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Magic State Distillation via Codes over Binary Extension Fields

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Four noisy CS states produce one clean CS state at distance 2

desk verdict A substantial construction paper with explicit, checkable protocols; the headline performance claim is real only within an unvalidated cost model and should be softened. read the letter →

arxiv 2608.09727 v1 pith:AGUX5TQA submitted 2026-08-10 quant-ph

Anqi Gong , Christopher A. Pattison , Patrick Rall , Adam Wills This is my paper · ORCID
classification quant-ph MSC 81P6894B2711T7114G50 PACS 03.67.Pp03.67.Lx
keywords magicstatedistillationGaloisquditsbinaryextensionfieldsalgebraicgeometrycodestransversalnon-Cliffordgatesthree-orthogonalityCSandCCZstatesquantumerrorcorrection
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 claims that distilling the multi-qubit magic states $|\mathrm{CS}\rangle$ and $|\mathrm{CCZ}\rangle$ becomes much cheaper when the underlying quantum codes are defined over binary extension fields $\mathbb{F}_{2^s}$ rather than directly over qubits. Multi-qubit gates such as CS and CCZ can be repackaged as simple single-qudit gates over those fields, and a short algebraic condition---three-orthogonality of the code's generator rows---guarantees the gate is transversal, turning the code into a distillation protocol. The headline examples are a protocol that distills 4 $|\mathrm{CS}\rangle$ states into 1 $|\mathrm{CS}\rangle$ state at distance 2 using only 4 logical qubits, and one that distills 24 $|\mathrm{CCZ}\rangle$ states into 4 $|\mathrm{CCZ}\rangle$ states at distance 3 using the Klein quartic over $\mathbb{F}_8$. In a Pauli-based-computation cost model, concatenating these protocols outperforms all published alternatives for almost every distillation task at input error rates $10^{-3}$ and $10^{-6}$. If correct, these are the smallest known factories at their distances and shift the resource-cost frontier for fault-tolerant magic state production.

What carries the argument

The machinery is the X-generator matrix $G \in \mathbb{F}_{2^s}^{m \times n}$ whose first $k$ rows are logical operators and whose remaining rows are stabilizer generators, together with the generalized three-orthogonality identity $\sum_i g_{a,i}g_{b,i}g_{c,i} = \delta_{a=b=c}$ (and its twisted variant $\sum_i g_{a,i}^4 g_{b,i}^2 g_{c,i} = \delta_{a=b=c}$ for $U_7$). These identities are exactly what make the corresponding single-qudit diagonal gate transversal on the CSS code, so that applying the noisy gate transversally and measuring the syndrome gives a distillation protocol. Codes are built by evaluating monomials on points of low-genus maximal curves over $\mathbb{F}_4, \mathbb{F}_8, \mathbb{F}_{16}, \mathbb{F}_{32}, \mathbb{F}_{64}$ (projective line, elliptic curves, Hermitian curves, the Klein quartic), then puncturing selected columns to create logical rows, and finally binarizing through a self-dual basis to obtain qubit codes. The distance of the resulting protocol is the minimum number of input magic states whose failure can cause an undetectable logical error.

What would settle it

Run the proposed 4$|\mathrm{CS}\rangle \to 1|\mathrm{CS}\rangle$ protocol with independent errors on the four inputs and measure output error as a function of input error $p$: distance 2 predicts leading behavior $\propto p^2$, so any undetectable failure caused by a weight-two input error pattern, or a suppression exponent of one, refutes the claim. For the cost-frontier claim, an independent implementation of the time and space model with an explicitly costed CS-cultivation step that reverses the reported ordering would refute it.

