{"paper":{"title":"Quantum Walk on the Line","arxiv_id":"quant-ph/0010117","primary_cat":"quant-ph","submitted_at":"2000-10-31T23:10:21+00:00","version_compiled":"v1"},"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 v1","sentences":[{"i":0,"p":0.94,"id":"s0","own":false,"role":"background","text":"Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs.","hedge":{"p":0.65,"label":"asserted"},"own_p":0.15,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.94,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":1,"p":0.98,"id":"s1","own":true,"role":"contribution","text":"We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''.","hedge":{"p":0.85,"label":"asserted"},"own_p":0.91,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"result","text":"We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)].","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.36,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.84,"id":"s3","own":true,"role":"interpretation","text":"This implies that the same walk defined on the circle mixes in_linear_ time.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.87,"main_claim_p":0.35,"quantitative":true,"quantitative_p":0.6,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.16,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.84}},{"i":4,"p":0.82,"id":"s4","own":false,"role":"interpretation","text":"This is in direct contrast with the quadratic mixing time for the corresponding classical walk.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.44,"main_claim_p":0.24,"quantitative":true,"quantitative_p":0.93,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.17,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.82}},{"i":5,"p":1.0,"id":"s5","own":false,"role":"future_work","text":"We conclude by indicating how our techniques may be applied to more general graphs.","hedge":{"p":0.98,"label":"hedged_possibility"},"own_p":0.3,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.03,"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":0.78,"text":"This implies that the same walk defined on the circle mixes in_linear_ time.","location":"abstract s3","hedge_strength":{"p":0.99,"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":4816,"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":0.04,"states_limitation":0.14,"human_participants":0.01,"controlled_comparison":0.4,"randomized_assignment":0.02,"states_quantitative_result":0.96,"compares_with_prior_results":0.96,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.74,"text":"We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)].","location":"abstract s2"}},"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 v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":6,"sentences_in_abstract":6},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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