On approximate quantum error correction for symmetric noise

Gereon Ko{\ss}mann; Julius A. Zeiss; Mario Berta; Omar Fawzi

REVIEW 4 major objections 6 minor 1 cited by

On approximate quantum error correction for symmetric noise

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Outer SDP bounds for approximate quantum error correction can be rounded into concrete codes, and noise symmetries can be merged with extendability to make fixed-level relaxations computable.

desk verdict A genuinely useful symmetry-reduction framework for AQEC, with a likely-fixable proof gap in the central averaging lemma. read the letter →

arxiv 2507.12326 v1 pith:YD3UDZXT submitted 2025-07-16 quant-ph

Gereon Ko{ss}mann , Julius A. Zeiss , Omar Fawzi , Mario Berta This is my paper · ORCID

classification quant-ph MSC 81P6890C2220C3081P45

keywords approximatequantumerrorcorrectionsemidefiniteprogramminghierarchysymmetryreductionextendabilitychannelfidelityroundingschemedepolarizingdeFinettitheorem

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 targets the gap between outer bounds and real codes in approximate quantum error correction (AQEC): the extendability semidefinite programming (SDP) hierarchy gives converging upper bounds on the best channel fidelity, but its fixed levels are too large to evaluate. The authors claim two fixes work together. First, a measurement-based rounding scheme turns an optimal solution of the level-$n$ relaxation into a genuinely feasible encoder--decoder pair whose fidelity is within the same polynomial-in-dimension, $\sqrt{\ln d / n}$ gap as the outer bound. Second, they show that noise symmetries---permutation symmetry of identical copies, isotropic unitary symmetry, and Choi covariance---can be combined with the extendability permutation symmetry, provided the symmetries act diagonally on every extension copy. The payoff is that non-trivial outer bounds for one logical qubit encoded into three physical qubits under depolarizing or amplitude-damping noise become numerically accessible.

What carries the argument

The objects doing the work are the constrained $n$-extendable states, the sets $\Sigma^n_{\mathrm{prod}}(LP:\bar L\bar P)$, whose constraints (positivity, permutation invariance, and two partial-trace equalities) encode Choi's theorem. The rounding machinery is an informationally complete measurement on the extension copies, which projects an outer solution onto a convex combination of product states while preserving the partial-trace constraints. The symmetry machinery is the enlarged diagonal action of the iid-symmetry group $S_m$, the isotropic unitary group $U(2)$, and the Choi covariance group $V$ on all extension copies, together with the extendability group $S_n$; Schur--Weyl duality and branching rules decompose the commutant into small isotypic blocks, and Proposition 5.1 supplies the group averaging that keeps the hierarchy constraints intact. The general criterion for merging an objective symmetry $U$ with a variable symmetry $V$ is that $W=\{UV\}$ be a group.

What would settle it

Take the non-commuting action of Example 7.2 (two-copy iid symmetry versus two-column extendability on six subsystems), compute the symmetry-reduced level-2 SDP in the commutant of the generated group $W$, and compare its value with the un-reduced level-2 relaxation; Proposition 5.1 and Theorem 5.3 predict equality only for diagonal actions, so a mismatch would disprove the enlarged-invariance claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a pair of theorems. Theorem 3.1 states that from a level-$n$ optimizer of the outer hierarchy one can construct inner states $\sigma^{(n)}_{LP\bar L\bar P}\in\Sigma^n_{\mathrm{prod}}$ whose fidelity gap to the true channel fidelity closes as $c(d_{\bar L\bar P})\sqrt{2\ln 2\,\ln d_{LP}/n}$, matching the outer convergence rate; the construction measures the extended systems with an informationally complete measurement and forms a separable post-measurement state. Theorem 5.3 states that, when the noise symmetries are extended diagonally to all $n$ extension copies, the symmetry-reduced SDP in the commutant of the combined group $W$ has the same optimal value as the original level-$n$ relaxation; the proof rests on Proposition 5.1's group-averaging argument and a Hilbert--Schmidt basis of the commutant. The paper also shows that for a global depolarizing channel the reduction degenerates to a linear program at low levels, and Section 7 identifies a necessary condition for combining symmetry groups: the product set $W=\{UV\}$ must itself be a group.

