{"id":"ba3c2da7-166d-4a9a-8ba0-9ff6684db99e","arxiv_id":"2411.09210","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A classical error-rectification algorithm recovers heavy Fourier coefficients from noisy quantum Fourier sampling, enabling efficient noise-tolerant agnostic parity learning and its classical verification.","lead":"Quantum Fourier sampling circuits with bit-flip measurement noise can still support agnostic parity learning, because a classical post-processing step can recover the heavy Fourier coefficients from noisy samples. This matters for classical verification of quantum learning on near-term noisy devices, where untrusted quantum servers must be checked by classical clients.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Depolarizing-noise derivation in Eqs. (5)-(6) is incorrect: the effective flip rate is η_dep/2, not η_dep−η_dep^2/2, so the claimed extension to depolarizing channels is not established.","rationale":"Read in good faith, the paper's core bit-flip error-rectification (Theorem 1) appears mathematically solid: the prefix-recursive matching argument with mismatch probability ≤ η and the 2/θ slack is coherent, and the claimed sample complexity is plausible. The central quantum learning claim for bit-flip noise can stand. The most load-bearing weakness I find is the depolarizing extension, because it is one of the two advertised noise models and is used to argue general noise robustness. The derivation in the main text is not a minor typo: the effective flip rate is off by roughly a factor of two, and the core term p_{0,eff} is not the correct conditioned distribution. The correct calculation still yields a distribution of the same family with q = η_dep/2, so the main idea likely survives, but as written the theorem's proof for depolarizing noise does not follow. The reader's conditional verdict is appropriate; I would keep it conditional rather than rejecting, because the bit-flip theorem and verification protocol are independent and seem sound under their stated assumptions, including Fτ. The additional mismatch in Theorem 3 between O(ε^2) and τ^2/10 should also be corrected in revision.","tokens_in":17966,"tokens_out":41257,"duration_ms":481423,"concrete_test":"Analytically recompute p_dep(s) for the Fig. 1(c) circuit: apply Λ_dep to each of the n+1 qubits before measurement, derive the conditional distribution of the first n bits given y=1, and compare the result with Eqs. (5)-(6) for η_eff = η_dep − η_dep^2/2. Specifically, compute the effective single-qubit flip probability q from Λ_dep(ρ) on |0⟩ and |1⟩; the calculation yields q = η_dep/2. Then verify whether the claimed p_{0,eff} form is recovered. This single check settles whether the depolarizing extension is correct. A numerical simulation at η_dep = 0.1 comparing sampled QFS with depolarization against Eq. (5) with both η_eff and q would also expose the discrepancy.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Main text Eqs. (5)-(6) state that depolarizing QFS is equivalent to bit-flip noise with η_eff = η_dep − η_dep^2/2 and p_{0,eff} = (1−η_eff)p0 + η_eff δ0. This is not what the circuit in Fig. 1(c) produces. For Λ_dep(ρ) = (1−η_dep)ρ + η_dep I/2 applied before measurement, a computational-basis outcome b is preserved with probability 1−η_dep/2 and flipped with probability η_dep/2, so the correct single-qubit effective flip rate is q = η_dep/2. Conditioning on the noisy y-qubit being 1 gives p_dep(s) = ((1−q)p0 + q δ0) * Flip_q(s), i.e. p_dep(s) = Σ_{s'} q^{d_H(s,s')}(1−q)^{n−d_H(s,s')} [(1−q)p0(s') + q δ_{s',0}]. This is of the same form as Eqs. (5)-(6) only with q = η_dep/2, not η_eff = η_dep − η_dep^2/2; the claimed factor (1−η_dep)^2 for 'original result' is unjustified. Since the paper advertises depolarizing noise as a main model in the abstract and Fig. 1(c), this is a load-bearing gap: the proof that the same error-rectification applies to depolarizing channels rests on an incorrect calculation. The bit-flip result (Theorem 1) is not affected, and the depolarizing claim is likely patchable, but as written it is unsupported. A second related inconsistency: main-text Theorem 3's completeness condition 'noise strength below O(ε^2)' does not match the proof's η ≤ τ^2/10.