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The paper claims that in a large family of logical magic state preparation protocols, every circuit-level Pauli error propagates to a final Clifford error — even though the protocols contain many non-Clifford gates — enabling efficient clas

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2026-08-03 13:34 UTC pith:QX3AAS7U

load-bearing objection Real advance in simulating magic-state preparation, but the formal theorem covers only no-reuse standard gadgets, and the flag-based part of the advertised scope is a remark plus one worked example. the 2 major comments →

arxiv 2512.23799 v2 pith:QX3AAS7U submitted 2025-12-29 quant-ph

Efficient simulation of logical magic state preparation protocols

Samyak Surti , Lucas Daguerre , Isaac H. Kim This is my paper
classification quant-ph MSC 81P68
keywords magic state preparationClifford simulationcircuit-level noisePauli-Square-Root Cliffordstabilizer rankPauli rankerror propagationfault-tolerant quantum computing
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 introduces a scalable classical simulation method for logical magic state preparation under circuit-level noise. The key claim is that in a broad family of protocols — including magic state cultivation, distillation, and code switching — every circuit-level Pauli error propagates to a Clifford error at the end of the circuit, despite the presence of many non-Clifford gates. This means a noisy protocol can be recast as a noiseless logical magic state acted on by a Clifford error, so the output fidelity can be computed with stabilizer simulators in time polynomial in the number of qubits and the stabilizer or Pauli rank of the target magic state. If correct, this replaces exponential state-vector simulations for these protocols and provides a direct, approximation-free way to benchmark their fidelities at large code distances.

Core claim

The central discovery is the error-propagation property and the mathematical tool used to prove it: Pauli-Square-Root Cliffords (PSCs), defined as non-Pauli Clifford unitaries whose square is a Pauli. The paper proves that controlled-PSCs lie in the third level of the Clifford hierarchy and gives a canonical form that yields circuit identities under which Pauli errors propagate as Clifford errors. For the standard, Shor-style, and partially flag-based measurement gadgets, any single-qubit circuit-level error propagates to a Clifford error with O(r^2 w_q (w_c + ℓ)) one- and two-qubit gates (Theorem 2), where r is the number of measurement rounds and w_c, w_q are the parity-check sparsity para

What carries the argument

Pauli-Square-Root Clifford (PSC): a non-Pauli Clifford U with U^2 a Pauli. Its canonical form U = α P exp(iπ/4 Σ Q_j) with commuting Paulis Q_j (Proposition 1) yields circuit identities (Theorem 1 and the propagation rules in Fig. 7) that systematically push Pauli errors past controlled-PSC and controlled-Pauli gates, producing only Clifford errors with bounded gate complexity. The stabilizer-rank and Pauli-rank decompositions of the target magic state then let each propagated Clifford error be simulated classically, with the cost set by the rank rather than by the number of non-Clifford gates.

Load-bearing premise

For the PSC-measurement family, the proof requires each tensor factor of the transversal Clifford to act on a constant number of qubits and requires that ancilla qubits are never reused; if either condition fails, a single Pauli error could propagate to a non-Clifford error outside the proven bound.

What would settle it

Propagate a single-qubit Pauli error inserted at an arbitrary circuit location through a flag-based or ancilla-reusing magic-state preparation protocol, using a stabilizer-state simulator to track the operator; if any propagated operator is not a Clifford transformation (i.e., it maps some stabilizer state to a non-stabilizer state), the central theorem fails and the polynomial complexity bound collapses.

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

If this is right

  • Simulation of magic state cultivation, distillation, and code-switching protocols becomes polynomial in code distance rather than exponential, whenever the target state has small stabilizer or Pauli rank (e.g., rank 2 for |T>).
  • Fidelity estimates require O(p^2 2^k / ε^2) stabilizer-circuit samples for the Pauli-rank method or O(q^2 n_tot^3 / ε) for the stabilizer-rank method, versus exponential state-vector cost.
  • The ratio-based heuristic used to extrapolate magic-state fidelity from stabilizer-state fidelity in cultivation protocols can now be tested directly at large code distances without approximation.
  • Because propagated errors are structured (entangling Clifford errors with known gate counts), flag-gadget design can be informed by exact error shapes rather than worst-case adversarial assumptions.

