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arXiv:quant-ph/0703132

Elaboration of linear-optical implementations of Deutsch algorithm utilizing entanglements

Koji Nagata, Sangkyung Lee, Jaewook Ahn

Published Mar 15, 2007• 0 citations
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Abstract

We propose multiple implementations of linear-optical processing of Deutsch algorithm using entangled state. Recently Oliveira {\\it et al.} [ J. Opt. B \\textbf{7}, 288 (2005) ] experimentally demonstrated the use of polarization and transverse spatial modes of the electromagnetic field as qubits for Deutsch algorithm, in which three Hadamard gates and a function are used. In our new implementations, the answer to the Deutsch problem is successfully determined even without the use of Hadamard gates. Instead, we use a polarization-momentum entangled bi-photon state. Coincidence measurements of photon arrivals coupled with polarization state analyzers identify answers to Deutsch problem. The entanglement-witness analysis is applied to validate the fidelity or the performance of the schemes.

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We propose multiple implementations of linear-optical processing of Deutsch algorithm using entangled state. Recently Oliveira {\it et al.} [ J. Opt. B \textbf{7}, 288 (2005) ] experimentally demonstrated the use of polarization and transverse spatial modes of the electromagnetic field as qubits for Deutsch algorithm, in which three Hadamard gates and a function are used. In our new implementations, the answer to the Deutsch problem is successfully determined even without the use of Hadamard gates. Instead, we use a polarization-momentum entangled bi-photon state. Coincidence measurements of photon arrivals coupled with polarization state analyzers identify answers to Deutsch problem. The entanglement-witness analysis is applied to validate the fidelity or the performance of the schemes.
A quantum computation algorithm is demonstrated using single photons and linear optics. Quantum registers are realized by modes and polarization of photons, and the unitary transformations for them are implemented with linear optics. For each quantum computation, the answer to the Deutsch Jozsa problem for any 4 bit digit is given by a single-photon detection signal with a small error rate. The origin of the error is also studied and categorized according to the sources: systematic errors of quantum gates, phase relaxation, amplitude dumping, and misreadout. It is suspected that a demonstration of 11 qubits using linear optics anda single photon with less than a 20 % error rate is achievable by the technique used in the experiment. As a further objective, an idea to construct controlled-NOT gate using quantum phase gate (QPG) given by Chuang et. al., is introduced and experiments on the QPGs are also reviewed.
Optical quantum computation was limited to a few photon applications such as quantum cryptography and partial teleportation until 2001. Then the possibility of performing arbitrary logic conditional on certain detection outcomes was introduced by Knill, Laflamme and Milburn ( Nature 2001 409 46). Simple circuits consisting of single photon sources, entangled pair sources, linear optical elements and high efficiency photon counting detectors can be used to develop probabilistic two-photon logic gates. Initially such gates show a low probability of success but can approach the success rate of deterministic (non-linear) logic at the cost of added complexity. These developments have led to the emergence of a novel field of research aimed at improving the efficiency of linear logic and developing the efficient sources and detectors for its implementation. This topical issue of Journal of Optics B: Quantum and Semiclassical Optics is devoted to recent advances in this field. Key to all linear optics gates is the interference effect between separate photons that have no previous history. In this issue we include the first experiment to show such an effect (Rarity et al ), on which many later experiments are based. These interference effects can then be used to entangle separate pairs of particles into higher order entangled states (Zou et al 2005 J. Opt. B: Quantum Semiclass. Opt. 7 119–121) and also to establish entanglement between matter qubits (Kok et al ). Quantum dot single photon sources suitable for linear optics quantum logic are discussed in the paper by Unitt et al while a waveguide source of pair photons is presented by Ducci et al . The difficulties in scaling linear optics quantum computing arise from the low success probabilities of the gates. Introducing a small amount of non-linearity could allow unit efficiency gates to be designed (Munro et al ) while large non-linearities generated by the collective effects in EIT could be used to implement a wide variety of quantum logic functions (Petrosyan). The development of algorithms suitable for optical implementation is addressed by Ellinas et al . Key contributors to this issue are collaborating in the project IST:38864 RAMBOQ which has a primary goal of developing linear optics computation tools and techniques.