 
Summary: Quantum Certificate Complexity
Scott Aaronson #
Abstract
Given a Boolean function f , we study two natural generalizations of the certificate complexity C (f ):
the randomized certificate complexity RC (f) and the quantum certificate 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 relation R0 (f) = O # Q 2 (f) 2 Q 0 (f) log n # for total f , where R0 , Q 2 ,
and Q 0 are zeroerror randomized, boundederror quantum, and zeroerror quantum query complexities
respectively. Finally we give asymptotic gaps between 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 Q 2 (f)
=# (n/ log n).
Most of what is known about the power of quantum computing can be cast in the query or decisiontree
model [1, 3, 4, 7, 6, 9, 10, 11, 20, 25, 24]. Here one counts only the number of queries to the input, not the
number of computational steps. The appeal of this model lies in its extreme simplicityin contrast to (say)
the Turing machine model, one feels the query model ought to be `completely understandable.' In spite of
this, open problems abound.
Let f : S # {0, 1} be a Boolean function with S # {0, 1} n , that takes input Y = y 1 . . . yn . Then the
deterministic query complexity D (f) is the minimum number of queries to the y i 's needed to evaluate f , if
Y is chosen adversarially and if queries can be adaptive (that is, can depend on the outcomes of previous
queries). Also, the boundederror randomized query complexity, R 2 (f ), is the minimum expected number
