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oai:drops-oai.dagstuhl.de:20213 2024-07-02T07:52:51Z ddc:004 open_access

BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting Gharibian, Sevag Kamminga, Jonas Approximate Counting Search to Decision Reduction BQP NP Oracle Complexity Class What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log n) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(log n) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(log n) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(log n) NP queries, then BPP^NP[o(n)] contains a 𝖯^NP-complete problem if the algorithm is classical and FBQP^NP[o(n)] contains an FP^NP-complete problem if the algorithm is quantum. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sevag Gharibian and Jonas Kamminga 2024 Is Part Of LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2024.70 urn:nbn:de:0030-drops-202134 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.70 eng https://creativecommons.org/licenses/by/4.0/legalcode