Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
J. Watrous , Dept. of Comput. Sci., Calgary Univ., Alta., Canada
The article considers 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.
group theory; theorem proving; quantum computing; computational complexity; succinct quantum proofs; finite group properties; quantum computational variant; nondeterminism; quantum proof; quantum state; certificate; NP-type proof; black-box groups; finite groups; strings; group operations; group oracle; Group Non-Membership problem; quantum computer; bounded error; polynomial time; simple proof; quantum proofs; classical proofs; group properties
J. Watrous, "Succinct quantum proofs for properties of finite groups," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 537.