TY - GEN

T1 - Hardness of approximation for quantum problems

AU - Gharibian, Sevag

AU - Kempe, Julia

PY - 2012

Y1 - 2012

N2 - The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Our work thus yields the first known hardness of approximation results for a quantum complexity class. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy (Umans 1999). We close by showing that a variant of the local Hamiltonian problem with hybrid classical-quantum ground states is complete for the second level of our quantum hierarchy.

AB - The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Our work thus yields the first known hardness of approximation results for a quantum complexity class. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy (Umans 1999). We close by showing that a variant of the local Hamiltonian problem with hybrid classical-quantum ground states is complete for the second level of our quantum hierarchy.

UR - http://www.scopus.com/inward/record.url?scp=84883794770&partnerID=8YFLogxK

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U2 - 10.1007/978-3-642-31594-7_33

DO - 10.1007/978-3-642-31594-7_33

M3 - Conference contribution

AN - SCOPUS:84883794770

SN - 9783642315930

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 387

EP - 398

BT - Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings

T2 - 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012

Y2 - 9 July 2012 through 13 July 2012

ER -