{"id":"185009bc-4285-4b3e-ab70-340eee4e978f","arxiv_id":"quant-ph/9511026","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":9.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Kitaev presents a polynomial quantum algorithm for the Abelian stabilizer problem based on measuring eigenvalues of unitary operators, generalizing Shor's factoring and discrete-log algorithms.","lead":"The paper gives a polynomial-time quantum algorithm for the Abelian stabilizer problem, covering factoring and discrete log via eigenvalue measurement of a unitary operator. A smart generalist should read it because the technique became the foundation for phase estimation, now used across quantum algorithms.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption matches the paper's explicit problem formulation. The derivation is parameter-free and reduces directly to standard quantum mechanics plus circuit complexity; no internal inconsistency or unstated assumption about group representation or precision scaling undermines the polynomial claim.","tokens_in":1517,"tokens_out":240,"duration_ms":39841,"concrete_test":"Re-derive the success probability and gate complexity of the eigenvalue measurement routine (Section on quantum measurements) for a cyclic group of order 2^k with exact controlled-U^{2^j} gates; confirm that O(k) measurements suffice to recover the stabilizer with probability >2/3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper derives a polynomial-time quantum algorithm for the abelian stabilizer problem (including factoring and discrete log) via eigenvalue measurement of the group-action unitary. This rests explicitly on the assumption that the unitary realizing the action is efficiently implementable as a quantum circuit, which is the standard oracle model for the problem. The phase-estimation construction and its application to extract stabilizer characters appear internally consistent with no unsupported steps or hidden precision blow-ups.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript presents a polynomial-time quantum algorithm for the Abelian stabilizer problem (encompassing factoring and discrete logarithm) via eigenvalue measurement of the unitary operator realizing the group action. It also derives a quantum Fourier transform over arbitrary finite Abelian groups and includes a detailed introductory overview of quantum computation.","tokens_in":1589,"tokens_out":320,"duration_ms":50841,"significance":"If the central claims hold, the work provides a unifying framework that generalizes Shor's algorithms and introduces the eigenvalue-measurement primitive as a reusable tool for quantum algorithms on group problems. The derivation follows directly from standard quantum postulates and circuit constructions under the efficient-oracle assumption, which is the conventional model for these problems.","major_comments":[],"minor_comments":[{"comment":"The abstract states the algorithm is 'polynomial' but does not explicitly note the dependence on the group order or the precision parameter; a single clarifying sentence would improve readability.","section":"Abstract"},{"comment":"In the description of the eigenvalue measurement procedure, the analysis of the number of repetitions needed to achieve sufficient precision for stabilizer extraction is sketched but could be expanded with an explicit bound on the failure probability.","section":"Section on eigenvalue measurement"},{"comment":"Notation for the group action unitary and its eigenvectors is introduced without a consolidated table of symbols; adding one would aid readers new to the stabilizer formulation.","section":"Introduction and preliminaries"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive review, accurate summary of the manuscript, and recommendation to accept. The significance assessment aligns with our intent to provide a unifying framework via eigenvalue measurement that generalizes Shor's algorithms.","responses":[],"tokens_in":1009,"tokens_out":61,"duration_ms":30711,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Kitaev's paper supplies a polynomial quantum algorithm for the Abelian stabilizer problem that covers both factoring and discrete log. The approach rests on a procedure to measure an eigenvalue of the unitary that implements the group action, then uses that to recover the stabilizer characters via a quantum Fourier transform over the group. This is the main new piece: the eigenvalue measurement is presented as a standalone primitive, and the QFT construction works for arbitrary finite Abelian groups rather than just cyclic ones from the earlier Shor papers. The derivation stays close to the basic postulates of quantum mechanics and standard circuit elements like controlled operations and phase kickback. The steps are laid out explicitly, and the paper includes a self-contained introduction to quantum computation that makes the argument easier to follow. The central assumption is that the unitary for the group action can be realized efficiently as a quantum circuit. That is the usual oracle model for these problems, so it does not create a hidden gap. Some concrete implementation details for particular oracles are left implicit, but that matches the level of other theoretical work in the area. No equations depend on fitted parameters or self-referential definitions, and the logic does not loop back on itself. The paper is aimed at people already working on quantum algorithms who want the general framework rather than just the factoring case. A reader who needs to see how phase estimation and group Fourier transforms fit together will find it useful. The constructions are internally consistent and the claims track the derivations without overreach. I would send this to peer review; the ideas are foundational enough and the supporting steps are clear enough to merit referee attention.","headline":"Kitaev gives a clean general method for eigenvalue measurement of unitaries that directly yields the Abelian stabilizer algorithm and a QFT over any finite Abelian group.","tokens_in":2071,"tokens_out":398,"would_cite":true,"duration_ms":36787,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith.Foundation.DimensionForcing","rs_theorem":"dimension_forced","paper_passage":"We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Our method is based on a procedure for measuring an eigenvalue of a unitary operator."