Watch

Extended reading notes

Core claim

The central discovery is that codes over $\mathbb{F}_{2^s}$, viewed as Galois qudit codes, reduce the transversality conditions for practically relevant multi-qubit gates to simple identities, and the resulting qubit protocols are unusually compact. The paper shows that the two-qubit CS gate is, in a self-dual basis of $\mathbb{F}_4$, the single-qudit phase gate $|\gamma\rangle \mapsto i^{\gamma^3-\mathrm{tr}(\gamma)}|\gamma\rangle$, and that the three-qubit CCZ gate appears as the three-qudit gate $|x\rangle|y\rangle|z\rangle \mapsto (-1)^{\mathrm{tr}(xyz)}|x\rangle|y\rangle|z\rangle$; the $U_7$ gate over $\mathbb{F}_8$ is a single CCZ. For each such gate the paper derives a linear-algebra condition on the X-generator matrix (three-orthogonality for CS and CCZ, twisted three-orthogonality for $U_7$) that makes the transversal physical gate induce the desired logical gate, and then constructs codes satisfying these conditions from Reed-Solomon codes, affine and projective monomial codes, and algebraic-geometry one-point codes, including punctured codes from maximal curves such as the Hermitian curve and the Klein quartic. Because distance is measured in input magic states, the framework handles the correlated errors present on multi-qubit states by construction.

Load-bearing premise

The claim that these protocols beat the state of the art rests on a specific cost accounting---one time unit per T-state injection, two per CS injection, capped concatenation depth of eight, a 75-logical-qubit space limit, and the untested assumption that CS states can be pre-cultivated to $10^{-6}$ error---so if a real architecture prices these operations differently the protocols still work but the frontier claim may not.

Editorial extensions

If this is right

  • The $(2k+2)|\mathrm{CS}\rangle \to k|\mathrm{CS}\rangle$ family achieves asymptotic overhead 2 at distance 2, and Appendix A.4 proves this overhead is optimal for distance-2 CS distillation from $\mathbb{F}_4$-linear triorthogonal codes.
  • The Klein quartic yields a 24-to-4 CCZ protocol at distance 3, reaching rate $1/6$ at distance 3 at small sizes---a regime for which the paper says no previous method is known.
  • Concatenations of the new protocols sit on the optimal Pareto frontier for almost every CS/CCZ distillation task at input error rates $10^{-3}$ and $10^{-6}$, for both time rate and spacetime volume, under the paper's cost model.
  • New protocols output exotic states such as TOF# directly (8 CS to 1 TOF# at distance 2, 16 CS to 1 TOF# at distance 3), which can be cheaper than distilling T states and then synthesizing the gate.

Reading between the lines

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

  • The same Galois-qudit packaging argument suggests a systematic search for compact distillation protocols for any diagonal real-phase multi-qubit gate, using the paper's classification of such gates in the Galois qudit Clifford hierarchy.
  • If CS cultivation turns out to be cheap in a real architecture, the $10^{-6}$ frontier may shift even further toward the new CS-input protocols; if cultivation is expensive, the T-input protocols remain a robust fallback.
  • The forbidden-hypergraph maximum-independent-set construction used for the U7 and Klein quartic logicals is an explicit recipe that becomes NP-hard to scale, pointing to SAT-based or symmetry-based searches as the natural next step for finding even smaller factories at larger distances.
  • Because distance is measured in input magic states rather than qubit codeword weight, a fair architectural comparison with surface-code or qLDPC implementations of the same logic is an open question beyond the paper's Pauli-based-computation model.
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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

2 major / 5 minor

Summary. The paper constructs magic-state distillation protocols for qubits from CSS codes over the binary extension fields F_{2^s}. It packages multi-qubit gates (CS, CCZ, TOF#) as single-Galois-qudit gates, derives algebraic orthogonality conditions for transversality, and constructs finite-length protocols from Reed-Solomon, one-point, Hermitian, elliptic, and Klein-quartic codes. The marquee results are a 4CS→1CS distance-2 protocol, a 24CCZ→4CCZ distance-3 protocol from the F8 Klein quartic, a 64T→2CCZ distance-4 protocol with leading error 2720p^4, and a family of (2k+2)CS→kCS protocols. The final sections benchmark concatenations of these protocols in a Pauli-based computation model and claim effectiveness in almost every considered regime.

