5 V. CONCLUSIONS ( s tending to zero), the term in parenthesis can be ap- proximated to 1 + O ( s $^{2}$), the terms of order 2 and higher being too small to be considered. The integral in t is now Optical quantum computation has been proven a wide field of possibilities for practical implementations of quantum algorithms. We explored the combined action of discrete qubits and CV states of light to perform the well known Deutsch-Jozsa algorithm, taking advantage of both schemes of computation and using only optical devices. In previous implementations of the algorithm, the main problem lies in the difficulty of producing the oracle operation. The hybrid model proposed here allevi- ates the problem by using classical feed-forward to create an ancillary state that performs the desired operation. which can be solved in terms of Gauss error functions [17, 18], finally yielding in the limit of high squeezing. The error function Γ$_{err}$( q, s ) is given by As we have emphasized before, the importance of this particular algorithm lies in its relative simplicity that en- ables us to implement it in many settings. The hybrid proposal that we present also tries to inspire the imple- mentation of algorithms based on the Fourier transform, with a nontrivial oracle, in order to find the possible new capabilities that such algorithms would acquire in contrast to the available implementations. For example, Shor’s algorithm for integer factorization satisfies both conditions for the optical hybrid model. In addition, we showed how the performance of the al- gorithm is diminished by considering (highly squeezed) Gaussian states in Alice’s register. Hence, in a realis- tic implementation, apart from measurement errors, we would not have the ideal evolution of the quadrature states | q 〉 either. Such implementation should provide some form of error correction together with repeated ap- plications of the algorithm for a successful, fault-tolerant computation that does not depend highly on the quality of squeezing. We sense that a better measurement stage would help to solve the problem. On the other hand, other than all-optical implemen- tations of hybrid computation exist, as the interaction of trapped atoms with a cavity mode of electromagnetic field [12]. These proposals may be incorporated together in a “super-hybrid” computer, and take advantage of all the schemes to achieve scalability in the computation. These Gauss error functions arise naturally, measuring the quality of the Gaussian signal in terms of the quality of squeezing associated to s . If the function is constant indeed, we should measure the initial state with total cer- tainty in the case of infinite squeezing as in (6), but due to the realistic Gaussian states used in practice, the prob- ability of measuring the initial state is given by | 〈 s | S 〉 | $^{2}$, and then solving the resulting Gaussian integrals using equation (16) we obtain a probability of 1 $_{2}$. This means that the determinism of the algorithm is lost when using realistic states and the algorithm becomes probabilistic, when the function is constant we have a 50% chance of getting the correct measurement at the output, we are no longer certain of the nature of the function with just one evaluation of the algorithm. However, its exponen- tial speedup over a classical algorithm still holds in this case. ˆ Π$_{Δ}$$_{s}$ | S 〉 ≈ ± 1 π ∫ d q$^{(}$$^{∫}$ s +Δ s/ 2 s − Δ s/ 2 d t t 2 e − q $^{2}$/t2 ) | q 〉 , (20) ˆ Π$_{Δ}$$_{s}$ | S 〉 = ± 1 √ 4 π ∫ d q Γ$_{err}$( q, s ) | q 〉 , (21) Γ$_{err}$( q, s ) = 1 q { erf ( q s − Δ s/ 2$^{)}$ − erf ( q s + Δ s/ 2$^{)}}$ . (22) [1] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409 , 46 (2001). [2] E. Knill, R. Laflamme, and G. J. Milburn, “Thresh- olds for linear optics quantum computation,” (2000), arXiv:quant-ph/0006120v1. [3] C. R. Myers and R. Laflamme, “Linear optical quan- tum computation: an overview,” (2005), arXiv:quant- ph/0512104v1. [7] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Phys. Rev. Lett. 88 , 097904 (2002). [8] D. Deutsch and R. Jozsa, Proc. R. Soc. London A 439 , 553 (1992). [9] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000). [10] A. K. Pati and S. L. Braunstein, in Quantum Information with Continuous Variables (Kluwer Academic Publisher, 2002) arXiv:quant-ph/0207108v1. [11] S. Lloyd, in Quantum Information with Continuous Vari- ables (Kluwer Academic Publisher, 2000) arXiv:quant- ph/0008057v1. [4] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79 , 135 (2007). [5] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77 , 513 (2005). [6] S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82 , 1784 (1999). [12] P. van Loock, W. J. Munro, K. Nemoto, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, Phys. Rev. A 78 ,