有限群性质的简洁量子证明
Succinct quantum proofs for properties of finite groups
摘要：
在本文中，我们考虑一个量子计算变量的非确定性的基础上的概念的量子证明，这是一个量子状态，在NP-型证明中扮演着类似于证书的角色。具体来说，我们考虑黑盒群的性质的量子证明，黑盒群是有限群，其元素被编码为给定长度的字符串，其群运算由群预言机执行。我们证明了对于任意的群预言机，群非成员问题存在简洁的（多项式长度的）量子证明，可以在量子计算机上用多项式时间内以很小的误差进行检验.经典上这是不可能的--证明了存在一个群预言机，相对于该群预言机，这个问题没有简洁的证明，可以在多项式时间内用有界误差经典地检查（即，问题不在相对于构造的群oracle的MA中）。通过考虑群非成员问题的一个子问题，我们得到了一个简单的证明，存在一个预言，相对于该预言，BQP不包含在MA中。最后，我们表明，非成员资格的量子证明和各种其他群属性的经典证明可以结合起来，以产生简洁的量子证明其他群属性没有简洁的证明在经典设置，如验证一个数划分的顺序组和验证一个组不是一个简单的组。
英文摘要：
In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NP-type proof. Specifically, we consider quantum proofs for properties of black-box groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomial-length) quantum proofs for the Group Non-Membership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossible--it is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group Non-Membership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for non-membership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