Where Pith is reading between the lines

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

  • The method likely extends to any protocol built only from controlled-PSCs and non-reused ancillas, since the same propagation rules apply; this would cover recently proposed Clifford-stabilizer-code gadgets if they can be expressed in that form.
  • Protocols that reuse ancilla qubits are the natural stress test: the C(Z) bookkeeping that keeps propagated errors Clifford no longer trivially commutes, so a single non-Clifford propagated error would break the polynomial guarantee and sharply delineate the method's boundary.
  • The phase-sensitive stabilizer-rank estimator could be combined with the propagated-error distribution to build an importance-sampled estimator, potentially reducing the observed O(1/√N) sampling overhead for magic-state fidelity.
  • Because PSCs are exactly characterized, the error-propagation result suggests a design rule for future magic-state gadgets: prefer transversal Cliffords that square to a Pauli, since those are precisely the gates for which noise stays Clifford-classical.

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

2 major / 4 minor

Summary. The paper proposes a classical simulation method for logical magic-state preparation (MSP) protocols under circuit-level Pauli noise. The central observation is that for a large family of protocols, every circuit-level Pauli error propagates to an end-of-circuit Clifford error, despite the presence of many non-Clifford gates. To formalize this, the authors introduce Pauli-Square-Root Cliffords (PSCs) and prove structural lemmas (Lemma 1, Propositions 1–3, Theorem 1) characterizing their behavior under controlled gates. Theorem 2 gives a polynomial gate-count bound for propagated Clifford errors in 'standard' PSC-measurement protocols, with extensions to Shor-style gadgets (Corollary 2) and a remark plus one example for flag-based gadgets. The paper also treats magic-state distillation and code-switching protocols based on transversal non-Clifford gates. A fidelity-estimation method is developed using Pauli-rank and stabilizer-rank decompositions. A proof-of-principle numerical simulation prepares |H> on the Steane code using a flag-based gadget and is compared with state-vector simulation. The claimed advantage is polynomial simulation cost in the number of qubits and the non-stabilizerness of the target state rather than exponential state-vector cost.

Significance. If the main propagation claim holds, this is a valuable and timely contribution. The PSC framework is a genuine structural insight: it explains why many practical MSP protocols admit stabilizer-based simulation without fitted parameters. The paper is also careful to separate rigorously proved statements from heuristic or example-based claims, and the numerical demonstration agrees with state-vector simulation. The main significance is conditional on the scope of the proved results: the strongest formal theorem is for standard no-reuse gadgets, while flag-based and reused-ancilla gadgets — which appear in many practical protocols and in the paper's own numerical example — are supported only by a remark and an example. The missing general proof is a real limitation of the advertised scope, not a mere presentation issue.

major comments (2)
  1. [Section V A; Theorem 2; Section VII B; Appendix D] The central theorem is restricted to 'standard' protocols in which each stabilizer/PSC measurement uses a fresh single ancilla and ancillas are never reused. The proof explicitly relies on this assumption ('Since we consider a circuit model in which ancillas are never reused, these C(Z) errors can trivially be commuted to the end'). Flag-based protocols are introduced as a third family, but the text explicitly declines a general proof ('since it is difficult to systematically cover all flag-based protocols, we only provide a general remark and an example'). However, the numerical proof-of-principle in Fig. 9 and Appendix D uses a flag-based gadget, so the demonstrated result is not covered by Theorem 2. Since the abstract and introduction advertise applicability to cultivation-style and other practical protocols that commonly use flags or ancilla reuse, the formal support is narrower tha
  2. [Appendix A 1; Section V A] The initialization error model for arbitrary PSC-stabilized magic states is not fully justified. Propositions 4 and 5 cover single-qubit order-2 PSCs and diagonal third-level states, but the paragraph immediately after Proposition 5 states: 'We conjecture that this property generalizes to any k-qubit magic state stabilized by k independent and commuting PSCs, which is left for future work.' The main method for PSC-measurement protocols assumes that input magic-state initialization errors can be treated as Pauli errors. Thus the general applicability of the simulation method to multi-qubit PSC-stabilized states rests on an unproved conjecture. Please either prove this statement, or explicitly state that the method's scope for such input states is conditional on this conjecture.
minor comments (4)
  1. [Section VI, Eq. (42)] Equation (42) appears to be missing the factor β_j from the decomposition of ρ. With the definitions in Eq. (34), one expects ⟨P_i⟩_{ρ_N} = (1/2^k) Σ_{j=1}^p β_j Σ_l (-1)^{γ_l} ⟨P_i⟩_{E_L(Π^j_l)}. The final expression in Eq. (43) contains β_i β_j and is consistent with this corrected form; please fix Eq. (42).
  2. [Section VII C; Appendix F] The claimed O(1/ε) sample complexity for the phase-sensitive method is not derived in the appendix and appears inconsistent with standard Monte Carlo estimation of a bounded random variable, which requires O(1/ε^2) samples for additive error ε. Also, the final displayed time complexity 'O(q^2 n_tot^3)' in Section VII C seems to have dropped the per-sample factors O(M r^2 w_q n_tot^3) that appear earlier in the same paragraph. Please clarify the estimator and the constants.
  3. [Section V A; Corollary 2] Corollary 2 for Shor-style protocols is asserted with a brief argument ('naturally upper-bounds the size') but no proof is given. Since Shor-style gadgets are standard and important, a short proof or a pointer to a precise argument would strengthen the paper.
  4. [Throughout] The notation C prop(E), C m, and 'stabilizer operation' is used somewhat interchangeably in the toy example; a single formal definition at first use would improve readability.

