High-Rate Amplitude-Damping Shor Codes with Immunity to Collective Coherent Errors

Ching-Yi Lai; En-Jui Chang

REVIEW 2 major objections 5 minor 37 references

High-Rate Amplitude-Damping Shor Codes with Immunity to Collective Coherent Errors

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

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Pith's one-line read A new Shor-code family approximately corrects up to $w$ amplitude-damping errors, and its dual-rail concatenation is immune to collective coherent errors.

desk verdict A real new AD code family with a solid AQEC theorem, but the efficient syndrome-only decoding claim is unproven for w≥2 due to non-injective syndromes. read the letter →

arxiv 2412.16450 v2 pith:764S77CN submitted 2024-12-21 quant-ph

En-Jui Chang , Ching-Yi Lai This is my paper · ORCID

classification quant-ph

keywords amplitude-dampingcodesapproximatequantumerrorcorrectionShordual-railcodecollectivecoherenterrorsconstant-excitationstabilizerhigh-rate

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 proposes a family of quantum error-correction codes, the $[[(w+1)(w+K), K]]$ AD Shor codes, and claims that they approximately correct up to $w$ amplitude-damping (energy-loss) errors while achieving a code rate approaching $1/(w+1)$. The supporting proof shows that the approximate quantum error-correction conditions hold: for any two error patterns of weight at most $w$, the overlap between error-affected codewords is a logical-state-independent coefficient plus a correction of order $O(\gamma^{w+1})$. Concatenating the same family with the dual-rail code produces $[[2(w+1)(w+K), K]]$ codes whose codewords have constant excitation, making them exactly immune to collective coherent errors while still correcting $w$ amplitude-damping events. A reader should care because realistic quantum channels suffer both energy relaxation and Hamiltonian drift, and this family addresses both with explicit local encoding, syndrome extraction, and recovery circuits.

What carries the argument

The engine of the argument is a CSS (Calderbank-Shor-Steane) stabilizer code, meaning a stabilizer code whose generators are each products of only $X$-type or only $Z$-type Pauli operators. Its codewords are parity-constrained superpositions of repetition-code blocks: each of the first $w$ blocks is either $|0\rangle^{\otimes(w+1)}$ or $|1\rangle^{\otimes(w+1)}$, and the last block encodes the $K$ logical bits as $|i\rangle^{\otimes(w+1)}$ or $|i'\rangle^{\otimes(w+1)}$. The inner $(w+1)$-qubit repetition code supplies two orthogonal block states, and the operator $A_k^\dagger A_k$ acts on each qubit as a diagonal operator with eigenvalue $1$ or $1-\gamma$, so any block where two logical words differ contributes a factor of order $\gamma$. With $w$ parity-constrained blocks, the accumulated difference between logical expectations is $O(\gamma^{w+1})$. Syndrome extraction measures only the local $Z\otimes Z$ stabilizers, yielding a single-bit syndrome adapted to amplitude damping, and recovery uses a lookup table followed by controlled-$Y$ rotations that implement an artificial damping channel.

What would settle it

Compute every inner product $\langle i|A_k^\dagger A_l|j\rangle$ for the $[[6,2]]$ 1-code and the $[[12,2]]$ 2-code, with $i,j$ ranging over all logical codewords and $k,l$ over all error patterns of weight at most $w$. If any off-diagonal term with $k\neq l$ or $i\neq j$ fails to vanish, or if any diagonal difference between $\langle 0|A_k^\dagger A_k|0\rangle$ and $\langle i|A_k^\dagger A_k|i\rangle$ is not of order $\gamma^{w+1}$ at small $\gamma$, then Theorem 1 is false. This is a finite check that settles the unproved orthogonality step.