Load-bearing premise

The load-bearing premise is that the noise symmetries can be made to act diagonally on every copy of the extended system, so that averaging an optimal solution over the enlarged group still satisfies all hierarchy constraints.

Editorial extensions

If this is right

Fixed level-2 outer bounds for one logical qubit encoded into three physical qubits under iid depolarizing or amplitude-damping noise become computable, including cases that previously required intractable dense SDPs.
The rounding theorem yields explicitly constructible encoder-decoder pairs whose fidelity is within the same $O(\sqrt{\ln d / n})$ gap as the outer bound, giving certified warm starts for see-saw optimization.
For global depolarizing channels, the symmetry reduction makes low levels linear programs; for $m$-fold single-qubit depolarizing without extensions, the reduced program is a linear program.
The symmetry-reduced SDP has exactly the same optimal value as the original level-$n$ relaxation, so the reduced computation is still a valid outer bound.
When the combined symmetry set $W=\{UV\}$ is not a group, joint symmetry reduction fails, so the diagonal-action condition is not just convenience but necessary.

Reading between the lines

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

The rounding scheme is not tied to AQEC: it should transfer to other constrained separability problems where the hierarchy constraints are partial-trace constraints and an informationally complete measurement is available.
A testable extension is to benchmark asymmetric noise (e.g., unequal depolarizing rates per qubit) where the iid symmetry is absent; the Section 7 criterion predicts that only diagonal actions of the remaining symmetries can be safely merged.
The linear-program reductions suggest that small symmetric codes can be exhaustively benchmarked against the best non-signaling-assisted bounds, which may reveal how much of the gap comes from the coding constraints rather than the noise model.
One could strengthen the rounding by constructing problem-adapted measurements with fewer outcomes; the paper notes such measurements give valid codes, though without the rigorous optimality guarantee.

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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 paper is worth engaging with. It does three real things: it constructs a rounding scheme that converts outer-hierarchy optimizers into certified inner codes, it builds a symmetry-reduction framework that combines iid, isotropic, Choi, and extendability symmetries, and it gives a clean group-theoretic criterion for when such combined reductions are valid. The representation-theoretic tables check out (block dimensions sum to 4096), and the numerical examples, while modest, show the method producing outer bounds that were previously out of reach. This is a genuine step toward making AQEC bounds computable for small codes.

The soft spots are real but addressable. The stress-test note is on target: the proof of Proposition 5.1 verifies the wrong partial-trace identities. Equation (64) uses tr_P[ρ] = 1_L/d_L ⊗ ρ, but the hierarchy constraint (63) requires tr_L[ρ] = 1_P/d_P ⊗ ρ and a matching condition on the ¯P^{(n)} trace. The same P/L swap appears in Lemma 5.2 and is inherited by Theorem 5.3. As written, the symmetry reduction is unproven. That said, the averaging idea itself is sound: because the symmetries act diagonally on the P and L systems, the correct constraints are preserved by cyclicity of the partial trace under the twirl. The error is likely a typo, not a conceptual break. Section 7's Example 7.2 actually supports this reading by showing that the diagonal action is essential; the proof just needs to be re-run with the right trace constraints.

Two smaller issues. Theorem 3.1's statement is looser than the proof: it claims the same performance guarantee as Proposition 2.1(a), but the bound in (31) has an extra d^2 factor and a square root of log that don't match the clean poly(d_L,d_P)√(ln d/n) in the abstract. That is a presentation flaw, not a broken result. And the numerics come without code, data, or error bars, so a referee cannot independently verify the plots; for a proof-of-principle section this is forgivable but worth flagging.

The central argument holds up, modulo the fixable proof gap. This deserves a serious referee. I would send it to review and ask the authors to correct the P/L swap, tighten Theorem 3.1's constants, and release the numerical code.