Significance. The algebraic constructions are the core contribution, and they appear sound: Lemma 4.2 and Appendix A give explicit, hand-checkable transversality arguments, and the marquee protocols are stated as explicit matrices whose orthogonality can be verified directly. The 4→1 CS and 24→4 CCZ protocols, if correct, are the smallest known factories at these distances, and the trace-code 64T→2CCZ construction improves the leading error coefficient over Haah-Hastings. The linked SageMath notebooks and explicit matrices are reproducibility strengths. The headline benchmarking claims, however, are model-dependent and should not be presented as intrinsic properties of the codes.

major comments (2)
  1. [Section 1.3.2 and footnote 2] The p=10^-6 benchmark relies on the assumption that |CS> states can be cultivated to error ~10^-6 at modest cost, a statement the authors explicitly say has not been studied directly. Because the 'outperform state-of-the-art' conclusion for all 10^-6 cases in Figure 2 rests on this premise, this is a load-bearing unsupported assumption. The authors should either provide a concrete cultivation construction or resource estimate, or explicitly rephrase the claim as conditional on that assumption.
  2. [Section 1.3.2 and Appendix D] The claimed optimality in 'almost every situation' is computed under fixed cost choices: one time unit per |T> injection, two time units per |CS> injection, concatenated distance capped at 8, and footprint capped at 75 logical qubits. These choices are reasonable but not derived from any architecture. A sensitivity analysis varying these parameters is needed; without it, Figures 1 and 2 establish optimality only within this particular cost model, not as an intrinsic property of the new codes.
minor comments (5)
  1. [Section 2.2] The sentence 'A subset of fields are integral domains, which are integral domains' contains a duplicated phrase and should read 'A subset of fields are integral domains.'
  2. [Section 1.2] The word 'purvue' should be 'purview' in 'although our protocols lie outside the purvue of all of these.'
  3. [Section 1.1 and Table 1] The phrase 'using only 4 logical qubits' for the 4CS→1CS protocol should be cross-referenced to the footprint definition in Section 1.3.2, since the binarized code itself has eight physical qubits and the reported number counts the rows of the generator matrix under the Pauli-based computation model.
  4. [Equation (42)] The index ranges for a, b, c in the twisted three-orthogonality condition should be stated explicitly: the condition is imposed over all rows of the generator matrix, with the nonzero diagonal value allowed only when all three indices coincide and lie among the first k logical rows.
  5. [Section 5, Table 3] The paper states that upper bounds on distances are found by QDistRnd and that upper bounds are brute-force certified when they exceed lower bounds, but the text does not list which table entries were certified in this way. Including the exact verification scripts or certificates in the repository would make the catalogued parameters fully reproducible.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the protocol constructions are verified directly against algebraic orthogonality conditions, and the practical-outperformance claim rests on an explicitly flagged cost model rather than on a circular fit.

full rationale

The paper's central protocol claims are constructed and checked within the paper itself: the transversality conditions for qudit-CCZ (Eq. 40), U7 (Eq. 42), norm (Eq. 63), and CS (Eq. 67) are proven or verified by direct calculation, and explicit generator matrices (e.g., Eq. 45, 68, 75) determine the logical action and distance. The headline 4-CS-to-1-CS and 24-CCZ-to-4-CCZ protocols are therefore not inferred from the paper's own benchmarks; they are explicit code parameters. The Pareto-frontier comparison in Section 1.3.2 and Appendix D is a benchmark against independent prior protocols (BK05, BH12, CH17, HH18a), and its dependence on the cost model—in particular treating |CS> cultivation to 10^-6 as 'entirely reasonable' although 'the cultivation of |CS> states has not been studied directly' (footnote 2)—is a modeling assumption that could affect the outperformance conclusion, not a circular derivation. Author-overlapping references ([Wil26] for the qudit-to-qubit mapping, [WHY25] for the U7 gate) are used as building blocks, and the U7 transversality claim is re-proven in Lemma 4.2; the CS and CCZ protocol checks do not reduce to those citations. Hence there is no demonstrated circularity; at most one notes minor self-citation in supporting material.