Watch

Extended reading notes

Core claim

The paper's central discovery is that blockwise orthogonality inside a Shor-code structure suffices to make a whole family of high-rate codes approximate amplitude-damping correctors. Theorem 1 states that for the codewords $|i\rangle^{(w,K)}_{\mathrm{AD}}$ and for any two Kraus operators $A_k, A_l$ acting on at most $w$ qubits, the inner product satisfies $\langle i|A_k^\dagger A_l|j\rangle = \delta_{ij} C_{kl} + O(\gamma^{w+1})$; this is exactly the approximate Knill-Laflamme condition needed to improve the raw transmission fidelity from $1-O(\gamma)$ to $1-O(\gamma^{w+1})$. The proof decomposes $A_k^\dagger A_k$ into a tensor product of single-qubit diagonal operators and shows that each parity-constrained repetition block where two logical words differ contributes one factor of order $\gamma$, so the full difference is $O(\gamma^{w+1})$. Corollary 2 adds that the dual-rail concatenation is constant-excitation and hence immune to collective coherent errors, and the paper supplies local $Z$-stabilizer syndrome extraction together with a recovery operation built from controlled-$Y$ artificial damping rotations, worked out in detail for the $[[6,2]]$ code.

Load-bearing premise

The proof assumes that distinct error patterns of weight at most $w$ map to exactly orthogonal subspaces inside each logical codeword, so that all off-diagonal Knill-Laflamme inner products vanish; this is stated as 'straightforward' in the appendix and is verified only in examples, not proved in general.

Editorial extensions

If this is right

For fixed $w$, the asymptotic code rate $1/(w+1)$ is a square-root improvement over the single-logical-qubit Shor-code rate $1/(w+1)^2$, so the family protects multiple logical qubits against weight-$w$ energy loss with proportionally fewer physical qubits.
The $[[2(w+1)(w+K), K]]$ dual-rail concatenation corrects $w$ amplitude-damping events and is immune to collective coherent rotations, so it applies to channels where Hamiltonian drift and $T_1$ decay act simultaneously.
Because only $Z$-type stabilizers are measured, the scheme avoids the fidelity loss that $X$-based measurements cause under genuine amplitude damping, where a damped state is not an eigenstate of the corresponding stabilizer.
The CSS structure and transversal logical Pauli gates make the codes compatible with fault-tolerant protocols that prevent logical operations from introducing new correlated errors.
In 12 of the 18 small parameter cases compared in the paper, the new codes require fewer physical qubits than concatenating optimal stabilizer codes with the dual-rail code.

Reading between the lines

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

Inference: The unproved orthogonality step — that distinct weight-at-most-$w$ error patterns land in orthogonal subspaces inside each codeword — is the point to probe first; a direct computation of all off-diagonal inner products for the $[[12,2]]$ code would confirm or refute the family's guarantee beyond the worked examples.
Inference: Since the proof relies only on the diagonal form of $A_k^\dagger A_k$ and on repetition-code block parity, the same construction may transfer to qudit amplitude damping or to bosonic loss channels, where the excited-state projector is replaced by the number operator.
Inference: The corrected infidelity estimate for the $[[8,1]]$ code ($6\gamma^2$ instead of $28\gamma^2$) suggests that adding collective-coherent-error immunity through the dual-rail code is far cheaper than earlier analyses indicated; repeated-channel numerical simulations could test this for larger members of the family.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is the [[(w+1)(w+K), K]] interpolation between Shor's AD codes and the Fletcher–Shor–Win family, with the multi-logical-qubit AQEC proof and the dual-rail concatenation giving CC immunity. That part is worth taking seriously. The rate improvement from O(1/(w+1)^2) to O(1/(w+1)) is real, the encoding circuits are explicit, and Theorem 1 is a derivation, not a fit. The block-wise orthogonality argument in Appendix A is sound: distinct codeword terms differ in enough qubits that at most w damping events cannot make their supports overlap, and the iteration to O(γ^(w+1)) works because each difference factor contributes one power of γ. The [[6,2]] example is worked out with enough care to check out. The paper also correctly stresses that X-stabilizers are ill-suited to true AD noise and that Pauli approximations miss the energy bias; those points are fair and useful.

