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  