Assumptions & free parameters 6 free parameters · 12 assumptions · 0 invented entities

The protocol derivations contain no fitted constants; all listed free parameters are benchmarking or modeling choices. The mathematical axioms are standard and imported with citations (Riemann-Roch, maximal-curve classifications, Lucas' theorem, Delsarte duality, self-dual bases). Two domain assumptions carry the pipeline: the [Wil26] qudit-to-qubit mapping and the Pauli-based-computation cost model. Ad-hoc-to-paper choices are flagged: the monomial-only search ansatz, the survey caps, and the untested CS cultivation premise. No new physical entities are postulated; the 'forbidden hypergraph' is a computational search device, not an entity.

free parameters (6)
  • Time unit (one |T> injection) = 1 time unit
    Modeling choice in Section 1.3.2 and Appendix D; the time-rate Pareto curves are measured in these units.
  • CS injection time cost = 2 time units
    Hand-assigned in Section 1.3.2; this affects the time-rate comparisons in favor of CS-input schemes.
  • Max concatenated distance in survey = 8
    Section 1.3.2: 'we restrict ourselves to concatenated distance at most 8', which excludes deeper distillation towers.
  • Max spatial footprint in survey = 75 logical qubits
    Section 1.3.2: cap modeling a moderate qubit sacrifice; excludes larger factories that could win at low target error rates.
  • Input error rates for benchmarks = 10^-3 and 10^-6
    Section 1.3.2: fixed injection scenarios; the 10^-6 branch assumes unstudied CS cultivation.
  • Error metric = per-output average error rate
    Appendix D: the paper computes average error rate per output rather than probability of any error; frontier positions depend on this choice.
assumptions (12)
  • standard math Riemann-Roch theorem and dimension formula ell(D) = deg(D) - g + 1 for deg(D) > 2g - 2.
    Invoked to compute one-point code dimensions in Section 5.2 (Eq. 100), underpinning Corollary 5.5 parameter formulas.
  • standard math Classification of maximal and defect-zero curves over F4, F8, F16, F32, F64 (Hasse-Weil, Serre, Ihara bounds; uniqueness of Hermitian curves; results of RS94, FT96, AT98, VV00, DM87).
    Section 5.1.2: the Table 3 curve list and the resulting protocol parameters depend on these classifications.
  • standard math Stichtenoth Prop. 6.4.1 (Thm 5.1): monomial bases for L(rQ_inf) for function fields y^h + mu*y = f(x).
    Used to construct one-point code bases, including the y^2 + y = x^5 curve whose projective closure is singular.
  • standard math Lucas' theorem and parity of multinomial coefficients (Lemma 3.7).
    Basis for the Clifford-hierarchy classification (Thm 3.6) and for the norm-gate phase polynomial expansion (Eq. 62).
  • standard math Delsarte duality (C|_k)^perp = tr(C^perp) (Eq. 113).
    Underlies the trace-code protocols of Section 5.5, including the 64T to 2CCZ protocol.
  • standard math Existence of self-dual bases for F_{2^s} over F_2 (Seroussi-Lempel).
    Used throughout Section 3.1 and 3.2 for binarizing qudit gates (Eq. 11 and trace-matrix expansions).
  • domain assumption The qudit-to-qubit mapping of [Wil26] preserves states, CSS codes, Clifford hierarchy, and transversality.
    Section 3.1: 'All of these isomorphisms are compatible with each other in the natural ways'. The entire pipeline from Galois qudit codes to qubit protocols rests on this cited mapping.
  • domain assumption Magic state errors can be twirled into a dephasing channel (generalized BK05/BH12 twirling, Prop 3.3); only Z-distance matters.
    Standard for distillation; proven in Prop 3.3, but the relevance of Z-distance to correlated multi-qubit errors is an assumption the paper argues for.
  • domain assumption Pauli-based computation with the stated time and footprint cost assignments is a faithful model for comparing distillation factories.
    Section 1.3.2 and App. C: the Pareto-frontier conclusion that novel protocols 'outperform the state-of-the-art in almost every situation' inherits this assumption.
  • ad hoc to paper Cultivation of CS states to about 10^-6 is achievable at modest cost, though unstudied.
    Section 1.3.2, footnote 2: 'the cultivation of |CS> states has not been studied directly, as far as we know, but is entirely reasonable'. Used for the 10^-6 injection regime.
  • ad hoc to paper Restriction of U7 and CCZ logical searches to monomial evaluations, with total degree 1 mod 7 for the Klein quartic search.
    Constructions 4.11 and 5.8: the authors 'restrict ourselves to codewords coming from monomial evaluations'; optimality over all subspaces is not established.
  • ad hoc to paper Benchmark caps of concatenated distance at most 8 and spatial footprint at most 75 logical qubits.
    Section 1.3.2: competitive protocols with larger footprint are excluded by fiat, which can only favor the new protocols near the frontier.

