The learnability of quantum states

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 463, Issue 2088, 2007, Pages 3089-3114

  Aaronson, Scott ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Occam's razor; PAC learning; Quantum communication; Quantum computing; Quantum state tomography; Sample complexity

Indexed keywords

EID: 36349030376     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2007.0113     Document Type: Article

References (29)

1. Aaronson, S. 2004 Limitations of quantum advice and one-way communication. In Proc. 19th. IEEE Conf. on Computational Complexity, pp. 320-332.
2. Aaronson, S. 2005 Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. A 461, 3473-3482. (doi:10.1098/rspa.2005.1546)
3. Aaronson, S. 2006 QMA/qpoly is contained in PSPACE/poly: de-Merlinizing quantum protocols. In Proc. 21st IEEE Conf. on Computational Complexity, pp. 261-273.
4. Aaronson, S. & Gottesman, D. 2004 Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328. (doi:10.1103/PhysRevA.70. 052328)
5. Aaronson, S. & Kuperberg, G. 2007 Quantum versus classical proofs and advice. In Proc. 22nd IEEE Conf. on Computational Complexity, pp. 115-128.
6. Alon, N., Ben-David, S., Cesa-Bianchi, N. & Haussler, D. 1997 Scale-sensitive dimensions, uniform convergence, and learnability. J. ACM 44, 615-631. (doi:10.1145/263867.263927)
7. Ambainis, A., Nayak, A., Ta-Shma, A. & Vazirani, U. V. 2002 Quantum dense coding and quantum finite automata. J. ACM 49, 496-511. (doi:10.1145/581771.581773)
8. Anthony, M. & Bartlett, P. L. 2000 Function learning from interpolation. Combinat. Prob. Comput. 9, 213-225. (doi:10.1017/S0963548300004247)
9. Bartlett, P. L. & Long, P. M. 1998 Prediction, learning, uniform convergence, and scale-sensitive dimensions. J. Comput. Syst. Sci. 56, 174-190. (doi:10.1006/jcss.1997.1557)
10. Bartlett, P. L., Long, P. M. & Williamson, R. C. 1996 Fat-shattering and the learnability of realvalued functions. J. Comput. Syst. Sci. 52, 434-452. (doi:10.1006/jcss.1996.0033)
11. Blumer, A., Ehrenfeucht, A., Haussler, D. & Warmuth, M. K. 1989 Learnability and the Vapnik-Chervonenkis dimension. J. ACM 36, 929-965. (doi:10.1145/76359.76371)
12. Bogdanov, A. & Trevisan, L. 2006 Average-case complexity. ECCC TR06-073.
13. Bužek, V. 2004 Quantum tomography from incomplete data via MaxEnt principle. In Quantum state estimation (eds M. G. A. Paris & J. Řeháček), pp. 189-234. Berlin, Germany: Springer.
14. Bužek, V., Drobný, G., Derka, R., Adam, G. & Wiedemann, H. 1999 Quantum state reconstruction from incomplete data. Chaos Soliton. Fract. 10, 981-1074. (doi:10.1016/S0960-0779(98)00144-1)
15. D'Ariano, G. M., De Laurentis, M., Paris, M. G. A., Porzio, A. & Solimeno, S. 2002 Quantum tomography as a tool for the characterization of optical devices. J. Opt. B: Quant. Semiclass. Opt. 4, S127-S132. (doi:10.1088/1464-4266/4/3/366)
16. Gavinsky, D., Kempe, J., Kerenidis, I., Raz, R. & de Wolf, R. 2007 Exponential separations for one-way quantum communication complexity, with applications to cryptography. In 39th Ann. ACM Symp. on Theory of Computing, pp. 516-525.
17. Goldreich, O. 2004 On quantum computing. See www.wisdom.weizmann.ac.il/ oded/on-qc.html.
18. Goldreich, O., Goldwasser, S. & Micali, S. 1984 How to construct random functions. J. ACM 33, 792-807. (doi:10.1145/6490.6503)
19. Häffner, H. et al. 2005 Scalable multiparticle entanglement of trapped ions. Nature 438, 643-646. (doi:10.1038/nature04279)
20. Kearns, M. J. & Schapire, E. 1994 Efficient distribution-free learning of probabilistic concepts. J. Comput. Syst. Sci. 48, 464-497. (doi:10.1016/S0022-0000(05)80062-5)
21. Klauck, H. 2000 Quantum communication complexity. In Proc. Int. Colloquium on Automata, Languages, and Programming (ICALP), pp. 241-252. ArXiv:quant-ph/0005032.
22. Levin, L. A. 2003 Polynomial time and extravagant models, in the table of one-way functions. Probl. Inform. Transm. 39, 92-103. (doi:10.1023/A:1023634616182)
23. Linial, N., Mansour, Y. & Nisan, N. 1993 Constant depth circuits, Fourier transform, and learnability. J. ACM 40, 607-620. (doi:10.1145/174130.174138)
24. O'Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. 2003 Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264-267. (doi:10.1038/nature02054)
25. Paris, M. G. A. & Řeháček, J. (eds) 2004. Quantum state estimation. Berlin, Germany: Springer.
26. Resch, K. J., Walther, P. & Zeilinger, A. 2005 Full characterization of a three-photon Greenberger-Horne-Zeilinger state using quantum state tomograph. Phys. Rev. Lett. 94, 070402. (doi:10.1103/ PhysRevLett.94.070402)
27. Skovsen, E., Stapelfeldt, H., Juhl, S. & Mølmer, K. 2003 Quantum state tomography of dissociating molecules. Phys. Rev. Lett. 91, 090406. (doi:10.1103/PhysRevLett.91.090406)
28. Terhal, B. M. & DiVincenzo, D. P. 2002 Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65, 032325. (doi:10.1103/PhysRevA.65.032325)
29. Valiant, L. G. 1984 A theory of the learnable. Commun. ACM 27, 1134-1142. (doi:10.1145/1968. 1972)