Hardness of Approximation for Quantum Problems

S. Gharibian, J. Kempe, in: A. Czumaj, K. Mehlhorn, A. Pitts, R. Wattenhofer (Eds.), International Colloquium on Automata, Languages, and Programming (ICALP 2012), Springer Berlin Heidelberg, Berlin, Heidelberg, 2012, pp. 387–398.

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Abstract
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.
Publishing Year
Proceedings Title
International Colloquium on Automata, Languages, and Programming (ICALP 2012)
Page
387-398
Conference
International Colloquium on Automata, Languages, and Programming (ICALP)
Conference Location
Warwick, UK
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Cite this

Gharibian S, Kempe J. Hardness of Approximation for Quantum Problems. In: Czumaj A, Mehlhorn K, Pitts A, Wattenhofer R, eds. International Colloquium on Automata, Languages, and Programming (ICALP 2012). Berlin, Heidelberg: Springer Berlin Heidelberg; 2012:387-398. doi:10.1007/978-3-642-31594-7_33
Gharibian, S., & Kempe, J. (2012). Hardness of Approximation for Quantum Problems. In A. Czumaj, K. Mehlhorn, A. Pitts, & R. Wattenhofer (Eds.), International Colloquium on Automata, Languages, and Programming (ICALP 2012) (pp. 387–398). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_33
@inproceedings{Gharibian_Kempe_2012, place={Berlin, Heidelberg}, title={Hardness of Approximation for Quantum Problems}, DOI={10.1007/978-3-642-31594-7_33}, booktitle={International Colloquium on Automata, Languages, and Programming (ICALP 2012)}, publisher={Springer Berlin Heidelberg}, author={Gharibian, Sevag and Kempe, Julia}, editor={Czumaj, Artur and Mehlhorn, Kurt and Pitts, Andrew and Wattenhofer, RogerEditors}, year={2012}, pages={387–398} }
Gharibian, Sevag, and Julia Kempe. “Hardness of Approximation for Quantum Problems.” In International Colloquium on Automata, Languages, and Programming (ICALP 2012), edited by Artur Czumaj, Kurt Mehlhorn, Andrew Pitts, and Roger Wattenhofer, 387–98. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. https://doi.org/10.1007/978-3-642-31594-7_33.
S. Gharibian and J. Kempe, “Hardness of Approximation for Quantum Problems,” in International Colloquium on Automata, Languages, and Programming (ICALP 2012), Warwick, UK, 2012, pp. 387–398.
Gharibian, Sevag, and Julia Kempe. “Hardness of Approximation for Quantum Problems.” International Colloquium on Automata, Languages, and Programming (ICALP 2012), edited by Artur Czumaj et al., Springer Berlin Heidelberg, 2012, pp. 387–98, doi:10.1007/978-3-642-31594-7_33.

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