or state explicitly that no physical realization of H_X is currently known.
  3. [Results, 'ET gate performance with error correction' (Fig. 4b); Methods, 'Numerical simulation'] The X(π/2) performance results inherit the implementation gap. The master equation (Eq. 21) is solved with the gate Hamiltonian H(t), and for the X gate this is H_X of Eq. (10) with a gate time T_G obtained from the GRF coefficient matching that is invalid per Major Comment 1; the claim that the ET gate performance is identical to the idle case is a property of the ideal H_X Hamiltonian, not of any demonstrated physical pulse. The manuscript should state explicitly that Fig. 4b simulates the hypothetical availability of H_X as a control Hamiltonian, and should separate this conditional result from the Z and CZ gates, whose physical routes (virtual SNAP phases, ESR-based multi-tone pulses) are concrete. This distinction matters for the abstract's claim of charting a concrete path toward full fault-tolerant quantum computation.
minor comments (6)
  1. [Results, Fig. 3 caption] The caption states that the ESR pulses are conditional on the support states of the |1_L> state, but the operation is the two-qubit CZ gate and the phases are conditional on the support states of |1_L,1_L>; the text (which lists the four two-qubit states) is correct, and the caption should match it.
  2. [Results, Eq. (12) and following paragraph] The notation switches from b_n to b_n^{(I)} without definition, and the sentence 'we first compare the first term, i.e., Ω c_1^{(I)} = γ_n B_1/2, to determine Ω' should also state the consistency conditions Ω c_n^{(I)} = b_n^{(I)} for n ≥ 3 that the matching requires.
  3. [Methods, Eq. (21)] The phrase 'Linbladian dissipative channel' should read 'Lindblad dissipative channel'; also, the recovery operation R is described as a piecewise-constant control Hamiltonian following Fig. 7, but the main text does not specify the step durations or the electron-reset timing that the SM states to be optimal.
  4. [SM Sec. A, Table I] The assumed values of f_Q for 121Sb and 209Bi are flagged in the text but not in the table itself, which could mislead readers who consult only the table; adding a footnote marker to those entries would make the assumption explicit.
  5. [SM Sec. D.2] The fault-tolerance proof for Knill-EC is terse: the step 'the faults in the output state amounted to only t^(O) and is independent of the fault occurred in the input state' should explicitly invoke the FT-measurement result of SM D.1 (that measurement outcomes are exact for s + t ≤ r) before the correction step, so that the sufficiency of s + t ≤ r is verifiable.
  6. [Results, 'ET gate performance with error correction'] The sentence 'we find that Γ_n T_G falls around 10^{-2} to 10^{-1}' should state whether T_G refers to the gate time alone or to the full error-correction cycle time, since Fig. 4b's horizontal axis is Γ_n T_G and the adjacent discussion refers to both.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the ET gate construction is self-contained and benchmarked against an external unencoded qubit; the only author-overlap citation is methodology, not load-bearing.

full rationale

The central derivation is self-contained. The ET X gate Hamiltonian is defined explicitly in Eq. (10) from the errorword basis, decomposed in Eq. (11), and the coefficients c_n^(I) are obtained by solving the r+1 linear equations in Methods Eq. (19), with explicit solutions in Eq. (20); no parameter is fitted to the target fidelities, and the ET criterion Eq. (7) is verified algebraically rather than imported. The Z and CZ constructions are diagonal operations whose ET property follows from the stated support identities, and the ET measurement reduction to spin parity in Eqs. (13)-(14) is a direct algebraic identity. The Knill-EC fault-tolerance argument in SM Sec. D.2 is an explicit fault-counting proof; the citation to Ref. [53] (which shares author H. K. Ng) supplies the circuit-level methodology, but the load-bearing inequalities are derived in the supplement and do not reduce to an unverified self-citation. The numerical performance claims are benchmarked against the unencoded spin-1/2 donor (break-even), so they are externally falsifiable rather than forced. The rotating-frame realization route in Eq. (12) is physically questionable — a frame transformation generated by functions of I_x cannot change the lab-frame propagator — but that is a correctness/realizability concern for one proposed implementation, not a circular reduction of the main theoretical result, and the paper itself flags the implementation routes as open challenges. Overall, no prediction or first-principles result is equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The central construction is a mathematical derivation that uses no fitted parameters. Quantitative performance claims depend on assumed simulation inputs: the drive amplitude B_d, the electron dephasing rate Gamma_e, and the quadrupole splitting f_Q for 121Sb and 209Bi, which are not experimentally measured. These inputs are disclosed and do not affect the existence or error-transparency of the gates, but they do affect the claimed operating regime.