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies whether classical verification of quantum learning can be made robust to noise in the quantum prover's hardware. It focuses on quantum Fourier sampling (QFS) and agnostic parity learning. The authors propose a classical error-rectification algorithm that, from O(log(n/δ)/θ^2) samples of a bit-flip-noisy QFS distribution, outputs a small set L containing all heavy Fourier coefficients (Theorem 1). They use this to estimate the Fourier coefficients (Corollary 1), to solve 1-agnostic parity learning with a noisy quantum prover and classical random examples (Theorem 2), and to give a one-round interactive proof system with a classical verifier and a noisy quantum prover under a spectral-gap promise on the target function (Theorem 3). The technical core is a recursive nearest-neighbor matching argument in SM Sec. II.","tokens_in":18394,"tokens_out":16820,"duration_ms":167013,"significance":"The bit-flip part of the paper is a solid and self-contained contribution: the matching lemma (Lemma S3) is correct, the recursion maintains the invariant H_m ⊆ L_m, and the sample complexity is explicit and logarithmic in n. If the advertised depolarizing-noise extension is repaired, the results would be a meaningful step toward practical delegation of quantum learning to NISQ-era servers, and the verification protocol naturally extends the noise-free framework of Caro et al. [20]. The soundness of the verification step is information-theoretic and does not depend on computational assumptions. However, the paper currently overstates its generality: the depolarizing-noise calculation contains a mathematical error, and the completeness statement of Theorem 3 does not match its proof.","major_comments":[{"comment":"The claimed reduction of depolarizing noise to bit-flip noise is not correct. For a single qubit, Λ_dep(ρ) = (1−η_dep)ρ + η_dep I/2 maps a computational-basis outcome b to b with probability 1−η_dep/2 and to 1−b with probability η_dep/2. Conditioning on the y-qubit outcome being 1 gives p_dep(s) = Σ_{s'} q^{d_H(s,s')}(1−q)^{n−d_H(s,s')}[(1−q)p0(s') + q δ_{s',0}] with q = η_dep/2. This is of the same form as Eq. (5), but with q, not with η_eff = η_dep − η_dep^2/2 as defined in Eq. (6). The factor (1−η_dep)^2 used for the 'original result' is therefore unjustified. Since the abstract and Fig. 1(c) advertise depolarizing noise as a main model, this error is load-bearing; the bit-flip Theorem 1 is unaffected, and the depolarizing claim appears patchable by replacing η_eff with q, but as written the extension is not established.","section":"QFS with noises, Eqs. (5)-(6)"},{"comment":"The completeness condition as stated in the main text, 'If the noise strength of P is below O(ε^2)', does not match the proof. The proof in SM Theorem S5 requires η ≤ τ^2/10, where τ is the spectral-gap parameter from Definition S3; Step 1 applies Theorem S1 with θ = τ^2. Since τ can be much smaller than ε, the stated O(ε^2) condition is neither the one used nor sufficient for the protocol as proved. The theorem should state the noise condition in terms of τ (e.g., η ≤ τ^2/10) or impose an explicit relation such as τ ≥ cε.","section":"Theorem 3, Completeness bullet"}],"minor_comments":[{"comment":"The two bad events in the soundness proof, H⊆L with Step 3 failure and H⊄L with L passing validation, are disjoint, so their probabilities add; the soundness error is at most 2δ/3, not δ/3. The final soundness bound δ still holds because 2δ/3 < δ, so this does not affect the theorem.","section":"SM Sec. V, Eq. (S26)"},{"comment":"Step 1 should explicitly state that the verifier runs Algorithm S1 on the samples received from the prover; this is what enforces the size bound |L| ≤ 2/τ^2 and keeps the verifier efficient against arbitrary provers. If instead the prover is expected to supply L, a verifier-side size check is needed.","section":"Theorem 3, Step 1"},{"comment":"The caption describing the depolarizing channel has the probabilities reversed; it should say that the channel applies ρ with probability 1−η_dep and I/2 with probability η_dep.","section":"Fig. 1(c) caption"},{"comment":"There are several typos: 'whether existed' should be 'whether existing' in the abstract; 'the the quantum Fourier sampling' should be 'the quantum Fourier sampling'; and 'Ryderger atoms' should be 'Rydberg atoms' in the Discussion.","section":"Abstract and Discussion"},{"comment":"The sentence 'with probability at least 1−δ/2×2 = 1−δ' is not a correct composition of the two success probabilities; the intended bound is (1−δ/2)^2 ≥ 1−δ, which is what the argument needs.","section":"SM Theorem S3, proof"}],"recommendation":"major_revision","confidential_remarks":"The bit-flip core (Theorem 1 and its Corollary) appears sound and is the main