TY - GEN

T1 - Quantum expanders

T2 - 23rd Annual IEEE Conference on Computational Complexity, CCC 2008

AU - Ben-Aroya, Avraham

AU - Schwartz, Oded

AU - Ta-Shma, Amnon

PY - 2008

Y1 - 2008

N2 - We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL{2, q) given by Lubotzky, Philips and Sarnak [27]. The second construction is combinatorial, and is based on a quantum variant of the Zig-Zag product introduced by Reingold, Vadhan and Wigderson [35]. Both constructions are of constant degree, and the second one is explicit. Using quantum expanders, we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed stales, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK-complele (where QSZK is the class of languages having a zero-knowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof resembles the classical proof that the entropy difference problem is SZK-complete, but crucially depends on the use of quantum expanders.

AB - We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL{2, q) given by Lubotzky, Philips and Sarnak [27]. The second construction is combinatorial, and is based on a quantum variant of the Zig-Zag product introduced by Reingold, Vadhan and Wigderson [35]. Both constructions are of constant degree, and the second one is explicit. Using quantum expanders, we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed stales, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK-complele (where QSZK is the class of languages having a zero-knowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof resembles the classical proof that the entropy difference problem is SZK-complete, but crucially depends on the use of quantum expanders.

UR - http://www.scopus.com/inward/record.url?scp=51749124686&partnerID=8YFLogxK

U2 - 10.1109/CCC.2008.23

DO - 10.1109/CCC.2008.23

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AN - SCOPUS:51749124686

SN - 9780769531694

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 292

EP - 303

BT - Proceedings - 23rd Annual IEEE Conference on Computational Complexity, CCC 2008

Y2 - 23 June 2008 through 26 June 2008

ER -