Estimating Jones polynomials is A complete problem for one clean qubit

Quantum Information and Computation

Volumn 8, Issue 8-9, 2008, Pages 0681-0714

  Shor, Peter W ^a   Jordan, Stephen P ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Jones polynomial; One clean qubit

Indexed keywords

POLYNOMIAL APPROXIMATION; QUANTUM THEORY;

COMPLETE PROBLEMS; COMPLEXITY CLASS; JONES POLYNOMIAL; MIXED STATE; ONE CLEAN QUBIT; QUANTUM CIRCUIT; TIME POLYNOMIALS;

QUBITS;

EID: 56249109842     PISSN: 15337146     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. Dorit Aharonov and Itai Arad. The BQP-hardness of approximating the Jones polynomial. 2006. arXiv:quant-ph/0605181.
2. Dorit Aharonov, Itai Arad, Elad Eban, and Zeph Landau. Polynomial quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane. arXiv:quant- ph/0702008, 2007.
3. Dorit Aharonov, Vaughan Jones, and Zeph Landau. A polynomial quantum algorithm for approximating the Jones polynomial. STOC 06, 2006. arXiv:quant-ph/0511096. (Pubitemid 44306577)
4. Andris Ambainis, Leonard Schulman, and Umesh Vazirani. Computing with highly mixed states. Journal of the ACM, 53(3):507-531, May 2006. arXiv:quant-ph/0003136. (Pubitemid 44180745)
5. Michael Artin. Algebra. Prentice Hall, 1991.
6. Adriano Barenco, Artur Ekert, Kalle-Antti Suominen, and Paivi Torma. Approximate quantum Fourier transform and decoherence. Phyical Review A, 54(1):139-146, 1996. arXiv:quant- ph/9601018. (Pubitemid 126583487)
7. David A. Barrington. Bounded-width polynomial-size brancing programs recognize exactly those languages in NC1. Journal of Computer and System Sciences, 38:150-164, 1989.
8. D. Coppersmith. An approximate fourier transform useful in quantum factoring. Technical report, IBM, 1994. arXiv:quant-ph/0201067. (Pubitemid 24214083)
9. Steven A. Cuccaro, Thomas G. Draper, Samuel A. Kutin, and David Petrie Moulton. A new quantum ripple-carry addition circuit. arXiv:quant-ph/0410184, 2004.
10. Animesh Datta, Steven T. Flammia, and Carlton M. Caves. Entanglement and the power of one qubit. Physical Review A, 72(042316), 2005. arXiv:quant-ph/0505213.
11. Thomas Draper. Addition on a quantum computer. arXiv:quant-ph/0008033, 2000.
12. Michael Freedman, Alexei Kitaev, and Zhenghan Wang. Simulation of topological field theories by quantum computers. Communications in Mathematical Physics, 227:587-603, 2002. (Pubitemid 36198340)
13. Michael Freedman, Michael Larsen, and Zhenghan Wang. A modular functor which is universal for quantum computation. 2000. arXiv:quant-ph/0001108.
14. W. Haken. Theorie der normalflachen, ein isotopiekriterium fur den kreisknoten. Acta Mathematica, (105):245-375, 1961.
15. Joel Hass, Jeffrey Lagarias, and Nicholas Pippenger. The computational complexity of knot and link problems. Journal of the ACM, 46(2):185-211, 1999. arXiv:math.GT/9807016.
16. F. Jaeger, D. L. Vertigan, and D. J. A. Welsh. On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings ofthe Cambridge Philosophical Society, (108):35-53,1990.
17. Vaughan F. R. Jones. Braid groups, Hecke algebras and type II1 factors. In Geometric methods in operator algebras, US-Japan Seminar, pages 242-273, Kyoto, July 1983.
18. Vaughan F. R. Jones. A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society, 12:103-111, 1985.
19. Louis H. Kauffman and Samuel J. Lomonaco Jr. q-deformed spin networks knot polynomials and anyonic topological quantum computation. arXiv:quant-ph/0606114, 2006.
20. Alexei Yu. Kitaev. Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6):1191-1249, 1997. (Pubitemid 128069764)
21. E. Knill and R. Laflamme. Power of one bit of quantum information. Physical Review Letters, 81(25):5672-5675, 1998. arXiv:quant-ph/9802037. (Pubitemid 128628238)
22. E. Knill and R. Laflamme. Quantum computating and quadratically signed weight enumerators.Information Processing Letters, 79(4):173-179, 2001. arXiv:quant-ph/9909094.
23. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.
24. Christos Papadimitriou. Computational Complexity. Addison Wesley, 1994.
25. David Poulin, Robin Blume-Kohout, Raymond Laflamme, and Harold Ollivier. Exponential speedup with a single bit of quantum information: Measuring the average fidelity decay. Physical Review Letters, 92(17):177906, 2004. arXiv:quant-ph/0310038.
26. C. A. Ryan, J. Emerson, D. Poulin, C. Negrevergne, and R. Laflamme Characterization of complex quantum dynamics with a scalable NMR information processor. Physical Review Letters, 95:250502, 2005. arXiv:quant-ph/0506085. (Pubitemid 43272723)
27. Dan Shepherd. Computation with unitaries and one pure qubit. 2006. arXiv:quant-ph/0608132.
28. Yasuhiro Takahashi and Noburu Kunihiro. A quantum circuit for Shor's factoring algorithm using 2n + 2 qubits. Quantum Information and Computation, 6(2):184-192, 2006.
29. Ingo Wegener. The Complexity of Boolean Functions. Wiley, 1987.
30. Edward Witten. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics, 121(3):351-399, 1989.
31. Pawel Wocjan and Jon Yard. The Jones polynomial: quantum algorithms and applications in quantum complexity theory. arXiv:quant-ph/0603069, 2006.