By Artur Ekert, Patrick Hayden and Hitoshi Inamori.

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75 346 (1995). [9] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1979). [10] T. Toffoli, Mathematical Systems Theory 14 13 (1981). [11] C. H. Papadimitriou, Computational Complexity (Addison-Wesley, 1994). [12] D. Coppersmith, IBM Research report (1994). [13] R. Solovay and V. Strassen SIAM J. Comp. 6 84 (1977). [14] R. Motwani and P. Raghavan, Randomised Algorithms (Cambridge University Press, 1995). [15] P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factoring” Proc.

105). It follows that F (t) = |α|4 + |β|4 + 2|α|2|β|2 Re[ m0 (t) | m1 (t) ] . (112) The expression above depends on the initial state | φ and clearly indicates that some states are more vulnerable to decoherence than others. In order to get rid of this dependence we consider the average fidelity, calculated under the assumption that any initial state | φ is equally probable. Taking into account the normalisation constraint the average fidelity is given by 1 F¯ (t) = 0 1 F (t) d |α|2 = (2 + Re[ m0 (t) | m1 (t) ]) .

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