{"paper":{"title":"The power of oracle access: Optimal sample and query complexity of the abelian state hidden subgroup problem","arxiv_id":"2609.35656","primary_cat":"quant-ph","submitted_at":"2026-09-28T17:21:41+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":1.0,"id":"s0","own":false,"role":"background","text":"In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model.","hedge":{"p":0.89,"label":"asserted"},"own_p":0.05,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.11,"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":0.57,"id":"s1","own":false,"role":"contribution","text":"Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state.","hedge":{"p":0.82,"label":"asserted"},"own_p":0.43,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.43,"limitation":0.0,"future_work":0.0,"contribution":0.57,"interpretation":0.0}},{"i":2,"p":0.82,"id":"s2","own":false,"role":"gap","text":"For finite abelian groups, existing Fourier-sampling algorithms use $O(\\log(|G|)/\\epsilon)$ copies of the state, but whether this scaling is optimal has remained open.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.95,"role_probabilities":{"gap":0.82,"method":0.0,"result":0.0,"background":0.18,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.99,"id":"s3","own":true,"role":"contribution","text":"We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.89,"main_claim_p":0.17,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":4,"p":0.93,"id":"s4","own":true,"role":"result","text":"In the query model, we give a time-efficient quantum algorithm using $O(\\log(|G/H|)/\\sqrt{\\epsilon})$ forward and inverse queries, and prove a matching $\\Omega(\\log(|G/H|)/\\sqrt{\\epsilon})$ lower bound which holds even in the stronger conjugate-query and controlled-query settings.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.34,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"By contrast, we show that in the sample model, $\\Theta(\\log(|G/H|)/\\epsilon)$ copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.24,"quantitative":true,"quantitative_p":0.97,"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":6,"p":1.0,"id":"s6","own":true,"role":"interpretation","text":"Thus, the quadratic improvement in $\\epsilon$ genuinely arises from coherent access to the preparation circuit.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.51,"quantitative":true,"quantitative_p":0.87,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":1.0}},{"i":7,"p":0.64,"id":"s7","own":true,"role":"contribution","text":"As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.34,"background":0.0,"limitation":0.0,"future_work":0.01,"contribution":0.64,"interpretation":0.01}}]},"compiled":{"main_claim":{"p":0.79,"text":"Thus, the quadratic improvement in $\\epsilon$ genuinely arises from coherent access to the preparation circuit.","location":"abstract 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