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Cite this review

Pith. "Pith review of Magic State Distillation via Codes over Binary Extension Fields." pith.science (2026). https://pith.science/paper/AGUX5TQA

@misc{pith2026260809727,
  author       = {Pith},
  title        = {Pith review of: Magic State Distillation via Codes over Binary Extension Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGUX5TQA}},
  note         = {Machine review of arXiv:2608.09727}
}
abstract

Fault-tolerant quantum computation architectures are frequently bottlenecked by the overhead of producing high-fidelity magic states. In this work, we use algebraic geometric techniques to construct codes over binary extension fields $\mathbb{F}_{2^s}$, thus discovering new protocols for the distillation of qubit magic states, where our focus is on the regime of practical qubit-based quantum computing architectures. To do this, we show that multi-qubit gates of interest such as $\text{CS}$, $\text{CCZ}$, and $\text{TOF}\# = \text{CCZ}_{123}\text{CCZ}_{345}$, can be packaged into simple gates over the larger fields, and we derive simple algebraic conditions in the extension fields allowing the distillation of these gates. Because they are derived from Galois qudits, the corresponding qubits codes naturally handle the correlated errors present on such multi-qubit states. Moreover, the protocols we discover are extremely compact; for example, we show that 4 $\text{CS}$ states can be distilled to 1 $\text{CS}$ state at distance 2, using only 4 logical qubits. For a case study, we consider the distillation of $\text{CS}$ and $\text{CCZ}$ states from injected $\text{T}$ and $\text{CS}$ states. When optimized for magic state production per unit time, or logical spacetime volume, we find that our protocols outperform the state-of-the-art in almost every situation, both at input error rates $10^{-3}$ (direct injection), and $10^{-6}$ (allowing some cultivation pre-injection).

Figures

Figures reproduced from arXiv: 2608.09727 by the authors.

Figure 1
Figure 1. Pareto frontiers for physical input error [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Pareto frontiers for physical input error [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Examples of Clifford equivalence. (a) and (b) are both Clifford equivalent to a single CCZ [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a) F4-qudit CCZ expanded as qubit CCZs. Expand x = x1ω + x2ω 2 , where x1, x2 ∈ F2, and similarly for y, z. (b) The magic state corresponding to F4-CCZ can be consumed to perform multiplication in F4 – one just needs to conjugate the third pair of qubits by Hadamard g…
Figure 5
Figure 5. Figure 5: U β 7 gate expanded in qubit CCZ gates up to Clifford equivalence. (a) U β 7 in F8 with β satisfying tr(β) ̸= 0. If tr(β) = 0, U β 7 is the identity. (b) In F16, all U β 7 for which β ̸= 0 are Clifford equivalent to a single CCZ. (c) In F32, all U β 7 for which β ̸= 0 …
Figure 6
Figure 6. Figure 6: Left: the resulting logical circuit when applying transversal [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

