{"paper":{"title":"A Polynomial Quantum Algorithm for Approximating the Jones Polynomial","arxiv_id":"quant-ph/0511096","primary_cat":"quant-ph","submitted_at":"2005-11-09T20:08:32+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v2","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"The Jones polynomial, discovered in 1984, is an important knot invariant in topology.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.02,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.8,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.8,"id":"s3","own":false,"role":"background","text":"These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e^{2\\pi i/5}, and moreover, that this problem is BQP-complete.","hedge":{"p":0.85,"label":"asserted"},"own_p":0.05,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.84,"role_probabilities":{"gap":0.15,"method":0.0,"result":0.0,"background":0.8,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.05}},{"i":4,"p":1.0,"id":"s4","own":false,"role":"gap","text":"Unfortunately, this important algorithm was never explicitly formulated.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.11,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":1.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.87,"id":"s5","own":false,"role":"gap","text":"Moreover, the results mentioned are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.13,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.87,"method":0.0,"result":0.0,"background":0.1,"limitation":0.03,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"contribution","text":"We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n-strands braid with m crossings at any primitive root of unity e^{2\\pi i/k}, where the running time of the algorithm is polynomial in m,n and k.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.93,"quantitative":true,"quantitative_p":0.85,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":7,"p":0.83,"id":"s7","own":true,"role":"method","text":"Our algorithm is based, rather than on TQFT, on well known mathematical results.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.83,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.17,"interpretation":0.0}},{"i":8,"p":0.73,"id":"s8","own":false,"role":"interpretation","text":"By the results of Freedman et. al., our algorithm solves a BQP complete problem.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.45,"main_claim_p":0.13,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.26,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.73}},{"i":9,"p":0.83,"id":"s9","own":true,"role":"future_work","text":"The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems.","hedge":{"p":0.56,"label":"hedged_possibility"},"own_p":0.66,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.83,"contribution":0.01,"interpretation":0.15}},{"i":10,"p":1.0,"id":"s10","own":false,"role":"future_work","text":"Candidates of particular interest are the approximations of other #P-hard problems, most notably, the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial.","hedge":{"p":0.55,"label":"hedged_possibility"},"own_p":0.17,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":1.0,"contribution":0.0,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n-strands braid with m crossings at any primitive root of unity e^{2\\pi i/k}, where the running time of the algorithm is polynomial in m,n and k.","location":"abstract s6","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":8222,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.3,"states_limitation":0.37,"human_participants":0.02,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.65,"compares_with_prior_results":0.89,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":0.74,"value":"proof or derivation","probabilities":{"argument":0.04,"literature":0.22,"proof_or_derivation":0.74,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"method or tool","probabilities":{"review":0.0,"method_or_tool":1.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n-strands braid with m crossings at any primitive root of unity e^{2\\pi i/k}, where the running time of the algorithm is polynomial in m,n and k.","location":"abstract s6"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v2"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":11,"sentences_in_abstract":11},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448,"review_comments_judged":{"judged_by":["gpt-5.6-luna","grok-4.7"],"judged_against":"Pith's own text copy of each paper: text extracted from the PDF, the first 90,000 characters","measured_on":"2026-10-04","accurate":113,"decided":149,"sampled_reviews":200,"split_by_lane":false,"by_lane":"https://pith.science/accuracy#by-lane"}}}