free parameters (3)
  • f_Q for 121Sb and 209Bi = assumed -66 kHz (same as 123Sb)
    SM Table I caption: no experimental f_Q values exist; authors assume the 123Sb value. Affects spin level spacings and simulated gate and error correction times.
  • Electron dephasing rate Gamma_e in X gate simulations = 10^4 s^-1
    Used in Eq. (21) and in the X(pi/2) performance simulations as a representative experimental value; the CZ gain plots sweep Gamma_e over a range.
  • Drive amplitude B_d = 0.1 mT
    Sets ESR and NMR pulse times (approximately 0.7 microseconds for ESR 2pi pulses and 0.74 ms for the ET X pi/2 gate); assumed throughout the numerical simulations.
assumptions (7)
  • standard math Knill-Laflamme conditions and ideal recovery achieve optimal fidelity for rank-r dephasing errors.
    Used in SM Sec. C to state the error correction protocol is optimal under ideal implementation; referenced to Ref. [32].
  • domain assumption Dephasing noise is fully captured by the error set E_z^[r] = {1, I_z, ..., I_z^r}.
    Results, Error model: stochastic fluctuations couple predominantly through I_z given the hierarchy gamma_n B0 >> Q_alpha,beta.
  • domain assumption High-field hierarchy gamma_e B0 >> gamma_n B0 >> A >> Q makes product eigenstates |s> tensor |I,m_I> valid and transitions individually addressable.
    Results, High-spin donor system, Eqs. (1)-(3) and accompanying discussion.
  • domain assumption Rotating wave approximation and slowly varying drive amplitudes hold for B_d = 0.1 mT.
    Methods, Numerical simulation: 'This puts the driving in the slow-driving regime and enables implementation in the rotating wave approximation.'
  • domain assumption ESR 2pi-pulses conditional on nuclear support states implement a CZ-type geometric phase, following Ref. [37].
    Results, Error-transparent CZ gate; the phase sequence on support states realizes the logical CZ.
  • standard math The set {I_x, I_y, h} with h containing a rank-2 irreducible spherical tensor generates su(d_n), per Merkel's theorem.
    Results, Error-transparent X gate, second implementation approach, citing Ref. [38].
  • standard math Spin cat-x code is equivalent to the spin binomial code, so bosonic ET constructions transfer.
    SM Sec. B, Eq. (24): explicit binomial coefficients match the spin cat-x codewords.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Towards fault-tolerance with universal phase-error-transparent gates for high-spin cat codes." pith.science (2026). https://pith.science/paper/YAZHOFOX

@misc{pith2026260805992,
  author       = {Pith},
  title        = {Pith review of: Towards fault-tolerance with universal phase-error-transparent gates for high-spin cat codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YAZHOFOX}},
  note         = {Machine review of arXiv:2608.05992}
}
abstract

