Impossibility of Succinct Quantum Proofs for Collision-Freeness

We show that any quantum algorithm to decide whether a function f : [n] → [n] is a permutation or far from a permutation must make Ω n1/3 /w queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that SZKA ⊂ QMAA , answering an eight-year-old open question of the author. Indeed, we show that relative to some oracle, SZK is not in the counting class A0PP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Scott Aaronson
• Keywords: CORR 2011, Education, Eight-year-old Open Question, Permutation, W-qubit Quantum Witness