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.
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References
Aharonov, D., Arad, I., Landau, Z., Vazirani, U.: The detectibility lemma and quantum gap amplification. In: 41st ACM Syposium on Theory of Computing, vol. 287, pp. 417–426 (2009)
Aaronson, S.: The quantum PCP manifesto (2006), http://scottaaronson.com/blog/?p=139
Aharonov, D., Naveh, T.: Quantum NP - A survey. Preprint at arXiv:quant-ph/0210077v1 (2002)
Arad, I.: A note about a partial no-go theorem for quantum PCP. Preprint at arXiv:quant-ph/1012.3319 (2010)
Fortnow, L., Lipton, R., van Melkebeek, D., Viglas, A.: Time-space lower bounds for satisfiability. Journal of the ACM 52, 835–865 (2005)
Fortnow, L.: Time-space tradeoffs for satisfiability. Journal of Computer and System Sciences 60(2), 337–353 (2000)
Hastings, M.B.: Trivial low energy states for commuting hamiltonians, and the quantum PCP conjecture. Preprint at arXiv:quant-ph/1201.3387 (2012)
Hemaspaandra, L.: SIGACT news complexity theory column 38. ACM SIGACT News 33(4) (2002); Guest column by Schaefer, M., Umans, C.
Kempe, J., Kitaev, A., Regev, O.: The complexity of the local Hamiltonian problem. SIAM Journal on Computing 35(5), 1070–1097 (2006)
Kempe, J., Regev, O.: 3-local Hamiltonian is QMA-complete. Quantum Information & Computation 3(3), 258–264 (2003)
Kitaev, A., Shen, A., Vyalyi, M.: Classical and Quantum Computation. American Mathematical Society (2002)
Lautemann, C.: BPP and the polynomial time hierarchy. Information Processing Letters 17, 215–218 (1983)
Sipser, M.: A complexity theoretic approach to randomness. In: 15th Symposium on Theory of Computing, pp. 330–335. ACM Press (1983)
Srinivasan, A., Zuckerman, D.: Computing with very weak random sources. In: 35th Symposium on Foundations of Computer Science, pp. 264–275 (1994)
Ta-Shma, A., Umans, C., Zuckerman, D.: Lossless condensers, unbalanced expanders, and extractors. Combinatorica 27(2), 213–240 (2007)
Umans, C.: Hardness of approximating \(\Sigma_2^p\) minimization problems. In: 40th Symposium on Foundations of Computer Science, pp. 465–474 (1999)
Watrous, J.: Quantum computational complexity. In: Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science, ch. 17, pp. 7174–7201. Springer (2009)
Yamakami, T.: Quantum NP and a quantum hierarchy. In: 2nd IFIP International Conference on Theoretical Computer Science, pp. 323–336. Kluwer Academic Publishers (2002)
Zuckerman, D.: On unapproximable versions of NP-complete problems. SIAM Journal on Computing 25(6), 1293–1304 (1996)
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Gharibian, S., Kempe, J. (2012). Hardness of Approximation for Quantum Problems. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_33
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DOI: https://doi.org/10.1007/978-3-642-31594-7_33
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