{"paper":{"title":"Quantum Algorithms for Modular Factorials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"quant-ph","authors_text":"Yann Tal","submitted_at":"2026-07-31T14:15:12Z","abstract_excerpt":"We give a bounded-error quantum algorithm that, given a prime $p$, a divisor $q\\mid(p-1)$, and an integer $0<n<p$, computes $n!\\bmod p$ in expected time $\\widetilde{O}(q^c+\\sqrt{p/q})$ for some absolute constant $c\\ge 1$. When $p-1$ has a divisor of size $q\\approx p^{1/(2c+1)}$, this gives the exponent $c/(2c+1)<1/2$. To our knowledge, this is the first algorithm to break the exponent $1/2$ barrier for modular factorials under such a divisor promise. The main technical ingredient is a quantum algorithm that reconstructs the relevant Jacobi sum exactly in compact algebraic form, with polynomial"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29453","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.29453/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}