High-dimensional nuclear spins offer a hardware-efficient route to quantum error correction (QEC), with the spin cat code providing intrinsic robustness against phase errors -- the dominant noise channel in donor-in-silicon architectures. However, realizing the full potential of this encoding requires gate operations that preserve its error-correcting properties. In this work, we construct a universal logical gate set that is error-transparent (ET) to phase errors, and discuss its practical implementations and challenges. The ET gates ensure that phase errors occurring stochastically during gate operations are propagated in a systematically traceable manner and remain correctable in a subsequent QEC step. Among the universal gate set constructed, we identify the logical $X$ gate as the primary challenge and discuss potential realization schemes. In addition, to fully leverage the spin cat code's advantage over an unencoded qubit, multi-tone microwave driving of the logical $CZ$ gate is essential. Our simulations show that ET gates significantly outperform non-ET gates and may be necessary to surpass the break-even point. We further show how logical measurement and recovery can be constructed from ET operations, and explain why state-preparation cannot be made ET. In particular, ET measurement in the computational basis is realizable via spin parity measurement, and that error correction circuits constructed from ET operations achieve optimal error correction capacity. Our work charts a concrete path toward full fault-tolerant quantum computation with high-dimensional nuclear spin systems.

Figures

Figures reproduced from arXiv: 2608.05992 by the authors.

Figure 1
Figure 1. FIG. 1: Diagram of the donor and electron system in sil [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) Dephasing errors act as spin ladder operations in the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Error-transparent implementation of logical [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Evolution and numerical performances of ET gates. (a) State evolution under an ET [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Knill error correction (Knill-EC) circuit based [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Knill-EC performance of several spin systems with various dual basis measurement setups in the data rail: [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Illustration of the circuit to implement the error correction protocol for spin- [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Faults and fault locations in Knill-EC. We assume the input state to have [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 32 canonical work pages

  1. [38]

    Quantum Control of d-Dimensional Quantum Systems with Application to Alkali Atomic Spins

    S. Merkel, Quantum control of d-dimensional quantum systems with application to alkali atomic spins (2009), arXiv:0906.4790 [quant-ph]

    work page Pith review arXiv 2009

  2. [1]

    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, Nature536, 441 (2016)

    2016

  3. [2]

    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, Nature616, 50 (2023)

    2023

  4. [3]

    B. L. Brock, S. Singh, A. Eickbusch, V. V. Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, Quan- tum error correction of qudits beyond break-even, Nature 641, 612 (2025)

    2025

  5. [4]

    A. M. Steane, Overhead and noise threshold of fault- tolerant quantum error correction, Phys. Rev. A68, 042322 (2003)

    work page 2003

  6. [5]

    D. P. DiVincenzo, Fault-tolerant architectures for super- conducting qubits, Physica Scripta2009, 014020 (2009)

    work page 2009

  7. [6]

    Takita, A

    M. Takita, A. W. Cross, A. D. C´ orcoles, J. M. Chow, and J. M. Gambetta, Experimental demonstration of fault- tolerant state preparation with superconducting qubits, Phys. Rev. Lett.119, 180501 (2017)

    2017

  8. [7]

    Chao and B

    R. Chao and B. W. Reichardt, Fault-tolerant quantum computation with few qubits, npj Quantum Information 4, 42 (2018)

    work page 2018

Show all 64 references
  1. [8]

    P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macro- scopically distinct quantum-superposition states as a bosonic code for amplitude damping, Phys. Rev. A59, 2631 (1999)

    1999

  2. [9]

    Gottesman, A

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

    2001

  3. [10]

    A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quan- tum computing with rotation-symmetric bosonic codes, Phys. Rev. X10, 011058 (2020)

    2020

  4. [11]

    X. Yu, B. Wilhelm, D. Holmes, A. Vaartjes, D. Schwien- bacher, M. Nurizzo, A. Kringhøj, M. R. v. Blankenstein, A. M. Jakob, P. Gupta,et al., Schr¨ odinger cat states of a nuclear spin qudit in silicon, Nature Physics21, 362 (2025)

    2025

  5. [12]

    Fern´ andez de Fuentes, T

    I. Fern´ andez 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,et al., Navigating the 16-dimensional hilbert space of a high-spin donor qudit with electric and magnetic fields, Nature communications 1...

