Query Complexity: WorstCase Quantum Versus AverageCase Classical
Abstract
In this note we investigate the relationship between worstcase quantum query complexity and averagecase classical query complexity. Specifically, we show that if a quantum computer can evaluate a total Boolean function f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^5) queries in the average case, under a uniform distribution of inputs. If f is monotone, we show furthermore that only O(T^3) queries are needed. Previously, Beals et al. (1998) showed that if a quantum computer can evaluate f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^6) queries in the worst case, or O(T^4) if f is monotone. The optimal bound is conjectured to be O(T^2), but improving on O(T^6) remains an open problem. Relating worstcase quantum complexity to averagecase classical complexity may suggest new ways to reduce the polynomial gap in the ordinary worstcase versus worstcase setting.
 Publication:

arXiv eprints
 Pub Date:
 January 2000
 arXiv:
 arXiv:cs/0001013
 Bibcode:
 2000cs........1013A
 Keywords:

 Computer Science  Computational Complexity;
 Quantum Physics;
 F.1.2
 EPrint:
 Withdrawn. The results in the paper only work for a certain subclass of Boolean functions, in which block sensitivity has properties similar to those of ordinary sensitivity. They don't work in general