 
Summary: Quantum Certificate Complexity
Scott Aaronson
Computer Science Division
University of California, Berkeley
Abstract
Given a Boolean function f, we study two natural gener
alizations of the certificate complexity C (f): the random
ized certificate complexity RC (f) and the quantum cer
tificate complexity QC(f). Using Ambainis' adversary
method, we exactly characterize QC (f) as the square root
of RC(f). We then use this result to prove the new rela
tion R0 (f) = O Q2 (f)
2
Q0 (f) log n for total f, where
R0, Q2, and Q0 are zeroerror randomized, bounded
error quantum, and zeroerror quantum query complexi
ties respectively. Finally we give asymptotic gaps be
tween the measures, including a total f for which C(f) is
superquadratic in QC (f), and a symmetric partial f for
which QC (f) = O (1) yet Q2 (f) = (n/ log n).
