Luis A. Garc´ıa ∗ and Jagdish R. Luthra † (Dated: October 30, 2018) Departament of Physics, Universidad de los Andes, Bogot´ a 111711, Colombia A hybrid model of the Deutsch-Jozsa algorithm is presented, inspired by the proposals of hybrid computation by S. Lloyd and P. van Loock et. al . The model is based on two observations made about both the discrete and continuous algorithms already available. First, the Fourier transform is a single-step operation in a continuous-variable (CV) setting. Additionally, any implementation of the oracle is nontrivial in both schemes. The steps of the computation are very similar to those in the CV algorithm, with the main difference being the way in which the qunats, or quantum units of analogic information, and the qubits interact in the oracle. Using both discrete and continuous states of light, linear devices, and photo-detection, an optical implementation of the oracle is proposed. For simplicity, infinitely squeezed states are used in the continuous register, whereas the optical qubit is encoded in the dual-rail logic of the KLM protocol. The initial assumption of ideal states as qunats will be dropped to study the effects of finite squeezing in the quality of the computation. PACS numbers: 03.67.Lx, 42.50.Ex, 03.67.Ac number of qubits [9]. The translation of the algorithm into a CV setting seems more feasible and practical, due to the ability to produce the superpositions needed with a very low number of resources, nonetheless, how to im- plement the oracle in such continuous scheme is still an unresolved problem [10]. The model proposed by Knill, Laflamme and Milburn [1, 2] presents an efficient way to implement a quantum computer using optical qubits, single-photon sources, lin- ear devices and photo-detection, known as the KLM pro- tocol. This well established model uses nonlinearities hidden in the measurement process to produce the inter- actions needed between the optical modes, together with offline state preparation and teleportation to enhance the probability of success of the conditional gates of the pro- tocol, and error correction codes, achieving a scheme that is both fault-tolerant and robust against errors [3, 4]. However, we can take advantage of both schemes by means of some sort of hybrid gates, i.e. a set of trans- formations that acts on both qubits and CV states [11]. In this paper we use a hybrid model of computation, in an all-optical setting, giving an explicit optical circuit for the action of the oracle on both registers involved in the computation. Such model is based on proposals by S. Lloyd [11] and P. van Loock et. al. [12], but differs from them by utilizing only linear optical devices, single- photon sources, offline squeezing and photo-detectors. In the first part of the work, we use idealized infinitely squeezed states as the computational CV basis for sim- plicity in our calculations. By the end, we shall drop such restriction to discuss the effects of realistic Gaus- sian states (that is states with finite squeezing) through- out the computational process. On the other hand, the unconditional nature in which entanglement with continuous variables can be obtained, and thus the unconditionalness of the operations per- formed over CV together with the need of just single- mode nonlinearities for universal quantum computation, are two of the most prominent reasons to adopt this set- ting over a discrete-variable one [5, 6]. Besides, the idea of studying quantum algorithms, generally designed for qubits, in a CV setting is at the same time challenging and of great importance for the development of quantum information theory. This generalization may bring into light new algorithms with a more natural formulation using CVs. For example, any algorithm based on the quantum Fourier transform, such as the Deutsch-Jozsa algorithm or Shor’s algorithm, is easily performed in a continuous setting, where such operation is a single-step operation [7]. The structure of the paper is as follows. First, in Sec- tion II we give a quick review of the original algorithm and its implementation in the continuous-variable case. Section III presents the algorithm in a hybrid and all- optical setting, paying special attention to the explicit implementation of the oracle. Section IV considers im- perfect CV states in the hybrid model of the algorithm, in order to analyze the possible distortions at the end of the computation. Finally, Section V closes the paper with some concluding remarks. Focusing on the Deutsch-Jozsa algorithm, it can be seen that its original discrete formulation [8] requires the production of superpositions of several qubit states via a quantum Fourier transform, a task that needs a lot of resources and computational power, even for a small In this section we present a review of the basic aspects of the Deutsch-Jozsa problem according to its original ∗ anib-gar@uniandes.edu.co † jluthra@uniandes.edu.co II. FUNDAMENTALS OF THE DEUTSCH-JOZSA ALGORITHM arXiv:1001.0187v1 [quant-ph] 31 Dec 2009 I. INTRODUCTION Hybrid and optical implementation of the Deutsch-Jozsa algorithm