{"id":"724481b3-10a4-46d0-ba51-574c855ae0af","arxiv_id":"2606.08690","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proposes encoding continuous variables in single-qubit amplitudes with amplitude estimation readout for qubit-efficient variational continuous optimization on standard qubit hardware.","lead":"The paper proposes encoding each continuous decision variable as the measurement probability of a single qubit rather than discretizing it into binary strings, then recovering the value via amplitude estimation instead of repeated sampling. This aims to achieve high-precision continuous optimization on standard gate-based quantum hardware with constant qubit cost per variable.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Error propagation from amplitude estimation through decision variables to objective value under regularity assumptions is the least-secured step for the claimed qubit-vs-precision tradeoff.","rationale":"The reader's weakest_assumption directly names the error-propagation step that must hold for the headline qubit-efficiency claim; no stronger internal inconsistency appears from the abstract or stated outline.","tokens_in":1718,"tokens_out":366,"duration_ms":27369,"concrete_test":"Take the one-dimensional case f(x) = x^{2} on [0,1] with x encoded as sin^{2}(\theta/2). Derive the end-to-end query complexity to reach objective error ε using amplitude estimation (M ≈ 1/δ queries per evaluation) versus a k-bit binary encoding with k = log_{2}(1/ε); if the AE route requires asymptotically more total oracle calls once parameter optimization cost is included, the tradeoff claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that encoding each continuous variable as a single-qubit measurement probability, read out via amplitude estimation, yields a distinct resource tradeoff versus binary discretization. This requires that AE error δ maps to variable error δ and then to objective error bounded by L·δ (Lipschitz or similar) without additional circuit-dependent error sources. The abstract states the propagation occurs “under standard regularity assumptions,” but supplies no explicit bound, no example objective, and no accounting for how the variational parameters that control the amplitudes are updated inside the hybrid loop. If the objective lacks the assumed regularity or if the state-preparation circuit introduces bias correlated with the amplitude, the claimed advantage does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proposes a variational framework for continuous optimization on standard qubit-based quantum computers. Each decision variable is encoded directly as the squared amplitude (measurement probability) of one qubit rather than via binary discretization. Amplitude estimation is used for readout, with the goal of achieving higher precision scaling. The paper outlines how estimation error propagates to variable error and then to objective error under standard regularity assumptions, claiming a distinct constant-qubit-per-variable versus logarithmic-qubit tradeoff relative to discretized QAOA-style approaches.","tokens_in":1845,"tokens_out":519,"duration_ms":11281,"significance":"If the error-propagation analysis can be made rigorous and the hybrid variational loop shown to be free of correlated bias, the approach would offer a genuinely new resource tradeoff for continuous optimization on gate-model hardware, replacing qubit-width costs with estimation-shot costs. The positioning against both discretized variational methods and CV-QAOA is clear and potentially useful.","major_comments":[{"comment":"Abstract (and throughout): the central claim that 'amplitude-estimation error propagates to decision-variable error and then to objective-value error under standard regularity assumptions' is stated without an explicit bound, Lipschitz constant, or derivation. No concrete mapping from AE error δ to objective error is supplied, so the asserted width-versus-precision advantage cannot yet be evaluated.","section":"Abstract"},{"comment":"Abstract: the manuscript supplies neither an example objective function nor any accounting of how the variational parameters that prepare the single-qubit amplitudes are updated inside the hybrid loop. Without this, it is impossible to verify that state-preparation bias remains uncorrelated with the amplitude being estimated.","section":"Abstract"},{"comment":"Abstract: the claimed 'distinct width-versus-precision tradeoff' is asserted by contrasting one qubit per variable against logarithmic qubits for binary precision, but no resource-counting table or circuit-depth comparison is provided to make the tradeoff quantitative.","section":"Abstract"}],"minor_comments":[{"comment":"The abstract refers to 'standard regularity assumptions' without naming them; a short list of the assumed conditions (e.g., Lipschitz continuity of the objective) would improve clarity.