    2024

  6. [13]

    Steinacker, G

    P. Steinacker, G. Goenka, R. Y. Su, T. Tanttu, W. H. Lim, S. Serrano, T. Botzem, J. D. Cifuentes, S. Q. Lim, J. C. McCallum, B. C. Johnson, F. E. Hudson, K. W. Chan, C. C. Escott, A. Saraiva, C. H. Yang, V. Mourik, A. Morello, A. S. Dzurak, and A. Laucht, Coupling a 73Ge nucle...

    2026

  7. [14]

    J. A. Gross, Designing codes around interactions: The case of a spin, Phys. Rev. Lett.127, 010504 (2021)

    2021

  8. [15]

    J. A. Gross, C. Godfrin, A. Blais, and E. Dupont-Ferrier, Hardware-efficient error-correcting codes for large nu- clear spins, Phys. Rev. Appl.22, 014006 (2024)

    2024

  9. [16]

    Chiesa, E

    A. Chiesa, E. Macaluso, F. Petiziol, S. Wimberger, P. Santini, and S. Carretta, Molecular nanomagnets as qubits with embedded quantum-error correction, The Journal of Physical Chemistry Letters11, 8610 (2020)

    2020

  10. [17]

    Mezzadri, A

    M. Mezzadri, A. Chiesa, L. Lepori, and S. Carretta, Fault-tolerant computing with single-qudit encoding in a molecular spin, Materials Horizons11, 4961 (2024)

    2024

  11. [18]

    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 Quantum5, 020355 (2024)

    2024

  12. [19]

    DeBry, N

    K. DeBry, N. Meister, A. V. Martinez, C. D. Bruzewicz, X. Shi, D. Reens, R. McConnell, I. L. Chuang, and J. Chi- averini, Error correction of a logical qubit encoded in a single atomic ion (2025), arXiv:2503.13908 [quant-ph]

    2025 arXiv

  13. [20]

    Kusano, K

    T. Kusano, K. Shibata, C.-H. Yeh, K. Saito, Y. Naka- mura, R. Yokoyama, T. Kashimoto, T. Takano, Y. Takasu, R. Takagi, and Y. Takahashi, Spin-cat qubit with biased noise in an optical tweezer array (2026), arXiv:2602.22883 [quant-ph]

    2026

  14. [21]

    J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, R. Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak,et al., Storing quantum infor- mation for 30 seconds in a nanoelectronic device, Nature nanotechnology9, 986 (2014)

    2014

  15. [22]

    J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. Morton, F. A. Zwanenburg, D. N. Jamieson, A. S. Dzu- rak, and A. Morello, High-fidelity readout and control of a nuclear spin qubit in silicon, Nature496, 334 (2013)

    2013

  16. [23]

    Morello, J

    A. Morello, J. J. Pla, P. Bertet, and D. N. Jamieson, Donor spins in silicon for quantum technologies, Ad- vanced Quantum Technologies3, 2000005 (2020)

    2020

  17. [24]

    Gottesman, An introduction to quantum error cor- rection and fault-tolerant quantum computation (2009), arXiv:0904.2557 [quant-ph]

    D. Gottesman, An introduction to quantum error cor- rection and fault-tolerant quantum computation (2009), arXiv:0904.2557 [quant-ph]

    2009 arXiv

  18. [25]

    S. Lim, J. Liu, and A. Ardavan, Fault-tolerant qubit en- coding using a spin-7/2 qudit, Phys. Rev. A108, 062403 (2023)

    2023

  19. [26]

    S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frat- tini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, S. T. Flammia, and S. M. Girvin, Bias-preserving gates with stabilized cat qubits, Science Advances6, eaay5901 (2020)

    2020

  20. [27]

    W.-L. Ma, M. Zhang, Y. Wong, K. Noh, S. Rosen- blum, P. Reinhold, R. J. Schoelkopf, and L. Jiang, Path- independent quantum gates with noisy ancilla, Phys. Rev. Lett.125, 110503 (2020)