Soft spots, in proportion. First, Appendix B's 'first and second steps follow straightforwardly' is where the general-K orthogonality actually needs proof. A Hamming-distance argument can fill it, but it should be written out rather than asserted. Second, and more serious, the main text claims a lookup table that maps a length-w Z-syndrome string to AD errors, enabling 'efficient detection using only local operations.' The stress-test note is right that for w≥2 this is not injective. In the [[9,1]] w=2 code, error patterns {0,2} and {1} in the same block share the same two-bit syndrome yet require different recoveries. A syndrome-only recovery that treats both as the single error at position 1 will apply the wrong correction to the {0,2} case, leaving a residual error of order γ^2 rather than the claimed γ^(w+1)=γ^3. The Appendix D recovery uses projectors onto individual error states, which does distinguish the patterns, but that is not the advertised local Z-measurement lookup. So the efficient decoding claim is unsubstantiated as stated. This does not refute Theorem 1, which is an existence statement about the code's AQEC conditions and appears correct, but it does mean the paper overclaims the decoding overhead.

No simulations or code are provided, and the recovery analysis for general (w,K) is sketched rather than fully derived. Still, the central construction and the main theorem are solid enough to merit serious referee time. The authors should be asked to prove the orthogonality explicitly, give a correct syndrome-conditioned recovery or clearly weaken the efficient-decoding claim, and ideally provide numerical checks for a few small (w,K) cases. This is a genuinely useful family of codes, and with those revisions it would be a good contribution.

Referee Report

2 major / 5 minor

Summary. The paper introduces a family of [[(w+1)(w+K), K]] amplitude-damping (AD) codes that generalize Shor codes, proves approximate Knill-Laflamme conditions for correcting up to w AD errors (Theorem 1), and concatenates with a dual-rail inner code to obtain [[2(w+1)(w+K), K]] constant-excitation codes immune to collective coherent errors (Corollary 2). The paper also gives explicit encoding circuits, logical operators, a two-dimensional layout, and a comparison with the Duan et al. construction. The main theorem is an existence result; the recovery described in Appendix D uses projectors onto individual error states rather than a syndrome-based decoder.

Significance. If the results are correct, the code family offers a rate improvement over prior AD codes (asymptotic rate 1/(w+1) versus 1/(w+1)^2 for the single-logical-qubit Shor code) and includes the Shor code and the Fletcher et al. codes as special cases. The proof of Theorem 1 is a genuine derivation with no fitted parameters, and the paper provides explicit encoding circuits, logical operator implementations, and a concrete qubit-count comparison (Table II). The dual-rail concatenation step is standard but applied cleanly. However, the advertised efficient local decoding based on Z-stabilizer syndromes is not established for w ≥ 2, which is a significant gap for the practical claims.

major comments (2)

[Syndrome extraction and recovery operations (paragraphs after Eq. (8))] The claimed lookup table that maps a length-w Z-syndrome string to an AD error is not injective for w ≥ 2, so the advertised local syndrome-based decoding is unsupported. For the [[9,1]] w=2 code (the K=1 case), the error patterns {0,2} and {1} acting on the first three-qubit block both flip the two stabilizers Z0Z1 and Z1Z2, giving the same syndrome string 11; yet on |0_L> they produce orthogonal block-one states |010> and |101> (compare Table III). A decoder that assigns a single correction to syndrome 11 cannot correct both; for instance, applying the correction appropriate for the single error at position 1 to the {0,2} state leaves a logical error whose probability is O(γ^2), so the recovered fidelity would be 1 − O(γ^2) rather than the claimed 1 − O(γ^{w+1}) with w=2. The recovery in Appendix D (Eqs. D1–D2) uses projectors onto individual error states |i'^(k)> and can distinguish these patterns, but that recovery requires knowledge of the exact error pattern k and is not equivalent to the local Z-stabilizer lookup table described in the main text. The abstract's claim of 'efficient detection of AD errors using only local operations and ancillary qubits' therefore needs either a correct syndrome-conditional recovery for all patterns sharing a syndrome, an explicit extension of the syndrome with additional local measurements, or a qualified statement of what the stabilizer measurements achieve.
[Appendix B, proof of Theorem 1 (paragraph beginning 'It can be shown that each difference term...')] The central estimate that the difference of diagonal terms is O(γ^{w+1}) is asserted rather than proved. The text states that each difference (α(a_w=0) − α(b_w=i)) is O(γ) and that 'by iterating this argument' the total difference is O(γ^{w+1}), but no induction is supplied. The same issue appears in Appendix A (Lemma 4), where the second step asserts orthogonality for k ≠ l from 'orthogonal error syndromes' without giving the Hamming-distance argument. Since these estimates are the core of the AQEC proof, the appendices should be expanded to include the missing formal details; as written, the phrases 'It can be shown' and 'By iterating' leave the central derivation incomplete.

