Coherent errors, syndrome extraction and the Robustness of Magic in stabiliser codes

Enrico Bozzetto; Jonte R. Hance

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Coherent errors, syndrome extraction and the Robustness of Magic in stabiliser codes

Enrico Bozzetto, Jonte R. Hance

Quantum Physics · Submitted 28 Oct 2026

Pith's verdict: The central additivity result is real and new; send to review.

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arXiv ID2610.38554 (version 2)

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CitationEnrico Bozzetto, Jonte R. Hance. "Coherent errors, syndrome extraction and the Robustness of Magic in stabiliser codes." arXiv:2610.38554 (2026). https://pith.science/paper/QOKE2G54

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Editor's note

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The thing to know: this paper has a real theorem, not a heuristic. Under condition (C), for a unitary error on a correctable region supporting no stabiliser, R(U ρ_enc U†) equals R(J_U ⊗ ρ_L), and ideal syndrome extraction returns R(ρ_L). That is a clean, code-independent accounting of how coherent errors add magic and how syndrome extraction removes exactly that added copy. I checked the flow of the proof; Lemma 3 (Bell-pair normal form) is the load-bearing step and it is proved, with the regime where the code has enough room made explicit via 4|A| ≤ 2(n−k). The authors also show the identity fails when a logical operator or stabiliser sits in the region, and Corollary 2 covers Pauli rotations outside N(S). No circular dependence, no fitted parameters.

Theorem 2 is a useful standalone result: appending any non-stabiliser state to any state lifts R+1 by a factor μ(σ)>1, tightly bounded by a linear program. The proof is short and sound, and the comparison with Piani and with multiplicativity failures in the literature is honest rather than padded.

What gets real credit: exact LP checks for the numerical tables, rational primal/dual certificates for the closed-form values, and a Zenodo notebook. The self-citation to the code-adapted Wigner representation is used as a tool and explicitly marked; not a problem.

Soft spots, in proportion. Several auxiliary claims rest on finite enumeration and floating-point LP rather than proof: the magic-capacity equality for two-qubit gates is proved for gates Clifford-equivalent to diagonal or single-qubit gates, and the paper openly leaves the remaining two-qubit classes open. The lifetime section is a code-capacity model with ideal syndrome extraction and a maximum-likelihood decoder; the 1.5× orientation gain is real within that model, but it should not be read past those assumptions. The Reed–Muller decoder is restricted to weight ≤5 Paulis with the omitted probability quantified at 4.6×10⁻⁹; the authors state this, and the claimed insensitivity is plausible. Tightness of Theorem 2 is shown for some σ in a range of R(ρ) and left open elsewhere; acknowledged.

Bottom line: this deserves serious refereeing. For people working on coherent noise in QEC and on magic resource theories, the central theorem and the appending bound are new, the proofs are mostly complete, and the caveats are stated rather than hidden. I would send it to review and expect acceptance after tightening the presentation of the numerical-dependent claims.

Referee report

Significance

If the main theorem is correct it is a substantive, code-independent statement: the magic added by a coherent error on a correctable region is exactly the Choi-state robustness of the error, and syndrome extraction removes precisely that amount. The result is parameter-free and derived rather than fitted. The paper also contains a large amount of machine-checkable support: Table I and Appendix F report exact or high-precision linear programs, and the accompanying notebook is explicitly referenced. The paper is honest about its boundaries: it acknowledges that regions containing logical operators are only bounded by magic capacity, that the capacity equality for general two-qubit gates remains open, and that syndrome extraction is treated as ideal. These are clearly stated limitations rather than hidden gaps.

