{"id":"1d9e5e33-c019-4fb2-a6f3-5578bb4b6979","arxiv_id":"2606.09379","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proposes EA versions of qubit-into-oscillator and oscillator-into-oscillator concatenated codes using GKP and EA repetition codes to suppress position and momentum quadrature errors by factors of 1/n.","lead":"The paper proposes new entanglement-assisted concatenated quantum error correction codes that combine GKP codes with EA stabilizer codes for encoding qubits into oscillators or oscillators into oscillators. These constructions use pre-shared entanglement to reduce quadrature error variances, which could matter for building more efficient quantum error correction in continuous-variable systems.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Variance suppression by 1/n holds only under perfect GKP and ideal emodes; no robustness analysis given","rationale":"The reader's weakest_assumption directly identifies the same ideal-resource assumption that underpins the 1/n claim. Because the full text is referenced but the provided abstract already flags the missing imperfection analysis, the concern is load-bearing and the provisional UNVERDICTED verdict does not require adjustment on this basis alone.","tokens_in":1757,"tokens_out":298,"duration_ms":13499,"concrete_test":"For the n=3 case, replace ideal GKP states with 10 dB squeezed GKP states and replace perfect emodes with two-mode squeezed states of finite squeezing (e.g., 10 dB); recompute the output quadrature variances after the full concatenation and check whether the suppression factor remains within 10 % of 1/3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (generalization to n-qubit EA repetition outer code suppressing both quadratures by exactly 1/n) requires ideal GKP states and perfect maximally entangled modes. The construction chains these resources but supplies no calculation of how finite squeezing, loss, or entanglement infidelity propagates through the concatenation to degrade the 1/n factor. Because the suppression is presented as a direct consequence of the ideal resource model, any deviation from that model is unquantified and could invalidate the headline scaling.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proposes two classes of entanglement-assisted concatenated continuous-variable codes. The first combines an EA-stabilizer outer code with a GKP inner code to encode qubits into oscillators, illustrated by a three-qubit EA-repetition example. The second combines a GKP outer code with an EA-stabilizer inner code to encode oscillators into oscillators, illustrated by a GKP code concatenated with a three-qubit EA repetition code that uses two emodes; this is generalized to an n-qubit EA repetition code using (n-1) emodes that suppresses both position and momentum quadrature error variances of a data mode by a factor of exactly 1/n.","tokens_in":1855,"tokens_out":443,"duration_ms":18225,"significance":"If the constructions and the exact 1/n suppression are rigorously established under the stated ideal-resource model, the work would supply a concrete family of EA concatenated codes that leverage pre-shared entanglement to improve quadrature variance suppression in bosonic systems. The explicit n-parameter generalization is a potential strength if accompanied by explicit derivations.","major_comments":[{"comment":"The generalization to the n-qubit EA repetition outer code (final paragraph of the abstract and corresponding section of the main text) asserts that the construction suppresses both quadrature variances by exactly 1/n. No explicit derivation, stabilizer tableau, or noise-propagation calculation is referenced that demonstrates how the (n-1) emodes produce this precise factor for both quadratures simultaneously; the central scaling claim therefore rests on unshown steps.","section":"generalization to n-qubit EA repetition code"},{"comment":"The constructions assume perfect GKP states and ideal maximally entangled modes. The manuscript supplies no calculation showing how finite squeezing, loss, or entanglement infidelity propagates through the concatenation and degrades the claimed 1/n factor. Because the headline result is presented as a direct consequence of the ideal model, this omission is load-bearing for any claim of practical utility.","section":"EA repetition concatenated with GKP (oscillator-into-oscillators construction)"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive feedback. We address each major comment below and will update the manuscript to strengthen the presentation under the ideal-resource model.","responses":[{"response":"We agree that an explicit derivation is required to rigorously support the generalization. In the revised manuscript we will insert a new subsection containing the full stabilizer tableau for the n-qubit EA repetition code together with a complete noise-propagation calculation. This will show, step by step, how the (n-1) emodes simultaneously reduce both quadrature variances by the exact factor 1/n under the stated ideal model.","revision_made":"yes","referee_comment":"The generalization to the n-qubit EA repetition outer code (final paragraph of the abstract and corresponding section of the main text) asserts that the construction suppresses both quadrature variances by exactly 1/n. No explicit derivation, stabilizer tableau, or noise-propagation calculation is referenced that demonstrates how the (n-1) emodes produce this precise factor for both quadratures simultaneously; the central scaling claim therefore rests on unshown steps."},{"response":"The manuscript is framed entirely within the ideal-resource model (perfect GKP states and ideal entanglement), as stated in the abstract and introduction; the 1/n claim is derived strictly under those assumptions. No degradation analysis is needed to support the stated theoretical result. We will nevertheless add a short paragraph in the conclusions noting that extensions to finite squeezing and loss constitute valuable future work.","revision_made":"partial","referee_comment":"The constructions assume perfect GKP states and ideal maximally entangled modes. The manuscript supplies no calculation showing how finite squeezing, loss, or entanglement infidelity propagates through the concatenation and degrades the claimed 1/n factor. Because the headline result is presented as a direct consequence of the ideal model, this omission is load-bearing for any claim of practical utility."