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41st Annual Symposium on Foundations of Computer Science
Succinct quantum proofs for properties of finite groups
Redondo Beach, California
November 12November 14
ISBN: 0769508502
ASCII Text  x  
J. Watrous, "Succinct quantum proofs for properties of finite groups," 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 537, 41st Annual Symposium on Foundations of Computer Science, 2000.  
BibTex  x  
@article{ 10.1109/SFCS.2000.892141, author = {J. Watrous}, title = {Succinct quantum proofs for properties of finite groups}, journal ={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, volume = {0}, year = {2000}, issn = {02725428}, pages = {537}, doi = {http://doi.ieeecomputersociety.org/10.1109/SFCS.2000.892141}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2013 IEEE 54th Annual Symposium on Foundations of Computer Science TI  Succinct quantum proofs for properties of finite groups SN  02725428 SP EP A1  J. Watrous, PY  2000 KW  group theory; theorem proving; quantum computing; computational complexity; succinct quantum proofs; finite group properties; quantum computational variant; nondeterminism; quantum proof; quantum state; certificate; NPtype proof; blackbox groups; finite groups; strings; group operations; group oracle; Group NonMembership problem; quantum computer; bounded error; polynomial time; simple proof; quantum proofs; classical proofs; group properties VL  0 JA  2013 IEEE 54th Annual Symposium on Foundations of Computer Science ER   
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 NPtype proof. Specifically, we consider quantum proofs for properties of blackbox 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 (polynomiallength) quantum proofs for the Group NonMembership 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 NonMembership 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 nonmembership 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.
Index Terms:
group theory; theorem proving; quantum computing; computational complexity; succinct quantum proofs; finite group properties; quantum computational variant; nondeterminism; quantum proof; quantum state; certificate; NPtype proof; blackbox groups; finite groups; strings; group operations; group oracle; Group NonMembership problem; quantum computer; bounded error; polynomial time; simple proof; quantum proofs; classical proofs; group properties
Citation:
J. Watrous, "Succinct quantum proofs for properties of finite groups," focs, pp.537, 41st Annual Symposium on Foundations of Computer Science, 2000
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