contribution. The depolarizing-noise extension and the Theorem 3 completeness statement are formally broken but seem patchable; I therefore recommend major revision rather than reject. The authors should also tighten the protocol description in Step 1 of Theorem 3 so that the verifier's efficiency against malicious provers is explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. The bit-flip error-rectification algorithm (Algorithm S1) is the real contribution, and I believe the proof of Theorem 1 is sound: the recursive matching argument, Lemma S3's bound, and the union bound over n steps all line up. That is new relative to Caro et al., which assumed noiseless quantum examples. The Corollary 1 and Theorem 2 consequences follow cleanly. The verification protocol is a fairly direct adaptation of Caro et al. with this added rectification step; the completeness/soundness argument is standard and acceptable.\n\nThe soft spots are all in the noise modeling and the theorem statements. First, the depolarizing noise extension in Eqs. (5)-(6) is wrong. For Λ_dep(ρ) = (1-η)ρ + η I/2, a measurement outcome in the computational basis flips with probability η/2, not η - η^2/2. So the claimed equivalence to the bit-flip model with η_eff = η - η^2/2 is not what the circuit produces. The correct mapping is η_eff = η/2 (up to the conditioning on the y-qubit). This means the paper's advertised result for depolarizing channels is unsupported as written. It is patchable, but it is a real error, not a cosmetic one, because depolarizing noise is one of the headline models in the abstract and in Fig. 1(c).\n\nSecond, Theorem 3's completeness condition says 'noise strength below O(ε^2)' but the proof requires η ≤ τ^2/10, where τ is the spectral gap. That's a different parameter. The statement should be O(τ^2), or the relationship between ε and τ should be made explicit. As written, the mismatch would confuse readers.\n\nThird, the abstract oversells the practicality. The real condition is η ≤ O(θ) for finding heavy coefficients and η ≤ O(ε^2) for the parity learning guarantee. That means the tolerated noise shrinks quadratically as the target accuracy improves. For ε = 0.01 you need η ≤ 10^-5, which is not 'practical constant-level noise' in the NISQ sense. The constant is there, but small.\n\nFinally, the verification protocol relies on the spectral gap promise F_τ: no nonzero Fourier coefficient below τ. It is explicitly stated in Theorem 3, so I don't call it a hidden flaw, but the abstract's 'sparse Fourier coefficients' is weaker and understates the assumption.\n\nOverall: the core bit-flip result is solid and worth publishing after these fixes. The paper deserves a serious referee, not a desk reject. I'd send it to review with a note to check the depolarizing mapping and align Theorem 3's noise condition with its proof. If the authors fix those, the paper is a reasonable contribution to the delegated quantum learning literature.","headline":"The bit-flip error-rectification algorithm is a genuine new technique and the proof of Theorem 1 holds up, but the depolarizing-noise extension is miscalculated and Theorem 3's noise condition is mismatched with its proof; the core result is sound and the paper deserves refereeing after fixes.","tokens_in":18868,"tokens_out":5119,"would_cite":true,"duration_ms":81155,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q12","68Q32","81P68"],"pacs":["03.67.-a","03.67.Lx"],"model":"deepseek-v4-flash","headline":"The paper claims that noisy quantum Fourier sampling still supports efficient agnostic parity learning and its classical verification, because a classical error-rectification algorithm recovers heavy Fourier coefficients from…","keywords":["classical verification of quantum learning","quantum Fourier sampling","error rectification","agnostic parity learning","heavy Fourier coefficients","noisy intermediate-scale quantum (NISQ) devices","interactive proof systems","quantum example oracle"],"falsifier":"Run Algorithm S1 on a known function with a controlled Fourier spectrum, for example one parity with $\\hat{g}(s^*)=0.6$ and several coefficients just below the threshold, under bit-flip noise at rate $\\eta=0.01$ with $\\theta=0.1$, and check whether the true heavy set is contained in $L$ using $k=O(\\log(n/\\delta)/\\theta^2)$ samples; observing a miss at constant probability, or requiring $k$ to grow with $n$ beyond logarithmic, would contradict Theorem 1.","tokens_in":17784,"feed_emoji":"⚛️","tokens_out":12883,"duration_ms":117100,"temperature":0.7,"pith_summary":"Classical