Quantum certificate complexity

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 Q2 (f)2 Q0 (f) log n for total f, where R0, Q2, and Q0 are zero-error randomized, bounded-error quantum, and zero-error 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 Q2 (f) = (n/ log n). Most of what is known about the power of quantum computing can be cast in the query or decision-tree 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 si...

• Type: Journal
• Year: 2008
• Where: JCSS
• Authors: Scott Aaronson
• Keywords: Boolean Function, Certificate Complexity, JCSS 2008, Quantum