eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-05-31
2:1
2:23
10.4230/LIPIcs.TQC.2019.2
article
Quantum Distinguishing Complexity, Zero-Error Algorithms, and Statistical Zero Knowledge
Ben-David, Shalev
1
Kothari, Robin
2
https://orcid.org/0000-0001-6114-943X
University of Waterloo, Waterloo, ON, Canada
Quantum Architectures and Computation (QuArC) group, Microsoft Research, Redmond, WA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol135-tqc2019/LIPIcs.TQC.2019.2/LIPIcs.TQC.2019.2.pdf
Quantum query complexity
quantum algorithms