{"paper":{"title":"Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","quant-ph"],"primary_cat":"cs.DS","authors_text":"Andris Ambainis, J\\=anis Iraids, Jevg\\=enijs Vihrovs, Juris Smotrovs, Kamil Khadiev, Kaspars Balodis, Kri\\v{s}j\\=anis Pr\\=usis, Vladislavs K\\c{l}evickis, Yixin Shen","submitted_at":"2020-07-06T09:51:41Z","abstract_excerpt":"We study the quantum query complexity of two problems.\n  First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. We call this the $Dyck_{k,n}$ problem. We prove a lower bound of $\\Omega(c^k \\sqrt{n})$, showing that the complexity of this problem increases exponentially in $k$. Here $n$ is the length of the word. When $k$ is a constant, this is interesting as a representative example of star-free languages for which a surprising $\\tilde{O}(\\sqrt{n})$ query quantum algorithm was recently constructed by Aaro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.03402","kind":"arxiv","version":2},"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/2007.03402/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"}