2026-08-10T23:44:13Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:5853 2024-03-06T10:37:11Z ddc:004 open_access

Sculpting Quantum Speedups Aaronson, Scott Ben-David, Shalev Quantum Computing Query Complexity Decision Tree Complexity Structural Complexity Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. We show that a total function f can be restricted to a promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only if f has a large number of inputs with large certificate complexity. The proof uses some interesting techniques: for one direction, we introduce new relationships between randomized and quantum query complexity in various settings, and for the other direction, we use a recent result from communication complexity due to Klartag and Regev. We also characterize sculpting for other query complexity measures, such as R(f) vs. R_0(f) and R_0(f) vs. D(f). Along the way, we prove some new relationships for quantum query complexity: for example, a nearly quadratic relationship between Q(f) and D(f) whenever the promise of f is small. This contrasts with the recent super-quadratic query complexity separations, showing that the maximum gap between classical and quantum query complexities is indeed quadratic in various settings - just not for total functions! Lastly, we investigate sculpting in the Turing machine model. We show that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in PromiseBPP. Under a weaker assumption, that some problem in BQP is hard on average for P/poly, we show that every paddable language outside BPP is sculptable in this way. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Scott Aaronson and Shalev Ben-David 2016 Is Part Of LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2016.26 urn:nbn:de:0030-drops-58538 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.26 eng https://creativecommons.org/licenses/by/3.0/legalcode