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be a high-level proposal rather than a fully worked-out method; the absence of any derivation or numerical example in the supplied text makes it difficult to judge whether the framework is merely conceptual or ready for detailed analysis."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive comments, which identify opportunities to strengthen the clarity and rigor of the presentation. We address each major comment below and will revise the manuscript to incorporate the requested details while preserving the core contribution.","responses":[{"response":"We agree that an explicit bound strengthens the claim. The manuscript outlines the propagation under standard Lipschitz assumptions on the objective (variable error scales linearly with AE error δ via the amplitude-to-probability mapping, and objective error then scales with the Lipschitz constant L). In revision we will add a concise statement of the bound (e.g., objective error O(L δ)) to the abstract and ensure the full derivation appears with the regularity assumptions stated explicitly in the main text.","revision_made":"yes","referee_comment":"[Abstract] Abstract (and throughout): the central claim that 'amplitude-estimation error propagates to decision-variable error and then to objective-value error under standard regularity assumptions' is stated without an explicit bound, Lipschitz constant, or derivation. No concrete mapping from AE error δ to objective error is supplied, so the asserted width-versus-precision advantage cannot yet be evaluated."},{"response":"We will add a concrete example (e.g., a quadratic objective) together with an explicit description of the hybrid loop: variational parameters θ prepare the single-qubit state via rotation gates, the objective is evaluated via amplitude estimation on the prepared state, and a classical optimizer updates θ. Because state preparation is performed before estimation shots and is independent of the estimation circuit, the preparation bias remains uncorrelated with the estimated amplitude. This accounting will be inserted in the revised manuscript.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the manuscript supplies neither an example objective function nor any accounting of how the variational parameters that prepare the single-qubit amplitudes are updated inside the hybrid loop. Without this, it is impossible to verify that state-preparation bias remains uncorrelated with the amplitude being estimated."},{"response":"We agree that a quantitative comparison is needed. The revision will include a resource table that, for target precision ε, contrasts (i) qubit count (constant 1 per variable vs. O(log(1/ε))), (ii) circuit depth for state preparation, and (iii) total shot complexity (O(1/ε) for amplitude estimation versus sampling overhead in the discretized case). This will make the width-versus-precision tradeoff explicit.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the claimed 'distinct width-versus-precision tradeoff' is asserted by contrasting one qubit per variable against logarithmic qubits for binary precision, but no resource-counting table or circuit-depth comparison is provided to make the tradeoff quantitative."}],"tokens_in":1390,"tokens_out":583,"duration_ms":18213,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper sketches a way to handle continuous optimization on standard qubits by putting each variable into the amplitude of one qubit and using amplitude estimation to read it out. The headline is that the error analysis needed to support the qubit-saving claim is missing from the abstract.\n\nThe new part is the specific encoding that avoids both binary discretization and continuous-variable hardware. It keeps everything in the qubit model and shifts the precision work to the estimation procedure. That framing is clear and directly addresses the qubit blowup in discretized QAOA for continuous problems.\n\nThe paper does a decent job positioning the idea against existing approaches. It notes the logarithmic qubit cost for finer precision in binary encodings and suggests replacing it with one qubit per variable plus estimation cost.\n\nThe main weakness is in the error propagation. The abstract claims that amplitude estimation error turns into decision variable error and then objective error under standard regularity assumptions, but it gives no explicit bound, no sample objective function, and no check on whether the variational circuit itself adds correlated errors. The stress-test note is right on this point; without that step secured, the distinct tradeoff does not follow.\n\nThere's also no mention of how the parameters are optimized in the hybrid loop or any numerical demonstration. The full manuscript might have more, but based on what's here the claims rest on unshown derivations.\n\nThis is for researchers thinking about variational methods for continuous problems on gate-based hardware. A reader looking for new encoding ideas might find the direction useful to think about, but the lack of supporting math means it is not ready for serious refereeing.