minor comments (5)

[Appendix B, text near Eq. (B7)] The phrase 'Similarly, or the logical one state' should read 'Similarly, for the logical one state'.
[Tables III and IV] Several codeword expressions contain unmatched parentheses (e.g., the entries for (w,K)=(2,1)), which makes the definitions hard to parse.
[Paragraph after Table I] The phrase 'a significant square-root improvement' is misleading: the asymptotic rate improves from 1/(w+1)^2 to 1/(w+1), which is a factor of (w+1), not a square-root scaling.
[Statement of Theorem 3] Theorem 3 is a restatement of a result from Duan et al. [19] and should be attributed to that work or proved in the appendix, since it is currently presented as a new theorem without proof.
[Appendix C, Eqs. (C1)–(C2)] The weight-1 fidelity contribution appears to contain a typo: the term is written as 4[γ(1−γ)^3/2], but the standard expression is 4γ(1−γ)^3; please check the 1/2 factor and the analogous term in Eq. (C2).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1 is derived from explicit codeword expansions and direct AQEC computation; the two self-citations are contextual and not load-bearing.

full rationale

The paper's central claim is a derivation, not a fit. Theorem 1 states approximate Knill-Laflamme conditions for the [[(w+1)(w+K), K]] code family, and the proof in Appendix B computes the diagonal differences explicitly from the codeword expansion (B1) and the tensor-product form of A_k^dagger A_k (B3), showing the difference is O(gamma^{w+1}). The parameters w and K are free design choices, and the damping rate gamma is the physical channel parameter used in the recovery, not a fitted value. The only unproven assertion is the off-diagonal orthogonality in Appendix B ('The first and second steps follow straightforwardly, as in the single-logical-qubit case'), but that is an omitted verification, not a circular reduction: the claim is a mathematical condition to be checked, and it is not assumed as the conclusion. The recovery schemes cite Leung et al. [29] for the artificial-AD technique, which is external support. The two self-citations ([20] and [25]) are parallel recent constructions and are not used as inputs to Theorem 1 or Corollary 2. Corollary 2 follows from Theorem 1 by standard concatenation with the dual-rail code, which is an independent construction. Accordingly, there is no step in which a prediction reduces by construction to an input, and the central result remains self-contained modulo the noted proof gap.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The code construction depends on these standard channel models and QEC criteria; none are introduced ad hoc to force the result. The integers (w,K) are code design choices, not fitted parameters, and γ is a physical input to the recovery.