Circularity Check

0 steps flagged

No circularity: Theorem 2 is derived, not fit; self-citations are not load-bearing.

full rationale

The paper's central derivation is self-contained. Section IV defines PSC and proves Lemma 1 and Proposition 1 from the symplectic representation; Propositions 2 and 3 and Theorem 1 then follow algebraically from the canonical form Eq. (21). Theorem 2 is proved in Appendix C1 by explicitly tracking how a single-qubit error accumulates O(r^2 w_q(w_c + l)) Clifford gates through standard no-reuse gadgets; no parameter is fitted and the end-of-circuit Clifford property is not assumed for the standard family. The numerical section cross-checks against state-vector simulation (Stim vs Cirq), and the simulation algorithm computes C_prop deterministically from sampled error configurations (Appendix E), so there is no fitted-input-called-prediction. The self-citations to [28] and [43] are used for a prior code-switching simulation and a |T> fidelity expression, but the PSC propagation claim does not reduce to those citations. The acknowledged flag-based limitation ('we only provide a general remark and an example') is a scope restriction, not a circular step. The minor self-citation is not load-bearing, so a low score is appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The derivation relies on standard circuit-level noise, deferred measurements, no ancilla reuse, O(1)-qubit PSC factors, and Pauli twirling of magic-state initialization. No parameter is fitted to data; the physical error rate and the target state's stabilizer/Pauli rank are inputs. PSC is a mathematical class, not an invented physical entity.

axioms (6)
  • domain assumption Circuit-level Pauli noise model: every operation is noiseless followed by stochastic Pauli error with probability p (Appendix A).
    Central to error propagation; if coherent or non-Pauli noise is present, propagation to a Clifford error is not established.
  • domain assumption Measurements can be postponed to the end of the circuit and shots post-selected on +1 outcomes (Section V).
    Used to define the canonical protocol; the authors defer mid-circuit measurement and reset analysis to future work.
  • domain assumption Ancilla qubits are never reused in standard protocols (Appendix C1).
    Required for the C(Z)-error bookkeeping and for the O(r^2 w_q(w_c+ℓ)) gate-count bound in Theorem 2.
  • domain assumption Each PSC tensor factor V_i acts on O(1) qubits and the logical PSC has order 2 (Section V A).
    Needed to prevent high-weight error spread; the authors state all known protocols satisfy this.
  • domain assumption Initialization errors on physical magic states can be twirled into Pauli errors (Propositions 4–5, Appendix A1).
    Proved for single-qubit order-2 PSCs and diagonal V in C(3), conjectured for general k-qubit PSC-stabilized states.
  • standard math Symplectic representation and known stabilizer-rank update rules from Ref. [48] are correct.
    Used in the proof of Proposition 1 and in the phase-sensitive simulation method in Appendix F.

pith-pipeline@v1.3.0-alltime-deepseek · 38172 in / 16529 out tokens · 143736 ms · 2026-08-03T13:34:04.555290+00:00 · methodology

0 comments
read the original abstract

Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.

Figures

Figures reproduced from arXiv: 2512.23799 by Isaac H. Kim, Lucas Daguerre, Samyak Surti.

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 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: In this protocol, an encoded magic state |H¯ ⟩|H¯ ⟩ is first prepared, where |H¯ ⟩ is the +1 eigen￾state of the Hadamard operator. Then, this state is verified using two rounds of logical Clifford and stabilizer measurements. We remark that this cir￾cuit is not fault-tolerant in the sense that a weight-1 circuit-level error may propagate to a weight-2 er￾ror. Though this issue can be dealt with by using a … view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: specifies a complete set of recursive error propagation rules for circuit-level Pauli noise under these gates and shows that only certain Clifford er￾rors are propagated through to the end of a PSC measurement-based protocol. Thus, the propagated error is necessarily Clifford. A motivation for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

100 extracted references · 2 canonical work pages · cited by 2 Pith papers

  1. [1]

    Scheme for reducing decoherence in quantum computer memory,

    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,”Phys. Rev. A52 no. 4, (1995) R2493–R2496