verification of quantum learning has been studied in idealized noiseless settings, and this paper asks whether the advantage survives on the noisy machines available now. It claims that for quantum Fourier sampling under independent bit-flip noise at rate $\\eta$, a classical error-rectification algorithm can recover every heavy Fourier coefficient of the target function — every frequency component contributing probability mass at least $\\theta$ — using only $k = O(\\log(n/\\delta)/\\theta^2)$ noisy samples, provided $\\eta \\le \\theta/10$. Once those heavy coefficients are restored, the agnostic parity learning task of finding the parity function that best matches the target becomes efficient, and a classical verifier with access to a random example oracle can certify the noisy quantum prover's answer in one round of communication whenever the target function has a gap in its Fourier spectrum. The practical payoff is that delegation of learning to untrusted, noisy quantum cloud servers could be made reliable without full quantum error correction.","feed_headline":"Noise no longer blocks verified quantum-learning advantage","feed_subtitle":"Rebuilding heavy Fourier coefficients from logarithmically many noisy samples keeps delegated parity learning sound.","key_machinery":"The load-bearing object is the noisy quantum Fourier sampling distribution $p_\\eta(s)$, the convolution of the ideal squared Fourier spectrum $p_0(s)=|\\hat{g}(s)|^2$ with a bit-flip channel. The mechanism that carries the argument is the nearest-neighbor matching step in the recursive error-rectification algorithm: at each prefix length $m$, every noisy sample is assigned to the closest candidate prefix in Hamming distance, and the algorithm keeps the $2/\\theta$ prefixes with the largest empirical frequency. Lemma S3 bounds the probability of a mismatch by the noise rate $\\eta$, so a genuinely heavy coefficient is matched correctly with probability at least $3/5$; this is what converts a deconvolution problem into a heavy-hitter search with logarithmic sample complexity. In the verification protocol, the equivalent load-bearing test is the sum-of-squares check $\\sum_{s\\in L}\\tilde{g}(s)^2 \\ge 1-\\tau^2/2$, which separates the case where $L$ contains the full support of $\\hat{g}$ from the case where it misses at least one nonzero coefficient.","core_discovery":"On the paper's own terms, the central discovery is that measurement noise in quantum Fourier sampling is not a quantum obstacle to be corrected by quantum error correction, but a classical deconvolution problem. The noisy distribution is the ideal distribution convolved with an independent bit-flip channel, $p_\\eta(s) = \\sum_{s'} \\eta^{d_H(s,s')}(1-\\eta)^{n-d_H(s,s')}p_0(s')$, and Theorem 1 gives a recursive list-maintenance algorithm that, for $\\eta \\le \\theta/10$, returns a set $L$ of size at most $\\lfloor 2/\\theta\\rfloor$ containing every $s$ with $p_0(s)\\ge\\theta$, using $k=O(\\log(n/\\delta)/\\theta^2)$ noisy samples with probability at least $1-\\delta$. The same algorithm feeds Corollary 1, which produces an $\\ell^\\infty$-accurate estimate of the Fourier coefficients of the target function, and Theorem 2, which solves 1-agnostic parity learning with error $\\varepsilon$ under noise rate $\\eta\\le\\varepsilon^2/10$. Theorem 3 then provides a one-round interactive proof in which a classical verifier checks the prover's returned Fourier set and accepts only if the sum of squared estimated coefficients is close to $1$, a test that is complete and sound under the promise $F_\\tau$ that no nonzero Fourier coefficient is smaller than $\\tau$.","pith_inferences":["Editorial inference: the recursive matching routine is really a classical heavy-hitter algorithm for distributions under bit-flip noise, so it should apply to any learning problem whose oracles produce Fourier-sparse samples corrupted by bounded noise; label-noise robust sparse Fourier recovery is a natural neighbour.","Editorial inference: the spectral-gap promise $F_\\tau$ is the practical bottleneck: if real target functions have many tiny nonzero Fourier coefficients, the verifier's sum-of-squares test either rejects or forces $\\tau$ small, which in turn forces the device noise $\\eta$ below $\\tau^2/10$; this is a testable restriction on which real datasets the protocol can certify.","Editorial inference: one direct experimental check would be to run QFS on a known Fourier-sparse function, inject controlled bit flips