Abstract
AbstractMatrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computer Science Applications,General Computer Science
Reference35 articles.
1. Ryser, HJohn: Combinatorial Mathematics, vol. 14. American Mathematical Soc., Providence (1963)
2. Valiant, L.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979). https://doi.org/10.1016/0304-3975(79)90044-6
3. Ben-Dor, A., Halevi, S.: Zero-one permanent is # p-complete, a simpler proof. In: Proceedings of the 2nd Israel Symposium on the Theory and Computing Systems, (1993)
4. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51(4), 671–697 (2004). https://doi.org/10.1145/1008731.1008738
5. Gurvits, Samorodnitsky: A deterministic algorithm for approximating the mixed discriminant and mixed volume, and a combinatorial corollary. Discrete Comput. Geom. 27(4), 531–550 (2002). https://doi.org/10.1007/s00454-001-0083-2