Minor objections (6)

[Section IV] In the tightness paragraph, the line "R(F⊗F)=R(F⊗F⊥)=12+√3" should read "1/2+√3"; the following equality with (1+√3)μ(F)−1 confirms this, and the current typesetting is misleading.
[Section III, Corollary 1] The definition of J_A^U is typeset as "2a(1⊗UΠ_A)|Φ⟩⟨Φ|"; this should be 2^a (or an otherwise explicit normalisation) to avoid confusion with a factor 2a.
[Sections I and III] There are several small typesetting lapses: "Fattalet al" should be "Fattal et al.", and the phrase "The while second statement" should read "The second statement".
[Section VIII] When Theorem 1 and Lemma 3 are extended to qudit codes, the room condition analogous to 4|A|≤2(n−k) should be stated explicitly; for qudits the same surjectivity argument gives 2|A|≤n−k, and spelling this out would make the extension fully self-contained.
[Appendix F / Section VII] The note that the omitted 4.6×10⁻⁹ probability in the Reed–Muller coset sums changes no lifetime is reassuring; consider placing this validation next to Table III in the main text.
[Throughout] Several displayed expressions have lost superscript formatting, for example C^4T and F_2 in Section V and Appendix D; a careful normalization of the mathematical typesetting would remove avoidable ambiguity.

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The paper

The paper claims

A coherent error on a correctable, stabiliser-free region of any stabiliser code adds exactly the robustness of the error's Choi state to the encoded magic; syndrome extraction removes precisely that, restoring the logical robustness.

The reading

Stored magic states are the fuel of a fault-tolerant quantum computer, and between preparation and use they sit inside stabiliser codes where coherent over-rotation errors are a realistic threat. This paper proves that, in every stabiliser code, a unitary error confined to a region that is erasure-correctable and contains no stabiliser acts in a simple and exact way: the encoded state's Robustness of Magic rises by exactly the robustness of the error's Choi state, and syndrome extraction removes exactly that amount, restoring the logical robustness. Because the added and removed amount is independent of the code and is strictly positive for every non-Clifford error, ideal syndrome extraction never consumes the logical state's own magic after such errors; the loss is entirely the error's. The paper also equates the magic capacity of broad classes of one- and two-qubit gates with their Choi-state robustness, gives an $F_2$-linear test for when Clifford error layers turn syndrome extraction into a logical-operator measurement, and shows that choosing the storage plane of a magic state extends its lifetime by about 1.5x on rotated surface codes under low code-capacity depolarising noise.

What carries the argument

The argument runs on three objects. Lemma 1 (the sector identity) says $R(E_C\rho) = \sum_s P(s) R(\rho_L(s))$, so robustness is additive over the classical syndrome flag and syndrome extraction becomes a weighted average over decoded logical states. Lemma 2 says syndrome extraction twirls a region-local error: under (C), only the weights $|u_P|^2$ of the error's Pauli expansion survive. Lemma 3 (the Bell-pair normal form) is the load-bearing piece: under (C), a Clifford $W$ on the complement of $A$ sends every code state to $|\Phi\rangle\langle\Phi|_{AA'} \otimes V''_0 \rho_L V''_0^\dagger$, turning the error into a unitary on one half of Bell pairs whose output is exactly the Choi state $J_U$. Theorem 2 supplies the positivity: for any $m$-qubit $\sigma$ and $n$-qubit $\rho$, $R(\sigma\otimes\rho) + 1 \ge \mu(\sigma)(R(\rho)+1)$, with $\mu(\sigma) \ge 1$ and equality only for stabiliser mixtures. For errors that can touch logical operators, the magic capacity $C(U)$, the largest factor by which any stabiliser input's robustness can grow, replaces Theorem 1, and the paper proves $C(U) = R(J_U)$ for all single-qubit gates and for two-qubit gates Clifford-equivalent to diagonal or single-qubit gates.