},{"relation":"unclear","rs_module":"IndisputableMonolith.Foundation.PhiForcing","rs_theorem":"phi_equation","paper_passage":"Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group."}],"headline":"Quantum algorithm for Abelian stabilizer (factoring/discrete log) via unitary eigenvalue measurement is orthogonal to RS forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (phase estimation on group-action unitary, QFT on Abelian groups, polynomial-time stabilizer extraction) operates in the standard quantum circuit model with an efficient unitary oracle assumption. RS derives J-cost uniqueness, φ, 8-tick periodicity, D=3 linking, and ledger structure from a single distinction plus analytic regularity, but has no theorems on quantum algorithms, eigenvalue measurement, or stabilizer problems. No shared structure (e.g., no J-cost, no φ-ladder, no 8-tick clock) appears; the domains are disjoint.","tokens_in":278786,"confidence":"moderate","tokens_out":322,"duration_ms":43446,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"lean_confirmation":{"model":"grok-4.3","status":"unconfirmed","citations":[],"rationale":"The paper's core claim is an algorithmic result in quantum computation. Shape-of-logic has zero modules or theorems on quantum algorithms, stabilizers, or eigenvalue measurement. The premise is out of scope for this corpus.","tokens_in":278594,"confidence":"moderate","tokens_out":161,"duration_ms":32902,"inferential_bridge":"Shape-of-logic contains no theorem about quantum circuits, unitary eigenvalue measurement, Abelian stabilizers, or Shor-like algorithms. Its content is physics-from-logic forcing (reality_from_one_distinction, J-cost, phi, D=3, etc.). No bridge exists.","load_bearing_premise":"Polynomial-time quantum algorithm for Abelian stabilizer problem via eigenvalue measurement of unitary operators (extending Shor's factoring/discrete-log).","cache_read_input_tokens":64,"cache_creation_input_tokens":0},"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A quantum algorithm solves the Abelian stabilizer problem in polynomial time, covering factoring and discrete logarithm.","keywords":["quantum algorithm","Abelian stabilizer problem","factoring","discrete logarithm","quantum Fourier transform","eigenvalue measurement","unitary operator","phase estimation"],"falsifier":"An explicit superpolynomial lower bound on the quantum circuit complexity of either integer factoring or the Abelian stabilizer problem would show the algorithm cannot exist.","tokens_in":2425,"feed_emoji":"⚛","tokens_out":598,"duration_ms":33375,"temperature":0.7,"pith_summary":"The paper introduces a quantum procedure that finds the stabilizer of an Abelian group action by measuring eigenvalues of the corresponding unitary operator. This yields an efficient algorithm for any problem reducible to the Abelian stabilizer task, including integer factoring and discrete logarithm computation. The same eigenvalue measurement also produces a quantum Fourier transform over an arbitrary finite Abelian group. A sympathetic reader would care because the result shows quantum computers can handle a natural class of algebraic problems beyond what Shor's earlier algorithms covered. The work includes a self-contained introduction to the basics of quantum computation.","feed_headline":"Quantum algorithm solves Abelian stabilizer problem in polynomial time","feed_subtitle":"Extends Shor's results on factoring and discrete log via eigenvalue measurement of unitaries and supplies QFT for any finite Abelian group.","key_machinery":"A procedure that measures an eigenvalue of a unitary operator by using controlled applications of the operator and an ancillary register to extract phase information.","core_discovery":"There exists a polynomial-time quantum algorithm for the Abelian stabilizer problem. The algorithm works by repeatedly measuring the eigenvalues of a unitary operator that encodes the group action; the measured phases reveal the stabilizer subgroup. The same eigenvalue measurement technique immediately supplies a quantum Fourier transform for any finite Abelian group.","pith_inferences":["The eigenvalue measurement technique may generalize to non-Abelian groups if suitable unitary representations can be constructed efficiently.","Problems whose solutions are hidden in the eigenspectrum of implementable unitaries become candidates for similar quantum speedups.","The method separates the algebraic structure of the problem from the details of the quantum circuit, suggesting a modular approach to algorithm design."],"forward_implications":["Factoring and discrete logarithm are solvable in polynomial time on a quantum computer.","A quantum Fourier transform can be performed over any finite Abelian group in polynomial time.","Any algebraic problem that reduces to finding the stabilizer of an efficiently implementable Abelian action inherits a polynomial quantum algorithm.","Quantum phase estimation becomes a reusable primitive for designing new algorithms."],"fun_headline_variants":["Quantum algorithm for Abelian stabilizer via eigenvalue measurement","Polynomial time quantum algorithm for Abelian stabilizer problem","Eigenvalue measurements of unitaries solve Abelian stabilizer problem","Quantum Fourier transform for any finite Abelian group"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The unitary operator corresponding to the group action or function can be realized by an efficient quantum circuit.","fun_headline_variants_meta":{"raw":{"variants":["Quantum algorithm for Abelian stabilizer via eigenvalue measurement","Polynomial time quantum algorithm for Abelian stabilizer problem","Eigenvalue measurements of unitaries solve Abelian stabilizer problem","Quantum Fourier transform for any finite Abelian group"]},"model":"grok-4.3","cost_usd":0.01256,"raw_usage":{"total_tokens":5294,"prompt_tokens":486,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":125603000,"prompt_tokens_details":{"text_tokens":486,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4749,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":486,"tokens_out":59,"duration_ms":89155,"temperature":1.0,"reasoning_tokens":4749,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-13T05:03:44.059564+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit superpolynomial lower bound on the quantum circuit complexity of either integer factoring or the Abelian stabilizer problem would show the algorithm cannot exist.","supporting_citations":[],"review_version":1}