    1995

  2. [2]

    Fault-Tolerant Quantum Computation with Constant Error Rate,

    D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation with Constant Error Rate,”SIAM J. Comput.38no. 4, (2008) 1207–1282,arXiv:quant-ph/9906129

    Pith/arXiv arXiv 2008

  3. [3]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,”Journal of Mathematical Physics43no. 9, (09, 2002) 4452–4505.https://doi.org/10.1063/1.1499754

    doi:10.1063/1.1499754 2002

  4. [4]

    Surface code quantum computing by lattice surgery,

    D. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, “Surface code quantum computing by lattice surgery,”New Journal of Physics14no. 12, (Dec, 2012) 123011.https: //doi.org/10.1088/1367-2630/14/12/123011

    doi:10.1088/1367-2630/14/12/123011 2012

  5. [5]

    Topological quantum distillation,

    H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,”Phys. Rev. Lett.97(Oct, 2006) 180501.https://link.aps. org/doi/10.1103/PhysRevLett.97.180501

    doi:10.1103/physrevlett.97.180501 2006

  6. [6]

    Low-overhead fault-tolerant quantum computing using long-range connectivity,

    L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, “Low-overhead fault-tolerant quantum computing using long-range connectivity,”Sci. Adv.8no. 20, (2022) abn1717,arXiv:2110.10794 [quant-ph]

    Pith/arXiv arXiv 2022

  7. [7]

    Partitioning qubits in hypergraph product codes to implement logical gates,

    A. O. Quintavalle, P. Webster, and M. Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates,”Quantum7(Oct.,

  8. [8]

    Universal quantum computation with ideal clifford gates and noisy ancillas,

    S. Bravyi and A. Kitaev, “Universal quantum computation with ideal clifford gates and noisy ancillas,”Physical Review A71no. 2, (Feb., 2005) .http: 20 //dx.doi.org/10.1103/PhysRevA.71.022316

    doi:10.1103/physreva.71.022316 2005

  9. [9]

    Magic-state distillation with the four-qubit code,

    A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code,”Quant. Inf. Comput.13no. 3-4, (2013) 0195–0209, arXiv:1204.4221 [quant-ph]

    Pith/arXiv arXiv 2013

  10. [10]

    Magic-state distillation with low overhead,

    S. Bravyi and J. Haah, “Magic-state distillation with low overhead,”Physical Review A86no. 5, (Nov., 2012) .http: //dx.doi.org/10.1103/PhysRevA.86.052329

    doi:10.1103/physreva.86.052329 2012

  11. [11]

    Low-overhead constructions for the fault-tolerant toffoli gate,

    C. Jones, “Low-overhead constructions for the fault-tolerant toffoli gate,”Physical Review A87 no. 2, (Feb., 2013) .http: //dx.doi.org/10.1103/PhysRevA.87.022328

    doi:10.1103/physreva.87.022328 2013

  12. [12]

    Surface code implementation of block code state distillation,

    A. G. Fowler, S. J. Devitt, and C. Jones, “Surface code implementation of block code state distillation,”Scientific Reports3no. 1, (June,

  13. [13]

    Multilevel distillation of magic states for quantum computing,

    C. Jones, “Multilevel distillation of magic states for quantum computing,”Physical Review A87 no. 4, (Apr., 2013) .http: //dx.doi.org/10.1103/PhysRevA.87.042305

    doi:10.1103/physreva.87.042305 2013

  14. [14]

    Distillation of nonstabilizer states for universal quantum computation,

    G. Duclos-Cianci and K. M. Svore, “Distillation of nonstabilizer states for universal quantum computation,”Physical Review A88no. 4, (Oct.,

  15. [15]

    Reducing the quantum-computing overhead with complex gate distillation,

    G. Duclos-Cianci and D. Poulin, “Reducing the quantum-computing overhead with complex gate distillation,”Physical Review A91no. 4, (Apr.,

  16. [16]

    Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost,

    E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost,”Physical Review A95no. 2, (Feb., 2017) . http: //dx.doi.org/10.1103/PhysRevA.95.022316

    doi:10.1103/physreva.95.022316 2017

  17. [17]

    1103/PhysRevA.88.042325

    042325.https://link.aps.org/doi/10. 1103/PhysRevA.88.042325. Publisher: American Physical Society

  18. [18]

    Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates,

    J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates,” Quantum2(June, 2018) 71. https://doi.org/10.22331/q-2018-06-07-71

    doi:10.22331/q-2018-06-07-71 2018

  19. [19]