\n\nI would not recommend sending it to peer review in its current form. It needs at least the error bounds and a small example worked out first.","headline":"The abstract proposes encoding continuous variables in single-qubit amplitudes with amplitude estimation, but supplies no derivations or checks on the claimed error propagation.","tokens_in":2301,"tokens_out":421,"would_cite":false,"duration_ms":19876,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Continuous decision variables can be encoded directly in single-qubit amplitudes for variational optimization on standard gate-based quantum computers.","keywords":["continuous optimization","amplitude estimation","variational algorithms","qubit efficiency","quantum optimization","QAOA"],"falsifier":"An experiment or calculation showing that objective error does not follow the predicted scaling from amplitude-estimation bounds for a given objective, or that equivalent precision requires more total resources than the discretized baseline.","tokens_in":2616,"feed_emoji":"","tokens_out":561,"duration_ms":16444,"temperature":0.7,"pith_summary":"The paper proposes a variational framework that maps each continuous decision variable to the squared amplitude of one qubit instead of discretizing it across multiple qubits in binary. Amplitude estimation is then used to recover the encoded value, aiming for improved precision scaling without leaving the standard qubit circuit model. This replaces the usual logarithmic growth in qubit count with a fixed cost of one qubit per variable, shifting the precision burden to the estimation step. A reader would care if this tradeoff holds because it could make continuous optimization feasible on hardware with limited qubits.","feed_headline":"One qubit per variable encodes continuous optimization","feed_subtitle":"Amplitude estimation recovers values without binary discretization, shifting precision cost from circuit width to estimation shots.","key_machinery":"Encoding each decision variable into the squared amplitude (measurement probability) of a single qubit state, recovered via amplitude estimation.","core_discovery":"By encoding each decision variable into the squared amplitude of a single qubit state and recovering it via amplitude estimation, the framework performs variational continuous optimization qubit-efficiently without explicit discretization, remaining inside the standard gate-based qubit model and providing a distinct width-versus-precision tradeoff relative to binary encodings.","pith_inferences":["The constant qubit cost per variable could extend the reachable dimensionality of continuous problems on fixed-width hardware.","Pairing the encoding with standard error-mitigation methods might further reduce the shots needed for target precision.","Direct comparisons of total resource cost versus achieved objective accuracy in simulation would test the claimed tradeoff."],"forward_implications":["Qubit count scales as one per decision variable rather than logarithmically with required binary precision.","Accuracy demands move from circuit width into the number of shots or estimation repetitions.","The method stays compatible with existing qubit-based variational algorithms such as QAOA.","It offers an alternative to continuous-variable hardware approaches that rely on qumodes."],"fun_headline_variants":["Continuous optimization encoded in single qubit amplitudes","Amplitude estimation for qubit-efficient continuous optimization","One qubit per variable without binary discretization","Variational continuous optimization via qubit amplitudes","Qubit amplitude encoding bypasses discretization"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Amplitude-estimation error propagates to decision-variable error and then to objective-value error under standard regularity assumptions on the objective function.","fun_headline_variants_meta":{"raw":{"variants":["Continuous optimization encoded in single qubit amplitudes","Amplitude estimation for qubit-efficient continuous optimization","One qubit per variable without binary discretization","Variational continuous optimization via qubit amplitudes","Qubit amplitude encoding bypasses discretization"]},"model":"grok-4.3","cost_usd":0.003576,"raw_usage":{"total_tokens":1857,"prompt_tokens":638,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":35762000,"prompt_tokens_details":{"text_tokens":638,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1160,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":638,"tokens_out":59,"duration_ms":6340,"temperature":1.0,"reasoning_tokens":1160,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T18:31:27.659465+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An experiment or calculation showing that objective error does not follow the predicted scaling from amplitude-estimation bounds for a given objective, or that equivalent precision requires more total resources than the discretized baseline.","supporting_citations":[],"review_version":1}