Extended reading notes

Core claim

Under condition (C), that the errored region $A$ supports no logical operator and no stabiliser ($N(S) \cap P_A = \{1\}$), Theorem 1 states that for every logical state $\rho_L$ and every unitary $U$ on $A$, $R(U\rho_{\mathrm{enc}}U^\dagger) = R(J_U \otimes \rho_L)$ and $R(E_C(U\rho_{\mathrm{enc}}U^\dagger)) = R(\rho_L)$, where $J_U$ is the Choi state of $U$. The first identity holds because a Clifford acting only outside $A$ factors every code state into Bell pairs on $A$ plus a smaller encoded logical state (Lemma 3), so $U$ simply manufactures its own Choi state. The second identity holds because syndrome extraction twirls $U$ into a Pauli mixture that decodes to $\rho_L$ in every sector (Lemmas 1 and 2). The removed amount $\Delta = R(J_U \otimes \rho_L) - R(\rho_L)$ is the same for every code in which (C) holds, and Theorem 2 shows it is strictly positive whenever $U$ is non-Clifford: appending any non-stabiliser state multiplies $1+R$ by a factor $\mu(\sigma) > 1$ given by a linear program. The same statements extend to channels on $A$ through their Choi states, positive exactly when the channel is not completely stabiliser-preserving, and throughout, syndrome extraction is treated as ideal.

Load-bearing premise

The load-bearing premise is Lemma 3's Bell-pair normal form: that under condition (C) a Clifford acting only outside the errored region $A$ can factor every code state into shared Bell pairs on $A$ next to a smaller encoded logical state, a factorization that needs the complement to have enough room ($4|A| \le 2(n-k)$) and that is known to fail as soon as $A$ contains a logical operator or a stabiliser.

Editorial extensions

If this is right

The classical simulation overhead for the segment between the error and syndrome extraction is set exactly: because the sampling cost grows with the square of the robustness, that segment costs the same as simulating with $J_U \otimes \rho_L$ as input, in every code satisfying (C).
The magic removed by syndrome extraction is never magic of the logical state and is the same for every code in which (C) holds, so no choice of code changes this particular loss; it is positive for every non-Clifford error.
The magic capacity equals the Choi-state robustness for every single-qubit gate and for every two-qubit gate Clifford-equivalent to a diagonal or single-qubit gate, making the capacity bound tight across these families.
On rotated surface codes of odd distance, full layers of $\pi/4$ rotations about $ZZ$ or $XX$ axes, and full layers of CZ gates at distances 3, 5 and 7, never make syndrome extraction measure a logical operator, while every horizontal or vertical CNOT layer tested does; the criterion is $F_2$-linear.
Storing a single-qubit magic state in the logical $XZ$ plane extends its magic lifetime by about 1.5x on rotated surface codes under low code-capacity depolarising noise, and under a uniform rotation repeated every round the decoded logical state's robustness oscillates, so a state can re-enter the stabiliser polytope and leave it again.

Reading between the lines

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

The normal-form Clifford $W$ isolates the error's Choi state before it is replaced by fresh Bell pairs, so a known calibrated over-rotation on a known region could be harvested by gate teleportation during the brief window in which the logical state is less protected; a natural engineering question is how to schedule that window into idle time.
The oscillation of robustness under repeated uniform rotations means single-round 'enter the polytope' estimates are misleading for coherent noise, and suggests a testable memory-scheduling strategy: inject or re-purify magic states at the phase of the oscillation where the state sits outside the polytope.
Theorem 2's proof uses only four cone properties of the free set, which the paper notes also hold for entanglement and coherence; the same 'appending a non-free state strictly increases the resource' statement should transfer to other convex resource theories whose free cone satisfies those four properties.
Because the removed robustness is code-independent under (C), protection against coherent errors on correctable regions should concentrate on the error itself, twirling, refocusing, or correcting before extraction, rather than on code distance; a testable consequence is that two codes of very different distance but the same region $A$ show identical $\Delta$.