    Efficient magic state factories with a catalyzed|CCZ⟩to 2|T⟩ transformation,

    C. Gidney and A. G. Fowler, “Efficient magic state factories with a catalyzed|CCZ⟩to 2|T⟩ transformation,”Quantum3(Apr., 2019) 135. https://doi.org/10.22331/q-2019-04-30-135

    doi:10.22331/q-2019-04-30-135 2019

  20. [20]

    Elucidating reaction mechanisms on quantum computers,

    M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, “Elucidating reaction mechanisms on quantum computers,”Proc. Nat. Acad. Sci.114 no. 29, (2017) 7555–7560,arXiv:1605.03590 [quant-ph]

    Pith/arXiv arXiv 2017

  21. [21]

    Quantum computation with realistic magic-state factories,

    J. O’Gorman and E. T. Campbell, “Quantum computation with realistic magic-state factories,” Physical Review A95no. 3, (Mar., 2017) .http: //dx.doi.org/10.1103/PhysRevA.95.032338

    doi:10.1103/physreva.95.032338 2017

  22. [22]

    The surface code with a twist,

    T. J. Yoder and I. H. Kim, “The surface code with a twist,”Quantum1(Apr., 2017) 2. http://dx.doi.org/10.22331/q-2017-04-25-2

    doi:10.22331/q-2017-04-25-2 2017

  23. [23]

    Quantum error correction with only two extra qubits,

    R. Chao and B. W. Reichardt, “Quantum error correction with only two extra qubits,”Physical Review Letters121no. 5, (Aug., 2018) .http: //dx.doi.org/10.1103/PhysRevLett.121.050502

    doi:10.1103/physrevlett.121.050502 2018

  24. [24]

    Fault-tolerant quantum computation with few qubits,

    R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits,”npj Quantum Information4no. 1, (Sept., 2018) . http://dx.doi.org/10.1038/s41534-018-0085-z

    doi:10.1038/s41534-018-0085-z 2018

  25. [25]

    A magic state’s fidelity can be superior to the operations that created it,

    Y. Li, “A magic state’s fidelity can be superior to the operations that created it,”New Journal of Physics17no. 2, (Feb., 2015) 023037.http: //dx.doi.org/10.1088/1367-2630/17/2/023037

    doi:10.1088/1367-2630/17/2/023037 2015

  26. [26]

    Efficient magic state distillation by zero-level distillation,

    T. Itogawa, Y. Takada, Y. Hirano, and K. Fujii, “Efficient magic state distillation by zero-level distillation,”PRX Quantum6no. 2, (June, 2025) . http://dx.doi.org/10.1103/thxx-njr6

    doi:10.1103/thxx-njr6 2025

  27. [27]

    Magic state cultivation: growing T states as cheap as CNOT gates,

    C. Gidney, N. Shutty, and C. Jones, “Magic state cultivation: growing T states as cheap as CNOT gates,”arXiv:2409.17595 [quant-ph]

    Pith/arXiv arXiv

  28. [28]

    Code switching revisited: Low-overhead magic state preparation using color codes,

    L. Daguerre and I. H. Kim, “Code switching revisited: Low-overhead magic state preparation using color codes,”Phys. Rev. Res.7no. 2, (2025) 023080,arXiv:2410.07327 [quant-ph]

    Pith/arXiv arXiv 2025

  29. [29]

    Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits,

    C. Chamberland and K. Noh, “Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits,”npj Quantum Inf.6(2020) 91

    2020

  30. [30]

    Cultivating T states on the surface code with only two-qubit gates,

    J. Claes, “Cultivating T states on the surface code with only two-qubit gates,”arXiv:2509.05232 [quant-ph]

    arXiv

  31. [31]

    Magic State Cultivation on the Surface Code,

    Y. Vaknin, S. Jacoby, A. Grimsmo, and A. Retzker, “Magic State Cultivation on the Surface Code,”arXiv:2502.01743 [quant-ph]

    arXiv

  32. [32]

    Efficient Magic State Cultivation onRP 2,

    Z.-H. Chen, M.-C. Chen, C.-Y. Lu, and J.-W. Pan, “Efficient Magic State Cultivation onRP 2,” arXiv:2503.18657 [quant-ph]

    Pith/arXiv arXiv

  33. [33]

    Fold-transversal surface code cultivation,

    K. Sahay, P.-K. Tsai, K. Chang, Q. Su, T. B. Smith, S. Singh, and S. Puri, “Fold-transversal surface code cultivation,”arXiv:2509.05212 [quant-ph]

    arXiv

  34. [34]

    Experimental demonstration of logical magic state distillation,

    P. S. Rodriguezet al., “Experimental demonstration of logical magic state distillation,” Nature645no. 8081, (2025) 620–625, arXiv:2412.15165 [quant-ph]

    arXiv 2025

  35. [35]

