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oai:drops-oai.dagstuhl.de:10394 2024-03-06T10:46:44Z ddc:004 open_access

Quantum Distinguishing Complexity, Zero-Error Algorithms, and Statistical Zero Knowledge Ben-David, Shalev Kothari, Robin Quantum query complexity quantum algorithms We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable. Using this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, improving on the previously known sixth power relationship. We also define a query measure based on quantum statistical zero-knowledge proofs, QSZK(f), which is at most Q(f). We show that QD(f) in fact lower bounds QSZK(f) and not just Q(f). QD(f) also upper bounds the (positive-weights) adversary bound, which yields the following relationships for all f: Q(f) >= QSZK(f) >= QD(f) = Omega(Adv(f)). This sheds some light on why the adversary bound proves suboptimal bounds for problems like Collision and Set Equality, which have low QSZK complexity. Lastly, we show implications for lifting theorems in communication complexity. We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Shalev Ben-David and Robin Kothari 2019 Is Part Of LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.TQC.2019.2 urn:nbn:de:0030-drops-103944 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019.2 eng https://creativecommons.org/licenses/by/3.0/legalcode