Original abstract

Coherent errors leave stabiliser-code-encoded magic states in superpositions of syndrome sectors. We ask how much magic these coherent errors add, and whether syndrome extraction depletes logical magic. Using Robustness of Magic, we show that, for a unitary error U on a correctable region which supports no stabiliser, the encoded state has the same robustness as the logical state plus one copy of the Choi state of U , in every stabiliser code encoding. Syndrome extraction removes this copy, returning the logical robustness. The removed magic is positive for every non-Clifford error: adding a non-stabiliser state to any state multiplies one plus the robustness by a (tightly- bounded) factor above one. The magic capacity of two-qubit gates Clifford-equivalent to diagonal or single-qubit gates is equal to their Choi-state robustness. We give a linear-algebraic test for when a Clifford error makes syndrome extraction measure a logical operator, and give magic lifetimes of stored single-qubit magic states under Pauli noise. Choosing the plane of the stored state extends these lifetimes about 1.5x on rotated surface codes under low code-capacity depolarising noise. We identify states and channels in odd-prime dimension for which Wigner negativity and robustness show equal lifetimes.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 6,976 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.29 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Coherent errors leave stabiliser-code-encoded magic states in superpositions of syndrome sectors. We ask how much magic these coherent errors add, and whether syndrome extraction depletes logical magic. Using Robustness of Magic, we show that, for a unitary error U on a correctable region which supports no stabiliser, the encoded state has the same robustness as the logical state plus one copy of the Choi state of U , in every stabiliser code encoding. Syndrome extraction removes this copy, returning the logical robustness. The removed magic is positive for every non-Clifford error: adding a non-stabiliser state to any state multiplies one plus the robustness by a (tightly- bounded) factor above one. The magic capacity of two-qubit gates Clifford-equivalent to diagonal or single-qubit gates is equal to their Choi-state robustness. We give a linear-algebraic test for when a Clifford error makes syndrome extraction measure a logical operator, and give magic lifetimes of stored single-qubit magic states under Pauli noise. Choosing the plane of the stored state extends these lifetimes about 1.5x on rotated surface codes under low code-capacity depolarising noise. We identify states and channels in odd-prime dimension for which Wigner negativity and robustness show equal lifetimes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Coherent errors leave stabiliser-code-encoded magic states in superpositions of syndrome sectors.	background 0.94	0.21	0.07	asserted 1.00
s1	We ask how much magic these coherent errors add, and whether syndrome extraction depletes logical magic.	gap 0.53	0.17	0.05	unclear 0.92
s2	Using Robustness of Magic, we show that, for a unitary error U on a correctable region which supports no stabiliser, the encoded state has the same robustness as the logical state plus one copy of the Choi state of U , in every stabiliser code encoding.	result 0.96	0.93	0.54	asserted 1.00
s3	Syndrome extraction removes this copy, returning the logical robustness.	result 0.98	0.77	0.25	asserted 1.00
s4	The removed magic is positive for every non-Clifford error: adding a non-stabiliser state to any state multiplies one plus the robustness by a (tightly- bounded) factor above one.	result 0.91	0.82	0.17	asserted 1.00
s5	The magic capacity of two-qubit gates Clifford-equivalent to diagonal or single-qubit gates is equal to their Choi-state robustness.	result 0.99	0.85	0.09	asserted 1.00
s6	We give a linear-algebraic test for when a Clifford error makes syndrome extraction measure a logical operator, and give magic lifetimes of stored single-qubit magic states under Pauli noise.	contribution 0.93	0.95	0.07	asserted 1.00
s7	Choosing the plane of the stored state extends these lifetimes about 1.5x on rotated surface codes under low code-capacity depolarising noise.	result 0.99	0.89	0.06	asserted 0.99
s8	We identify states and channels in odd-prime dimension for which Wigner negativity and robustness show equal lifetimes.	result 0.90	0.89	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Schematic of Theorem 1, for a region

**Figure 2.** FIG. 2. Robustness ∆ removed by syndrome extraction after

**Figure 3.** FIG. 3. A layer of disjoint horizontal CNOTs (arrows, control

Figures reproduced from arXiv:2610.38554 by the authors.

Discussion

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