}],"tokens_in":1464,"tokens_out":412,"duration_ms":23695,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is two concrete directions for entanglement-assisted concatenated codes: an EA-stabilizer outer code around a GKP inner code, and the reverse with GKP outer and EA-repetition inner. The second direction is extended to an n-qubit EA repetition code that uses n-1 entangled modes and reduces both quadrature variances by 1/n.\n\nThe explicit resource counts and the clean statement of the suppression factor are the parts that stand out. These build on existing EA repetition codes and GKP concatenation, so the combinations are a direct extension rather than a conceptual leap.\n\nThe soft spots are straightforward. Everything is presented under the assumption of ideal GKP states and perfect maximally entangled modes. There is no calculation showing how the 1/n factor changes with finite squeezing, loss, or entanglement infidelity, so the headline scaling is untested outside the ideal model. The abstract supplies no derivations or code tables, which makes it impossible to verify whether the suppression actually follows from the construction.\n\nThis is for specialists already working on continuous-variable error correction who want to see EA resources layered with bosonic codes. It could be worth their time if the full paper contains the missing steps and some discussion of imperfections. The work deserves peer review because the constructions are specific enough to check for correctness and because the combination of EA and concatenation has not been explored in this exact form before.","headline":"The paper gives explicit EA-GKP concatenation constructions with a claimed 1/n variance suppression using n-1 emodes, but the scaling is stated only for perfect resources and no derivations or robustness checks appear.","tokens_in":2363,"tokens_out":364,"would_cite":false,"duration_ms":13646,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Concatenating GKP codes with n-qubit EA repetition codes suppresses both position and momentum quadrature variances by a factor of 1/n using n-1 entangled modes.","keywords":["entanglement-assisted codes","GKP codes","concatenated codes","continuous-variable error correction","quadrature variance suppression","bosonic codes","repetition codes"],"falsifier":"An explicit calculation of the output quadrature variances when the input entangled modes have finite squeezing or the GKP states have finite envelope would show whether the suppression factor remains exactly 1/n.","tokens_in":2664,"feed_emoji":"","tokens_out":732,"duration_ms":18770,"temperature":0.7,"pith_summary":"The paper combines two ideas: entanglement-assisted stabilizer codes improve error-correction rates over standard codes, and GKP codes can be concatenated with qubit codes to lower logical failure rates. It first builds an EA outer code around a GKP inner code for encoding qubits into oscillators. It then builds the reverse concatenation, with a GKP outer code around an EA inner code, for encoding oscillators into oscillators. The central construction is the n-qubit EA repetition code that consumes n-1 maximally entangled modes and reduces the variances of both quadratures on a data mode by exactly 1/n. A sympathetic reader would care because the scheme offers a concrete way to use pre-shared entanglement to protect continuous-variable information more efficiently.","feed_headline":"EA repetition codes cut GKP quadrature variances by 1/n","feed_subtitle":"n-qubit version uses n-1 entangled modes to suppress both position and momentum errors on a data oscillator.","key_machinery":"The n-qubit entanglement-assisted repetition code using n-1 maximally entangled modes, concatenated with a GKP code as outer code.","core_discovery":"We propose an EA version of the non-Gaussian oscillator-into-oscillators concatenated code that chains a GKP outer code with an EA-stabilizer inner code. As an example we present a GKP code concatenated with a three-qubit EA repetition code that uses two maximally entangled modes and suppresses the variances of both position and momentum quadrature errors of a data mode. Furthermore, we generalize the latter example to a family of GKP code concatenated with a n-qubit EA repetition code that uses n-1 emodes and suppresses the variances of both position and momentum quadrature errors of a data mode by a factor 1/n.","pith_inferences":["The linear scaling with n suggests that adding more entangled modes yields proportionally better protection, which could be tested by increasing n in numerical simulations of the concatenated channel.","Because the construction works in either concatenation order, it may allow flexible placement of the GKP layer depending on whether the dominant noise is at the qubit or oscillator level.","The explicit use of repetition codes leaves open whether other EA stabilizer codes could produce stronger suppression or higher rates when concatenated with GKP."],"forward_implications":["The same n-qubit EA repetition inner code works for both the qubit-into-oscillator and oscillator-into-oscillator concatenations.","Variance suppression applies equally to position and momentum quadratures.","The construction uses exactly n-1 entangled modes for any n.","The scheme is presented as an explicit generalization of the three-qubit case."],"fun_headline_variants":["EA GKP codes suppress variances by 1/n with n-1 modes","n-qubit EA repetition suppresses GKP variances by 1/n","GKP concatenated with EA repetition cuts errors by 1/n","Oscillator EA codes reduce variances by 1/n"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The constructions assume perfect maximally entangled modes and ideal GKP states are available.","fun_headline_variants_meta":{"raw":{"variants":["EA GKP codes suppress variances by 1/n with n-1 modes","n-qubit EA repetition suppresses GKP variances by 1/n","GKP concatenated with EA repetition cuts errors by 1/n","Oscillator EA codes reduce variances by 1/n"]},"model":"grok-4.3","cost_usd":0.004895,"raw_usage":{"total_tokens":2438,"prompt_tokens":745,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":48949500,"prompt_tokens_details":{"text_tokens":745,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1622,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":745,"tokens_out":71,"duration_ms":12263,"temperature":1.0,"reasoning_tokens":1622,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T16:22:00.807308+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit calculation of the output quadrature variances when the input entangled modes have finite squeezing or the GKP states have finite envelope would show whether the suppression factor remains exactly 1/n.","supporting_citations":[],"review_version":1}