Referee Report

4 major / 6 minor

Summary. The paper revisits the extendability-based SDP hierarchy for approximate quantum error correction (AQEC) introduced by Berta et al. and contributes two main developments. First, it proposes a measurement-based rounding scheme that converts outer-hierarchy optimizers into feasible inner encoder-decoder pairs with a convergence guarantee comparable to the outer hierarchy. Second, it develops a symmetry-reduction framework for AQEC that combines the permutation symmetry of n-extendable states with symmetries of the noise model, specifically iid, isotropic, and Choi symmetries, and applies it to compute outer bounds for encoding one logical qubit into three physical qubits for depolarizing and amplitude-damping noise. The paper also contains a general discussion of when two symmetry groups can be combined for SDP reduction, formulated through the condition that the set UV is a group. The numerical section reports Level-1 and Level-2 outer bounds for the three-qubit examples.

Significance. If the proof defects described below are repaired, this is a useful and timely contribution. The rounding theorem is an original mechanism with explicit convergence bounds, and the explicit representation-theoretic block decomposition in Section 5.4.1 is a valuable concrete data point: the block dimensions sum to 4096, showing internal consistency. The numerical examples demonstrate that nontrivial Level-2 bounds for small symmetric codes can be obtained, which was previously intractable with generic SDP solvers. The general criterion for combining symmetry groups in Section 7 addresses a genuine methodological question. However, the load-bearing symmetry-reduction claim currently rests on a proof that verifies the wrong constraint identities, so the mathematical core of the paper needs a substantial repair before the claims can be accepted.

major comments (4)

[Section 5.1, Proposition 5.1] The averaging argument proves the wrong constraint identities. Equation (64) verifies tr_P[tilde rho] = 1_L/d_L ⊗ tilde rho_{barP barL} and Eq. (65) verifies tr_PL[tilde rho] = tilde rho_{barP barL}, but the level-n constraints in Eq. (63) require tr_L[rho] = 1_P/d_P ⊗ rho_{barP barL} and tr_{barP^(n)}[rho_{barP barL}] = rho_{barP barL}^{(1..n-1)} ⊗ 1_{barL^(n)}/d_{barL^(n)}. The proof therefore does not establish that the averaged state lies in Sigma^n_prod, and without that Proposition 5.1 and hence Theorem 5.3 are unproven. Since the averaging unitaries act on the systems over which the required partial traces are taken, the intended preservation is plausible, but the proof must be rerun with the correct constraints.
[Section 5.2, Lemma 5.2 and Theorem 5.3] Equations (68)-(69) in Lemma 5.2 state tr_P[rho] = 1_L/d_L ⊗ rho_{barP barL} and tr_L[rho_{barP barL}] = rho_{barP barL}^{(1..n-1)} ⊗ 1_P/d_P; neither is a hierarchy constraint. The hierarchy requires tr_L on the full state with 1_P/d_P on the P-side and tr_{barP^(n)} on the reduced state with 1_{barL^(n)}/d_{barL^(n)}. The constraints in Eq. (71) inherit this swap, so the claimed equivalence between the symmetry-reduced SDP and the level-n relaxation is not established. These statements are load-bearing for the numerical results in Section 6 and must be corrected.
[Section 3.1, Theorem 3.1] The dimension factor in the proof is inconsistent: Eq. (25) and the original program use d_P^2, while Eq. (31) repeatedly uses d_{barL barP}^2. The measurement parameter m is also not fixed: the proof constructs an m-fold product measurement and then states that all 2≤m≤n are iterated, but the object sigma[n] is never defined as a function of m, and the relationship between m and the extension level n is left implicit. The rounding scheme is conceptually sound, but the proof needs a clean statement with consistent dimensions before Theorem 3.1's guarantee is certified.
[Section 7.2, Theorem 7.6] The proof of the group extension structure is not rigorous as written. The map φ: G → Sym(barP_{1...m} × ... × barP^(n)_{1...m}) is stated to be surjective onto S_m≀S_n without a construction, and the kernel computation refers to 'ker(G→S_m≀S_{n-1})' although the preceding map is to S_m≀S_n. The conclusion G ≅ A_m ⋊ (S_m≀S_n) may be correct, but as written the proof does not establish the order of the kernel, which Corollary 7.7 relies on. This should be repaired for the general framework of Section 7 to be reliable.

minor comments (6)

