Quantum formulas: A lower bound and simulation

SIAM Journal on Computing

Volumn 31, Issue 2, 2001, Pages 460-476

  Roychowdhury, Vwani P ^a   Vatan, Farrokh ^a,b  

^a University of California   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Density matrix; Lower bound; Mixed state; Quantum formula

Indexed keywords

BOOLEAN FUNCTIONS; CIRCUIT THEORY; COMPUTATIONAL METHODS; COMPUTER SCIENCE; COMPUTER SIMULATION; GRAPH THEORY; POLYNOMIALS;

QUANTUM FORMULAS;

QUANTUM THEORY;

EID: 0036242960     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539700370965     Document Type: Article

References (27)

1. D. AHARONOV, A. KITAEV, AND N. NISAN, Quantum circuits with mixed states, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 1998, pp. 20-30.
2. A. BARENCO, C. BENNETT, R. CLEVE, D. DIVINCENZO, N. MARGOLUS, P. SHOR, T. SLEATOR, J. SMOLIN, AND H. WEINFURTER, Elementary gates for quantum computation, Phys. Rev. A (3), 52 (1995), pp. 3457-3467.
3. R. BEALS, H. BUHRMAN, R. CLEVE, M. MOSCA, AND R. DE WOLF, Quantum lower bounds by polynomials, in Proceedings of the 39th IEEE Annual Symposium on Foundations of Computer Science, 1998, IEEE Computer Society Press, Los Alamitos, CA, pp. 352-361.
4. C. H. BENNETT, Time/space trade-offs for reversible computation, SIAM J. Comput., 18 (1989), pp. 766-776.
5. E. BERNSTEIN AND U. VAZIRANI, Quantum complexity theory, SIAM J. Comput., 26 (1997), pp. 1411-1473.
6. R. B. BOPPANA AND M. SIPSER, The complexity of finite functions, in Handbook of Theoretical Computer Science, Vol. A, Algorithms and Complexity, J. van Leeuwen, ed., Elsevier Science, New York, MIT Press, Cambridge, MA, 1990, pp. 757-804.
7. M.- D. CHOI, Completely positive linear maps on complex matrices, Linear Algebra Appl., 10 (1975), pp. 285-290.
8. C. COHEN-TANNOUDJI, B. DIU, AND F. LALOË, Quantum Mechanics. I, John Wiley and Sons, New York, 1977.
9. D. DEUTSCH, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97-117.
10. D. DEUTSCH, Quantum computational networks, Proc. Roy. Soc. London Ser. A, 425 (1989), pp. 73-90.
11. P. E. DUNNE, The Complexity of Boolean Networks, Academic Press, London, 1988.
12. L. GROVER, A fast quantum mechanical algorithm for database search, in Proceedings of the 28th ACM Symposium on Theory of Computing, ACM Press, New York, 1996, pp. 212-219.
13. A. KITAEV, Quantum computations: Algorithms and error correction, Russian Math. Surveys, 52 (1997), pp. 1191-1249.
14. A. KITAEV, private communication, 2000.
15. E. KNILL, Approximating by Quantum Circuits, Los Alamos National Laboratory e-print quant-ph/9508006, 1995.
16. K. KRAUS, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer-Verlag, Berlin, New York, 1983.
17. R. Y. LEVINE AND A. T. SHERMAN, A note on Bennett's time-space tradeoff for reversible computation, SIAM J. Comput., 19 (1990), pp. 673-677.
18. M. A. NIELSEN AND I. L. CHUANG, Quantum Computation and Quantum Information, Cambridge University Press, New York, 2000.
19. A. SCHÖNHAGE AND V. STRASSEN, Schnelle multiplikation grosser zahlen, Computing, 7 (1971), pp. 281-292.
20. B. SCHUMACHER, Sending entanglement through noisy quantum channels, Phys. Rev. A (3), 54 (1996), pp. 2614-2628.
21. P. W. SHOR, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput., 26 (1997), pp. 1484-1509.
22. D. R. SIMON, On the power of quantum computation, SIAM J. Comput., 26 (1997), pp. 1474-1483.
23. GY. TURÁN AND F. VATAN, On the computation of Boolean functions by analog circuits of bounded fan-in, J. Comput. System Sci., 54 (1997), pp. 199-212.
24. L. G. VALIANT, Short monotone formulae for the majority function, J. Algorithms, 5 (1984), pp. 363-366.
25. H. E. WARREN, Lower bounds for approximation by nonlinear manifolds, Trans. Amer. Math. Soc., 133 (1968), pp. 167-178.
26. I. WEGENER, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987.
27. A. YAO, Quantum circuit complexity, in Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352-361.