    Experimental fault-tolerant code switching,

    I. Pogorelov, F. Butt, L. Postler, C. D. Marciniak, P. Schindler, M. M¨ uller, and T. Monz, “Experimental fault-tolerant code switching,” Nature Phys.21no. 2, (2025) 298–303, arXiv:2403.13732 [quant-ph]

    Pith/arXiv arXiv 2025

  36. [36]

    Magic State Injection on IBM Quantum Processors Above the Distillation Threshold,

    Y. Kim, M. Sevior, and M. Usman, “Magic State Injection on IBM Quantum Processors Above the Distillation Threshold,”arXiv:2412.01446 [quant-ph]

    Pith/arXiv arXiv

  37. [37]

    Scaling and logic in the color code on a superconducting quantum processor,

    N. Lacroixet al., “Scaling and logic in the color code on a superconducting quantum processor,” arXiv:2412.14256 [quant-ph]

    arXiv

  38. [38]

    Encoding a magic state with 21 beyond break-even fidelity,

    R. S. Guptaet al., “Encoding a magic state with 21 beyond break-even fidelity,”Nature625no. 7994, (2024) 259–263,arXiv:2305.13581 [quant-ph]

    Pith/arXiv arXiv 2024

  39. [39]

    Demonstration of fault-tolerant universal quantum gate operations,

    L. Postler, S. Heuben, I. Pogorelov, M. Rispler, T. Feldker, M. Meth, C. D. Marciniak, R. Stricker, M. Ringbauer, R. Blatt, P. Schindler, M. M¨ uller, and T. Monz, “Demonstration of fault-tolerant universal quantum gate operations,”Nature605 no. 7911, (May, 2022) 675–680.http: //dx.doi.org/10.1038/s41586-022-04721-1

    doi:10.1038/s41586-022-04721-1 2022

  40. [40]

    Realization of real-time fault-tolerant quantum error correction,

    C. Ryan-Andersonet al., “Realization of real-time fault-tolerant quantum error correction,”Phys. Rev. X11(Dec, 2021) 041058,arXiv:2107.07505 [quant-ph]

    Pith/arXiv arXiv 2021

  41. [41]

    Logical Magic State Preparation with Fidelity beyond the Distillation Threshold on a Superconducting Quantum Processor,

    Y. Yeet al., “Logical Magic State Preparation with Fidelity beyond the Distillation Threshold on a Superconducting Quantum Processor,”Phys. Rev. Lett.131no. 21, (2023) 210603, arXiv:2305.15972 [quant-ph]

    Pith/arXiv arXiv 2023

  42. [42]

    Magic state cultivation on a superconducting quantum processor,

    E. Rosenfeldet al., “Magic state cultivation on a superconducting quantum processor,” arXiv:2512.13908 [quant-ph]

    arXiv

  43. [43]

    Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching,

    L. Daguerre, R. Blume-Kohout, N. C. Brown, D. Hayes, and I. H. Kim, “Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching,”Phys. Rev. X15 no. 4, (2025) 041008,arXiv:2506.14169 [quant-ph]

    arXiv 2025

  44. [44]

    The Heisenberg representation of quantum computers,

    D. Gottesman, “The Heisenberg representation of quantum computers,”22nd International Colloquium on Group Theoretical Methods in Physics(7, 1998) 32–43,arXiv:quant-ph/9807006

    Pith/arXiv arXiv 1998

  45. [45]

    Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate,

    S. Dasu, S. Burton, K. Mayer, D. Amaro, J. A. Gerber, K. Gilmore, D. Gresh, D. DelVento, A. C. Potter, and D. Hayes, “Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate,”arXiv:2506.14688 [quant-ph]

    Pith/arXiv arXiv

  46. [46]

    Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates,

    S. Bravyi and D. Gosset, “Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates,”Phys. Rev. Lett.116no. 25, (2016) 250501

    2016

  47. [47]

    Trading Classical and Quantum Computational Resources,

    S. Bravyi, G. Smith, and J. A. Smolin, “Trading Classical and Quantum Computational Resources,”Phys. Rev. X6no. 2, (2016) 021043

    2016

  48. [48]

    Simulation of quantum circuits by low-rank stabilizer decompositions,

    S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, “Simulation of quantum circuits by low-rank stabilizer decompositions,”Quantum3(2019) 181, arXiv:1808.00128 [quant-ph]

    Pith/arXiv arXiv 2019

  49. [49]

    Improved simulation of stabilizer circuits,

    S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits,”Phys. Rev. A70 no. 5, (2004) 052328,arXiv:quant-ph/0406196