    2020

  21. [28]

    O. Vy, X. Wang, and K. Jacobs, Error-transparent evo- lution: the ability of multi-body interactions to bypass decoherence, New Journal of Physics15, 053002 (2013)

    2013

  22. [29]

    Kapit, Error-transparent quantum gates for small log- ical qubit architectures, Phys

    E. Kapit, Error-transparent quantum gates for small log- ical qubit architectures, Phys. Rev. Lett.120, 050503 (2018)

    2018

  23. [30]

    J. M. Radcliffe, Some properties of coherent spin states, Journal of Physics A: General Physics4, 313 (1971)

    1971

  24. [31]

    F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Atomic coherent states in quantum optics, Phys. Rev. A 6, 2211 (1972)

    1972

  25. [32]

    Knill and R

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

    1997

  26. [33]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, New York, 2000)

    2000

  27. [34]

    Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. Song, C.-L. Zou,et al., Error-transparent operations on a logical qubit protected by quantum error correction, Nature Physics16, 827 (2020)

    2020

  28. [35]

    O. C. Wetherbee, S. Roy, B. Royer, and V. Fatemi, A Mathematical Structure for Amplitude-Mixing Error- Transparent Gates for Binomial Codes, Quantum9, 1890 15 (2025)

    2025

  29. [36]

    D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficientzgates for quantum comput- ing, Phys. Rev. A96, 022330 (2017)

    2017

  30. [37]

    M. T. Madzik, S. Asaad, A. Youssry, B. Joecker, K. M. Rudinger, E. Nielsen, K. C. Young, T. J. Proctor, A. D. Baczewski, A. Laucht,et al., Precision tomography of a three-qubit donor quantum processor in silicon, Nature 601, 348 (2022)

    2022

  31. [39]

    J. Ma, X. Wang, C. Sun, and F. Nori, Quantum spin squeezing, Physics Reports509, 89 (2011)

    2011

  32. [40]

    Gupta, A

    P. Gupta, A. Vaartjes, X. Yu, A. Morello, and B. C. Sanders, Robust macroscopic schr¨ odinger’s cat on a nu- cleus, Phys. Rev. Res.6, 013101 (2024)

    2024

  33. [41]

    S. Roy, A. Senanian, C. S. Wang, O. C. Wetherbee, L. Zhang, B. Cole, C. P. Larson, E. Yelton, K. Arora, P. L. McMahon, B. L. T. Plourde, B. Royer, and V. Fatemi, Synthetic high angular momentum spin dy- namics in a microwave oscillator, Phys. Rev. X15, 021009 (2025)

    2025

  34. [42]

    Campbell, B

    E. Campbell, B. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017)

    2017

  35. [43]

    G. A. Paz-Silva, G. K. Brennen, and J. Twamley, Fault tolerance with noisy and slow measurements and prepa- ration, Phys. Rev. Lett.105, 100501 (2010)

    2010

  36. [44]

    C. A. Collett, S. M. Davvetas, A. Alsuhaymi, and G. A. Timco, A configurable two-tone electron spin reso- nance spectrometer, Review of Scientific Instruments95, 10.1063/5.0231566 (2024)

    2024 doi

  37. [45]

    Asaad, V

    S. Asaad, V. Mourik, B. Joecker, M. A. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Madzik, V. Schmitt, J. J. Pla, F. E. Hudson,et al., Coherent electrical control of a single high-spin nucleus in silicon, Nature579, 205 (2020)

    2020

  38. [46]

    Khaneja, T

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, Optimal control of coupled spin dy- namics: design of nmr pulse sequences by gradient as- cent algorithms, Journal of Magnetic Resonance172, 296 (2005)

    2005

  39. [47]

    V. F. Krotov, Global methods in optimal control the- ory, inAdvances in Nonlinear Dynamics and Control: A Report from Russia, edited by A. B. Kurzhanski (Birkh¨ auser Boston, Boston, MA, 1993) pp. 74–121