[Section 2.2, Eq. (5)] The notation S(H_LP, barLbarP) is not standard; the states should be density matrices on H_LP and H_barLbarP, respectively.
[Section 3.1, Eq. (26)] The informationally complete measurement M is not defined as a POVM; the normalization and the distortion constant c(d) should be stated explicitly.
[Section 5.4.1, Table (94)] The text says '12 blocks of sizes' but the table lists multiplicities; the actual block dimensions should be given explicitly, or the entries should be called multiplicities.
[Proposition 5.5] The sentence 'a several linear variables and four 2×2 SDP variables' is ungrammatical, and the proof of the LP claim in the trivial case shown in Eq. (101) is only sketched.
[Example 7.2, Eq. (121)] The braced expressions in Eq. (121) are notationally garbled and should be cleaned up; the notation V in (113) also conflicts with the Choi-symmetry group V used elsewhere.
[Figure 4] No details are given on how the Level 2 curves were obtained, such as solver, precision, or confirmation of the block decomposition; adding this information would strengthen the numerical claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the outer bounds, inner rounding, and symmetry reduction are derived from external published theorems and standard SDP arguments, not from fitted inputs or self-referential definitions.

full rationale

The claimed derivation chain is not circular. The outer hierarchy is taken from the published, peer-reviewed work of Berta et al. [3], and the rounding construction of Theorem 3.1 invokes the published de Finetti-type bound [3, Thm. 2.3] and the measurement constant of [22]; these are parameter-free results whose stated assumptions do not include the target fidelities, so the author overlap in the citations does not make the reasoning circular. No parameter is fitted to data and then renamed as a prediction: all bounds are SDP optima, and the inner points are constructed from outer optimizers with explicit convergence estimates. The symmetry reduction in Sections 5 and 7 also does not reduce to its inputs by definition: Proposition 5.1 attempts a group-averaging argument that the averaged optimizer remains feasible, and Theorem 5.3 states an equivalence between the original and symmetry-reduced SDPs. A legitimate concern is that the written proof of Proposition 5.1 appears to verify the swapped identities (64)-(65), namely tr_P[rho]=1_L/d_L⊗rho and tr_PL[rho]=rho, rather than the level-n constraints in (63) requiring tr_L[rho]=1_P/d_P⊗rho and trace over the n-th extension; this is a possible gap or typo in the proof, not a circular step, because the reduced SDP is not constructed to equal the target value by fiat. Similarly, the joint-symmetry criterion W=UV being a group in Definition 7.3 is a stated sufficient condition, and Proposition 7.5(d) explicitly proves that when it holds, twirling preserves the objective and constraints; the claim is not assumed as its own conclusion. There is no self-definitional reduction, no fitted input renamed as a prediction, and no load-bearing reliance on an unverified self-citation. The paper may have proof gaps, but those are correctness risks, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The ledger is light on fitted parameters: the theory introduces none, and the only hand-chosen numbers are the noise parameters p and gamma used to sweep the numerical examples. The load-bearing background consists of four published or standard ingredients: the linear-constraint de Finetti bound of [3] (authors overlap with this paper, but it is a peer-reviewed, parameter-free theorem), Schur-Weyl duality and the symmetry-reduction machinery of [12] and [9], the covariance properties of the specific noise models, and the Reimpell-Werner bilinear reformulation. No invented entities. The correctness risk sits in whether the constraint handling in Lemma 5.2 and Thm. 5.3 matches the stated hierarchy (63), not in the axioms themselves.

free parameters (1)

Noise parameters p (depolarizing) and gamma (amplitude damping) = p in [0, 0.7], gamma in [0, 1], grid inputs
Hand-chosen physical noise parameters for the Figure 4 examples. They are inputs of the numerical experiments, not fitted to data and not used to set any constant in the theorems; the theory itself has no fitted free parameters.