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Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

  1. [1]

    The coefficient ofxaya modulo two is nX i=1 tr(gaiω2)tr(gai) +tr(g ai)tr(gaiω) +tr(g aiω)tr(gaiω2) = nX i=1 (gaiω2 + (gai)2ω)(gai + (gai)2) + (gai + (gai)2)(gaiω+ (g ai)2ω2) + (gaiω+ (g ai)2ω2)(gaiω2 + (gai)2ω) = nX i=1 (gai)3

    work page

  2. [2]

    For the triple intersections (a̸=bora, b, cpairwise distinct):

    The coefficient ofxayb modulo two (a̸=b) is Pn i=1tr(gai(gbi)2ω2). For the triple intersections (a̸=bora, b, cpairwise distinct):

    work page

  3. [3]

    The coefficient ofxaxbxc modulo two isPn i=1tr(gaigbigci)

    work page

  4. [4]

    The coefficient ofxayaxb modulo two isPn i=1tr(gai(gbi)2ω2)

    work page

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    The coefficient ofxayayb modulo two isPn i=1tr(gai(gbi)2ω)

    work page

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    We know that for1≤a̸=b≤manda, bnot both in{1,

    The coefficient ofyaybyc modulo two isPn i=1tr(gaigbigci). We know that for1≤a̸=b≤manda, bnot both in{1, . . . , k}, we need the coefficient ofxayb,x ayaxb, xayayb to be even. This enforcesPn i=1 gai(gbi)2 = 0. Similarly, for1≤a, b, c≤mpairwise-distinct and a, b, cnot all in{1, . . . , k}, the coefficient ofxaxbxc, . . . , yaybyc needs to be even, soPn i=...

    work page

  7. [7]

    The coefficient ofxaxbyc modulo two isPn i=1tr(gaigbigciω)

    work page

  8. [8]

    The coefficient ofxaybyc modulo two isPn i=1tr(gaigbigciω2). 56

    work page

Show all 31 references
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    Write theXlogical and stabilizer rows asg a,a∈[m], and the firstkare logical rows

    Since we allow Clifford correction between stabilizer and logical rows, we can ignore thei−tr(u) part and only focus on applying transversaliu3 . Write theXlogical and stabilizer rows asg a,a∈[m], and the firstkare logical rows. A classical codeword can be written as Pm a=1 ua...

  2. [11]

    mX a=1 u3 a(gai)3 + 2 X 1≤a<b≤m u3 au3 b (gai)3(gbi)3 + X 1≤a<b≤m tr(u2 aub(gai)2gbi)(120) + 2 X 1≤a<b<c≤m tr(uaubucgaigbigci) # mod 4(121) ≡ mX a=1 u3 a

    =tr(a 2 1a2); similarly, a3 2tr(a2 1a2) =tr(a 2 1a2). Assuming the formula is true form, let us prove it form+ 1. CallA m :=a 1 ˆ+· · ·ˆ+am and Am+1 :=A m ˆ+am+1. (a1 ˆ+· · ·ˆ+am| {z } Am ˆ+am+1)3 ≡A 3 m +a 3 m+1 + 2A3 ma3 m+1 +tr(A ma2 m+1) mod 4, where we used tr(a2b) =tr(ab...