at various rates, and compare the recovered heavy set with the prediction of Theorem 1; current cloud quantum platforms should be able to do this with a handful of qubits."],"forward_implications":["Noisy quantum Fourier sampling remains useful for learning: heavy Fourier coefficients of the target function can be restored from logarithmically many noisy samples, with no additional qubits or gates and no quantum error correction.","The error-rectification step plus classical random examples yields an $\\ell^\\infty$ approximation to the full Fourier spectrum, which directly gives a near-optimal parity hypothesis for agnostic parity learning.","A classical verifier can interact once with an untrusted noisy quantum prover and either accept a near-optimal parity or reject, with soundness error at most $\\delta$, whenever the target function has no nonzero Fourier coefficient below $\\tau$ and the noise rate is below $\\tau^2/10$.","The same rectification argument extends to depolarizing noise and, more generally, to any single-qubit error that acts as an effective measurement-outcome flip, so the result is not tied to the specific bit-flip model.","Theorem 1's algorithm is stated as a general classical routine for distributions corrupted by bounded per-bit flip noise, so it can be reused outside QFS whenever the same convolution structure appears."],"supporting_citations":[{"why":"It supplies the noiseless interactive-proof framework and the empirical-distribution lemma that the noisy protocol extends and reuses.","marker":"[20]"},{"why":"It defines the quantum Fourier sampling circuit whose noisy output distribution is the object being rectified.","marker":"[26]"},{"why":"It introduces the quantum example oracle and the Fourier-sampling approach to quantum learning used throughout.","marker":"[15]"},{"why":"They provide the recursive heavy-Fourier-coefficient search methods that Algorithm S1's prefix matching and pruning generalizes.","marker":"[45, 46]"},{"why":"They establish that agnostic parity learning is at least as hard as learning parity with noise, the background hardness that makes the quantum route significant.","marker":"[47, 48]"}],"fun_headline_variants":["Classical error rectification unblocks noisy quantum learning","Logarithmic samples fix noisy quantum parity learning","Classical verifier checks noisy quantum prover's learning","Noisy quantum hardware still yields verified learning advantage","Classical deconvolution restores quantum learning proofs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the promise that the target function's frequency content is gapped: every nonzero Fourier coefficient has magnitude at least $\\tau$, and the device noise stays below $\\tau^2/10$; without that gap, a returned list that misses some small Fourier mass can still pass the verifier's check and lead to a suboptimal parity.","fun_headline_variants_meta":{"raw":{"variants":["Classical error rectification unblocks noisy quantum learning","Logarithmic samples fix noisy quantum parity learning","Classical verifier checks noisy quantum prover's learning","Noisy quantum hardware still yields verified learning advantage","Classical deconvolution restores quantum learning proofs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000593,"raw_usage":{"total_tokens":2823,"prompt_tokens":1037,"completion_tokens":1786,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":653,"completion_tokens_details":{"reasoning_tokens":1711}},"tokens_in":653,"tokens_out":1786,"duration_ms":14177,"temperature":1.0,"reasoning_tokens":1711,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:56:33.531187+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm S1 on a known function with a controlled Fourier spectrum, for example one parity with $\\hat{g}(s^*)=0.6$ and several coefficients just below the threshold, under bit-flip noise at rate $\\eta=0.01$ with $\\theta=0.1$, and check whether the true heavy set is contained in $L$ using $k=O(\\log(n/\\delta)/\\theta^2)$ samples; observing a miss at constant probability, or requiring $k$ to grow with $n$ beyond logarithmic, would contradict Theorem 1.","supporting_citations":[{"cited_title":"Atici and R","cited_arxiv_id":null,"evidence_quote":"It defines the quantum Fourier sampling circuit whose noisy output distribution is the object being rectified."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It introduces the quantum example oracle and the Fourier-sampling approach to quantum learning used throughout."}],"review_version":1}