无碰撞性的简洁量子证明的不可能性
Impossibility of Succinct Quantum Proofs for Collision-Freeness
摘要：
我们证明，任何量子算法，以决定是否一个函数f：[n]->[n]是一个置换或远离一个置换必须作出欧米茄（n^{1/3}/w）查询f，即使该算法被赋予一个w-量子比特量子证人支持f是一个置换。这意味着存在一个神谕A，使得SZK^A不包含在QMA^A中，回答了作者八年前的一个公开问题。事实上，我们表明，相对于一些甲骨文，SZK是不是在计数类A0 PP定义的Vyalyi。这个证明是碰撞问题的量子下限的一个相当简单的扩展。
英文摘要：
We show that any quantum algorithm to decide whether a function f:[n]->[n] is a permutation or far from a permutation must make Omega(n^{1/3}/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 SZK^A is not contained in QMA^A, 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.
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