    Pith/arXiv arXiv 2004

  50. [50]

    Stabilizer Tensor Networks: Universal Quantum Simulator on a Basis of Stabilizer States,

    S. Masot-Llima and A. Garcia-Saez, “Stabilizer Tensor Networks: Universal Quantum Simulator on a Basis of Stabilizer States,”Phys. Rev. Lett. 133no. 23, (2024) 230601,arXiv:2403.08724 [quant-ph]

    Pith/arXiv arXiv 2024

  51. [51]

    Classical Simulability of Quantum Circuits with Shallow Magic Depth,

    Y. Zhang and Y. Zhang, “Classical Simulability of Quantum Circuits with Shallow Magic Depth,” PRX Quantum6no. 1, (2025) 010337, arXiv:2409.13809 [quant-ph]

    Pith/arXiv arXiv 2025

  52. [52]

    STABSim: A Parallelized Clifford Simulator with Features Beyond Direct Simulation,

    S. Garner, C. Liu, M. Wang, S. Stein, and A. Li, “STABSim: A Parallelized Clifford Simulator with Features Beyond Direct Simulation,” arXiv:2507.03092 [quant-ph]

    Pith/arXiv arXiv

  53. [53]

    Efficient Classical Simulation of Clifford Circuits with Nonstabilizer Input States,

    K. Bu and D. E. Koh, “Efficient Classical Simulation of Clifford Circuits with Nonstabilizer Input States,”Phys. Rev. Lett.123no. 17, (2019) 170502

    2019

  54. [54]

    Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code,

    H. Goto, “Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code,”Sci. Rep.6no. 1, (2016) 19578

    2016

  55. [55]

    Flag fault-tolerant error correction with arbitrary distance codes,

    C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes,”Quantum2(Feb., 2018) 53. http://dx.doi.org/10.22331/q-2018-02-08-53

    doi:10.22331/q-2018-02-08-53 2018

  56. [56]

    Fault-tolerant magic state preparation with flag qubits,

    C. Chamberland and A. W. Cross, “Fault-tolerant magic state preparation with flag qubits,” Quantum3(2019) 143,arXiv:1811.00566

    Pith/arXiv arXiv 2019

  57. [57]

    Classical Simulations of Low Magic Quantum Dynamics,

    K. Aziz, H. Pan, M. J. Gullans, and J. H. Pixley, “Classical Simulations of Low Magic Quantum Dynamics,”arXiv:2508.20252 [quant-ph]

    Pith/arXiv arXiv

  58. [58]

    Cutting stabiliser decompositions of magic state cultivation with ZX-calculus,

    K. H. Wan and Z. Zhong, “Cutting stabiliser decompositions of magic state cultivation with ZX-calculus,”arXiv:2509.01224 [quant-ph]

    arXiv

  59. [59]

    How to simulate magic state cultivation with around 8 Clifford terms on average,

    K. H. Wan and Z. Zhong, “How to simulate magic state cultivation with around 8 Clifford terms on average,”arXiv:2509.08658 [quant-ph]

    Pith/arXiv arXiv

  60. [60]

    Procedurally optimised zx-diagram cutting for efficient t-decomposition in classical simulation,

    M. Sutcliffe and A. Kissinger, “Procedurally optimised zx-diagram cutting for efficient t-decomposition in classical simulation,”Electronic Proceedings in Theoretical Computer Science406 (Aug., 2024) 63–78. http://dx.doi.org/10.4204/EPTCS.406.3

    work page doi:10.4204/eptcs.406.3 2024

  61. [61]

    Fault-tolerant code-switching protocols for near-term quantum processors,

    F. Butt, S. Heußen, M. Rispler, and M. M¨ uller, “Fault-tolerant code-switching protocols for near-term quantum processors,”PRX Quantum5 no. 2, (May, 2024) .http: //dx.doi.org/10.1103/PRXQuantum.5.020345

    work page doi:10.1103/prxquantum.5.020345 2024

  62. [62]

    Dimensional jump in quantum error correction,

    H. Bomb ´ ın, “Dimensional jump in quantum error correction,”New J. Phys.18no. 4, (2016) 043038, arXiv:1412.5079 [quant-ph]

    Pith/arXiv arXiv 2016

  63. [63]

    Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes,

    M. E. Beverland, A. Kubica, and K. M. Svore, “Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes,”PRX Quantum2(Jun, 2021) 020341.https://link.aps.org/doi/10.1103/ PRXQuantum.2.020341

    2021

  64. [64]

    Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,

    D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature402no. 6760, (Nov., 1999) 390–393. http://dx.doi.org/10.1038/46503

    doi:10.1038/46503 1999

  65. [65]

    Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes,

    J. T. Anderson, G. Duclos-Cianci, and D. Poulin, “Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes,”Phys. Rev. Lett.113no. 8, (2014) 080501,arXiv:1403.2734 [quant-ph]

    Pith/arXiv arXiv 2014

  66. [66]

    Fault-tolerant quantum computation,

    P. W. Shor, “Fault-tolerant quantum computation,”Proceedings of 37th Conference on Foundations of Computer Science(5, 1996) , arXiv:quant-ph/9605011

    Pith/arXiv arXiv 1996

  67. [67]

    Topological computation without braiding,

    H. Bombin and M. A. Martin-Delgado, 22 “Topological computation without braiding,” Physical Review Letters98no. 16, (Apr., 2007) . http: //dx.doi.org/10.1103/PhysRevLett.98.160502

    doi:10.1103/physrevlett.98.160502 2007

  68. [68]

    Controlled Gates in the Clifford Hierarchy,

    J. T. Anderson and M. Weippert, “Controlled Gates in the Clifford Hierarchy,” arXiv:2410.04711 [quant-ph]

    Pith/arXiv arXiv

  69. [69]

    Universal fault tolerant quantum computation in 2D without getting tied in knots,

    M. Davydova, A. Bauer, J. C. M. de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, “Universal fault tolerant quantum computation in 2D without getting tied in knots,” arXiv:2503.15751 [quant-ph]

    Pith/arXiv arXiv

  70. [70]

    From Magic State Distillation to Dynamical Systems,

    Y. Zheng and D. E. Liu, “From Magic State Distillation to Dynamical Systems,”Quantum9 (2025) 1858,arXiv:2412.04402 [quant-ph]

    arXiv 2025

  71. [71]

    On the Structure of Protocols for Magic State Distillation,

    E. T. Campbell and D. E. Browne, “On the Structure of Protocols for Magic State Distillation,”Lect. Notes Comput. Sci.5906 (2009) 20–32,arXiv:0908.0838 [quant-ph]

    Pith/arXiv arXiv 2009

  72. [72]

    Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes,

    H. Bomb ´ ın, “Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes,”New J. Phys.17no. 8, (2015) 083002,arXiv:1311.0879 [quant-ph]

    Pith/arXiv arXiv 2015

  73. [73]

    Transversal Clifford gates on folded surface codes,

    J. E. Moussa, “Transversal Clifford gates on folded surface codes,”Phys. Rev. A94no. 4, (2016) 042316,arXiv:1603.02286 [quant-ph]

    Pith/arXiv arXiv 2016

  74. [74]

    Three-dimensional surface codes: Transversal gates and fault-tolerant architectures,

    M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures,”Phys. Rev. A100no. 1, (2019) 012312,arXiv:1801.04255 [quant-ph]

    Pith/arXiv arXiv 2019

  75. [75]

    Restrictions on Transversal Encoded Quantum Gate Sets,

    B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets,”Phys. Rev. Lett.102no. 11, (2009) 110502, arXiv:0811.4262 [quant-ph]

    Pith/arXiv arXiv 2009

  76. [76]

    Classification of Topologically Protected Gates for Local Stabilizer Codes,

    S. Bravyi and R. Koenig, “Classification of Topologically Protected Gates for Local Stabilizer Codes,”Phys. Rev. Lett.110no. 17, (2013) 170503,arXiv:1206.1609 [quant-ph]

    Pith/arXiv arXiv 2013

  77. [77]

    Universal transversal gates with color codes: A simplified approach,

    A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach,”Phys. Rev. A91no. 3, (2015) 032330, arXiv:1410.0069 [quant-ph]

    Pith/arXiv arXiv 2015

  78. [78]

    Efficient fault-tolerant code switching via one-way transversal CNOT gates,

    S. Heußen and J. Hilder, “Efficient fault-tolerant code switching via one-way transversal CNOT gates,”Quantum9(2025) 1846, arXiv:2409.13465 [quant-ph]

    Pith/arXiv arXiv 2025

  79. [79]

    Stim: a fast stabilizer circuit simulator,

    C. Gidney, “Stim: a fast stabilizer circuit simulator,”Quantum5(July, 2021) 497. https://doi.org/10.22331/q-2021-07-06-497

    doi:10.22331/q-2021-07-06-497 2021

  80. [80]

    Developers,Cirq

    C. Developers,Cirq. Zenodo, Aug., 2025.https: //zenodo.org/doi/10.5281/zenodo.4062499

    doi:10.5281/zenodo.4062499 2025

Showing first 80 references.