assumptions (5)

standard math Knill-Laflamme approximate quantum error correction conditions are the correct criterion for assessing AD error correction.
The paper verifies Eq. (4) (⟨i| A†_k A_l |j⟩ = δ_ij C_kl + O(γ^(w+1))) and uses it to conclude the code is a w-code; this relies on the standard AQEC framework of Knill and Laflamme.
domain assumption The amplitude-damping channel is modeled by Kraus operators A0 = |0><0| + sqrt(1-γ)|1><1| and A1 = sqrt(γ)|0><1|, and the collective coherent error by identical single-qubit Z rotations.
Used throughout the paper (Section 'Composite error model') to define the noise model and to justify the correction of up to weight-w AD errors.
domain assumption For constant-excitation states, CC and AD errors commute and CE states are eigenstates of the CC unitary, making the composite channel valid.
The paper uses this to justify the composite error model Eq. (2) and to claim dual-rail concatenation yields CC immunity.
domain assumption Only transversal Z-type stabilizers can be measured for AD error syndrome extraction; X-type stabilizers are unreliable under AD noise.
The paper argues that X X stabilizers of the Bell state fail under AD noise and therefore restricts syndrome measurements to Z Z stabilizers; this determines the sequence of measurements in the recovery protocol.
domain assumption A controlled-Y rotation with angle θ implements an artificial AD channel with rate γ' = sin^2(θ/2), and this can be used to reverse the unwanted phase from the AD channel.
Eqs. (9)-(12) describe the recovery gadget; the recovery protocol relies on this identity to counteract the Y-rotation induced by damping.

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Cite this review

Pith. "Pith review of High-Rate Amplitude-Damping Shor Codes with Immunity to Collective Coherent Errors." pith.science (2026). https://pith.science/paper/764S77CN

@misc{pith2026241216450,
  author       = {Pith},
  title        = {Pith review of: High-Rate Amplitude-Damping Shor Codes with Immunity to Collective Coherent Errors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/764S77CN}},
  note         = {Machine review of arXiv:2412.16450}
}

abstract

We introduce a family of high-rate amplitude-damping (AD) Shor Codes, designed to effectively correct AD errors while maintaining immunity to collective coherent (CC) errors. The proposed $[[(w+1)(w+K), K]]$ AD codes can approximately correct up to $w$ AD errors, with flexible parameters $(w, K)$, and we provide a rigorous proof that these codes satisfy the approximate quantum error correction conditions. These codes leverage structured stabilizer measurements, enabling efficient detection of AD errors using only local operations and ancillary qubits. We further construct a family of CC-AD Shor codes by concatenating these AD codes with the dual-rail code.

Figures

Figures reproduced from arXiv: 2412.16450 by the authors.

**Figure 2.** FIG. 2. Encoding circuit for the [[( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Layout of the [[12 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Encoding circuit for the [[2( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.

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Every codeword must be an eigenstate of the sta- bilizer with eigenvalue +1

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Every stabilizer must either commute or anti- commute with each element in the error set. The first property is straightforward and independent of error models. However, the second property reveals a critical limitation: some stabilizer codes constructed to correct stochastic Pauli errors may not stabilize code- words under AD noise. To illustrate this di...

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Mathematically: ( ˆI ⊗ ⟨0|) ˆCY(θ)(|ψ⟩ ⊗ |0⟩) = (|0⟩ ⟨0| + cos θ 2 |1⟩ ⟨1|) |ψ⟩ , (11) ˆX( ˆI ⊗ ⟨1|) ˆCY(θ)(|ψ⟩ ⊗ |0⟩) = (sin θ 2 ˆY |0⟩ ⟨1|) |ψ⟩

when the ancilla is measured in |0⟩, and a damped event ( ˆA′1), followed by a conditional application of the Pauli ˆX correction on the resulting state. Mathematically: ( ˆI ⊗ ⟨0|) ˆCY(θ)(|ψ⟩ ⊗ |0⟩) = (|0⟩ ⟨0| + cos θ 2 |1⟩ ⟨1|) |ψ⟩ , (11) ˆX( ˆI ⊗ ⟨1|) ˆCY(θ)(|ψ⟩ ⊗ |0⟩) = (sin θ 2 ˆY |0⟩ ⟨1|) |ψ⟩ . (12) By setting √1 − γ′ = cos θ 2 and √γ′ = sin θ 2 , t...

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