  3. [12]

    Initialize a qubit ancilla in the|+⟩state

  4. [13]

    Apply the joint measurementZ⊗Pto|+⟩ ⊗ |ψ⟩to get outcomeα∈ {0,1}

  5. [14]

    Apply the map|a⟩ →eiϕ(a⊕α) |a⟩to the ancilla

  6. [15]

    Measure the ancilla in theXbasis to obtain outcomeγ∈ {0,1}

  7. [16]

    Here it was natural to considerexp iϕ 1+P 2 rather thanexp (iϕP)since the eigenvalues of1+P 2 live inF 2, corresponding to the computational basis states of a qubit

    ApplyPto the|ψ⟩register whenγ= 1. Here it was natural to considerexp iϕ 1+P 2 rather thanexp (iϕP)since the eigenvalues of1+P 2 live inF 2, corresponding to the computational basis states of a qubit. This motivates the notationˆP= 1+P 2 as a Hermitian operator onnqubits with e...

  8. [17]

    Initialize a qudit ancilla in the|+⟩state

  9. [18]

    Apply the measurement generated byZ⊗Pto|+⟩ ⊗ |ψ⟩to get outcomeα∈F 2s

  10. [19]

    Apply the map|η⟩ →exp(iπtr((η−α) 7))|η⟩to the ancilla

  11. [20]

    Measure the ancilla in theXbasis to obtain outcomeγ∈F 2s

  12. [21]

    IfP=i tr(ux·uz)X(u x)Z(u z), applyi tr(γ2ux·uz)X(γu x)Z(γu z)to the|ψ⟩register. As a specialization, we see that the gateUβ 7 can be written asexp(iπtr(β ˆZ(1) 7)), and is gener- alized to a larger family of operatorsUβ 7 [g] := exp(iπtr(β ˆZ(g) 7))forg∈F n 2s, which is the se...

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    Initializenqudits into a logical|+⟩ ⊗k state within the codespace,

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    ApplyU β 7 on allnqudits,

  15. [24]

    Measure theXstabilizers on the code and either post-select or decode,

  16. [25]

    Unencode from the code to obtain the final distilled U β 7 |+⟩ ⊗k . However, for the case ofF2, it is known that there exists a quantum circuit supported on onlym qubits (wheremis the number of rows in theX-generator matrix) that realizes the protocol [Lit19b]. In this section...

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    Initializemqudits in the state|+⟩ ⊗m

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    Fori= 1throughn, apply faulty twirledU β 7 [gi]gates, 51 whereg i is thei’th column ofG

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    kX a=1 uaga i #7    |¯u⟩(175) = exp  iπtr  β

    Measure quditsk+ 1throughmin theXbasis. Then, the measurement results correspond to theXstabilizers of the CSS code, and qudits 1 through kcontain the final distilled U β 7 |+⟩ ⊗k . This result enables a realization of magic state distillation on fewer qubits, but perhaps more...

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    This has the effect of uniformizing the probabilities of errors over their weights

    For every (noisy) multi-qubit magic state (like|CS⟩and|CCZ⟩), at every stage, we randomly permute the qubits of the multi-qubit magic state (2or3in the case of|CS⟩and|CCZ⟩, respectively). This has the effect of uniformizing the probabilities of errors over their weights. For e...

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    For example, in the(20T→3CCZ [CH17])·(8CCZ→2CCZ) concatenation, since the first level has three outputs, we run three copies of the second level 8CCZ→2CCZ factory

    Between different layers in a concatenation, we randomly route different magic states into differ- ent copies of the next-level factory. For example, in the(20T→3CCZ [CH17])·(8CCZ→2CCZ) concatenation, since the first level has three outputs, we run three copies of the second l...

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    best of both worlds

    For every copy of every distillation step in a concatenated scheme, that is not at the top level, we randomly permute the columns of the matrix describing that copy of the protocol. For example, in the case of a concatenation of(14T→2T)·(64T→2CCZ), there are two copies of the ...

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

    arXiv:2505.15917 [quant-ph](cit. on pp. 4, 6). [GSJ24] Craig Gidney, Noah Shutty, and Cody Jones. “Magic state cultivation: growing T states as cheap as CNOT gates”. In:arXiv preprint arXiv:2409.17595(2024) (cit. on p. 4). [GK09] Massimo Giulietti and Gábor Korchmáros. “A new ...

    2024 arXiv

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