Abstract
We introduce several new definitions of universality for sets of quantum gates, and prove separation results for these definitions. In particular, we prove that realisability with ancillas is different from the classical notion of completeness. We give a polynomial time algorithm of independent interest which decides if a subgroup of a classical group (SO n , SU n , SL n ...) is Zariski dense, thus solving the decision problem for the completeness. We also present partial methods for the realisability with ancillas.
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References
Post, E.: The two-valued iterative systems of mathematical logic. Annals Mathematical Studies, vol. 5. Princeton University Press, Princeton (1941)
Fredkin, E., Toffoli, T.: Conservative Logic. International Journal of Theoretical Physics 21, 219–253 (1982)
Lloyd, S.: Almost any quantum logic gate is universal. Physical Review Letters 75, 346–349 (1995)
Deutsch, D., Barenco, A., Ekert, A.: Universality in quantum computation. Proceedings of the Royal Society of London, Series A 449, 669–677 (1995)
Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N.H., Shor, P.W., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Physical Review A 52, 3457–3467 (1995)
Barenco, A.: A universal two-bit gate for quantum computation. Proceedings of the Royal Society of London, Series A 449, 679–683 (1995)
Kitaev, A., Shen, V.: Classical and Quantum computation. Graduate Studies in Mathematics, vol. 47. American Mathematical Society, Providence (2003)
Shi, Y.: Both Toffoli and Controlled-NOT need little help to do universal quantum computation. Quantum Information and Computation 3, 84–92 (2003)
Brylinski, J.L., Brylinski, R.: Universal Quantum Gates. In: Mathematics of Quantum Computation, Chapman & Hall, Boca Raton (2002)
Derksen, H., Jeandel, E., Koiran, P.: Quantum automata and algebraic groups. To appear in Journal of Symbolic Computation (2004)
Onishchik, A., Vinberg, E.: Lie groups and algebraic groups. Springer, Berlin (1990)
Beals, R.: Algorithms for Matrix Groups and the Tits Alternative. Journal of Computer and System Sciences 58, 260–279 (1999)
Graham, A.: Kronecker Products and Matrix Calculus: with Applications. Ellis Horwood Limited, England (1981)
Babai, L., Beals, R., Rockmore, D.N.: Deciding finiteness for matrix groups in deterministic polynomial time. In: ISSAC 1993, pp. 117–126. ACM Press, New York (1993)
Mneimné, R., Testard, F.: Introduction à la théorie des groupes de Lie classiques. Hermann (1986)
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Jeandel, E. (2004). Universality in Quantum Computation. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_67
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DOI: https://doi.org/10.1007/978-3-540-27836-8_67
Publisher Name: Springer, Berlin, Heidelberg
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