    1993

  40. [48]

    Castro, A

    A. Castro, A. Garc ´ ıa Carrizo, S. Roca, D. Zueco, and F. Luis, Optimal control of molecular spin qudits, Phys. Rev. Appl.17, 064028 (2022)

    2022

  41. [49]

    Castro, qocttools: A program for quantum optimal control calculations, Computer Physics Communications 295, 108983 (2024)

    A. Castro, qocttools: A program for quantum optimal control calculations, Computer Physics Communications 295, 108983 (2024)

    2024

  42. [50]

    Peham, E

    T. Peham, E. Weilandt, and R. Wille, Optimizing fault- tolerant cat state preparation, Phys. Rev. A113, 062426 (2026)

    2026

  43. [51]

    Z.-J. Chen, W. Cai, L.-X. Xie, Q.-X. Jie, X.-B. Zou, G.- C. Guo, L. Sun, and C.-L. Zou, Fault-tolerant prepara- tion of arbitrary logical states in the cat code (2026), arXiv:2602.17438 [quant-ph]

    2026

  44. [52]

    Holstein and H

    T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev.58, 1098 (1940)

    1940

  45. [53]

    L. D. My, S. Qin, and H. K. Ng, Circuit-level fault toler- ance of cat codes, Quantum9, 1810 (2025)

    2025

  46. [54]

    Aliferis, D

    P. Aliferis, D. Gottesman, and J. Preskill, Quantum accu- racy threshold for concatenated distance-3 codes, Quan- tum Info. Comput.6, 97–165 (2006)

    2006

  47. [55]

    Hillmann, F

    T. Hillmann, F. Quijandr ´ ıa, A. L. Grimsmo, and G. Fer- rini, Performance of teleportation-based error-correction circuits for bosonic codes with noisy measurements, PRX Quantum3, 020334 (2022)

    2022

  48. [56]

    Hausladen and W

    P. Hausladen and W. K. Wootters, A ‘pretty good’ mea- surement for distinguishing quantum states, Journal of Modern Optics41, 2385 (1994)

    1994

  49. [57]

    Q. Xu, P. Zeng, D. Xu, and L. Jiang, Fault-tolerant op- eration of bosonic qubits with discrete-variable ancillae, Phys. Rev. X14, 031016 (2024)

    2024

  50. [58]

    A. M. Steane, Active stabilization, quantum computa- tion, and quantum state synthesis, Phys. Rev. Lett.78, 2252 (1997)

    1997

  51. [59]

    Hashim, L

    A. Hashim, L. B. Nguyen, N. Goss, B. Marinelli, R. K. Naik, T. Chistolini, J. Hines, J. Marceaux, Y. Kim, P. Gokhale, T. Tomesh, S. Chen, L. Jiang, S. Ferracin, K. Rudinger, T. Proctor, K. C. Young, I. Siddiqi, and R. Blume-Kohout, Practical introduction to benchmark- ing and ...

    2025

  52. [60]

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

    2016

  53. [61]

    Petiziol, A

    F. Petiziol, A. Chiesa, S. Wimberger, P. Santini, and S. Carretta, Counteracting dephasing in molecular nano- magnets by optimized qudit encodings, npj Quantum In- formation7, 133 (2021)

    2021

  54. [62]

    L. C. Biedenharn and J. D. Louck,Angular Momentum in Quantum Physics: Theory and Application, Encyclo- pedia of Mathematics and its Applications (Cambridge University Press, 1984)

    1984

  55. [63]

    A. S. Holevo,Probabilistic and statistical aspects of quan- tum theory, Vol. 1 (Springer Science & Business Media, 2011)

    2011

  56. [64]

    C. W. Helstrom, Quantum detection and estimation the- ory, Journal of statistical physics1, 231 (1969). 16 Supplementary Material: Towards fault-tolerance with universal phase-error-transparent gates for high-spin cat codes A. System parameters We show in Table I the physical ...

    1969

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.