assumptions (4)

domain assumption Quantum de Finetti theorem with linear constraints ([3, Thm. 2.3]): the n-level relaxation gap closes as poly(d)/sqrt(n), and the measurement-based reduction bound ||rho* - sigma[n]||_1 <= c(d) sqrt(2 ln 2 ln(d)/n) holds for some m in [2, n].
This published theorem from [3] is the engine behind both the convergence of the outer hierarchy (Prop. 2.1) and the inner rounding (Thm. 3.1). The paper does not re-derive it; it is an unproved background result shared with the cited authors (self-citation, but peer-reviewed and parameter-free).
standard math Schur-Weyl duality for S_n and U(d), branching rules for S_n down to S_(n-1), and the regular *-representation symmetry reduction of [12] as adapted to AQEC by [9].
Used in Thm. 5.3 and Sec. 5.4 to decompose the 4096-dimensional state space into isotypic blocks; the internal consistency of the resulting tables was checked (block dimensions sum to 4096).
domain assumption Noise covariance and invariance assumptions: N = M^otimes m (iid), depolarizing channel covariance under single-qubit unitaries (Choi symmetry), and isotropic invariance of the maximally entangled state under U tensor conjugate-U.
These are stated in Sec. 2.3 and Appendix A and define the regimes where the symmetry reduction applies; the amplitude-damping example only uses the iid and isotropic parts.
domain assumption Choi isomorphism and the bilinear AQEC reformulation (Eqs. 3-6) from Reimpell-Werner [31], including the cSEP convexification.
The hierarchy and all bounds in the paper are bounds on this reformulated channel-fidelity objective.

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Pith. "Pith review of On approximate quantum error correction for symmetric noise." pith.science (2026). https://pith.science/paper/YD3UDZXT

@misc{pith2026250712326,
  author       = {Pith},
  title        = {Pith review of: On approximate quantum error correction for symmetric noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YD3UDZXT}},
  note         = {Machine review of arXiv:2507.12326}
}

read the original abstract

We revisit the extendability-based semi-definite programming hierarchy introduced by Berta et al. [Mathematical Programming, 1 - 49 (2021)], which provides converging outer bounds on the optimal fidelity of approximate quantum error correction (AQEC). As our first contribution, we introduce a measurement-based rounding scheme that extracts inner sequences of certifiably good encoder-decoder pairs from this outer hierarchy. To address the computational complexity of evaluating fixed levels of the hierarchy, we investigate the use of symmetry-based dimension reduction. In particular, we combine noise symmetries - such as those present in multiple copies of the qubit depolarizing channel - with the permutational symmetry arising from the extendability of the optimization variable. This framework is illustrated through basic, but already challenging numerical examples that showcase its practical effectiveness. Our results contribute to narrowing the gap between theoretical developments in quantum information theory and their practical applications in the analysis of small-scale quantum error-correcting codes.

Figures

Figures reproduced from arXiv: 2507.12326 by the authors.

**Figure 1.** The figure highlights the fundamental difference between the two hierarchies Proposition 2.1 (a) and (b). In the first hierarchy, linear constraints are imposed on the cones before applying the minimal tensor product for convexification, that is, the minimal tensor product plays the role of allowing for all convex combinations of products). In the second hierarchy, the minimal tensor product of the cones is constru… view at source ↗

**Figure 2.** This diagram illustrates the process of obtaining an inner point. Starting with a solution from the outer hierarchy in Proposition 2.1, which lies within the set of n-extendable states, we project this solution onto an inner point. The projection ensures that the resulting state satisfies the constraints while maintaining control over the objective value. Proof. Let ML¯P¯→Z be an informationally complete measurement… view at source ↗

**Figure 3.** The subsystem structure for the n-level relaxation is drawn. The extendibility symmetry enforces a Sn-symmetry within the entire n columns counting from the right, i.e. without the first column from the left. The iidsymmetry is given by a group Sm which permutes the rows of P’s in the first two columns from the left. The action we propose in this section is that, by considering [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: We consider outer bounds on the channel fidelity when encoding a single qubit into three iid qubits subject to depolarizing respectively amplitude damping noise. In addition to the PPT constraint, we include a non-signaling constraint preventing signaling from Bob to A…

**Figure 5.** Figure 5: In this figure, corresponding to Example 7.2, we consider a special case of [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]

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Fixed points in de Finetti hierarchies
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Fixed-point constraints on de Finetti hierarchies yield O(√(log n)/n) double-sided rates, block-structured dimension dependence, and poly-time certifiable separable inner approximations for fixed local dimensions.

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