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Journal of Optics B Quantum and Semiclassical Optics

Implementing the Deutsch algorithm with polarization and transverse spatial modes

A N de Oliveira, Stephen Patrick Walborn, Carlos Henrique Monken

Published Aug 23, 2005• 101 citations
Publisher

Abstract

In this paper we implement a simple quantum algorithm using polarization and transverse spatial modes of the electromagnetic field as qubits. The transverse spatial modes used are the Laguerre–Gaussian and Hermite–Gaussian beams of order N = 1. With these two qubits carried by the same photon, we have implemented the Deutsch algorithm using a simple linear optical setup.

DOI
10.1088/1464-4266/7/9/009

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In this paper we implement a simple quantum algorithm using polarization and transverse spatial modes of the electromagnetic field as qubits. The transverse spatial modes used are the Laguerre–Gaussian and Hermite–Gaussian beams of order N = 1. With these two qubits carried by the same photon, we have implemented the Deutsch algorithm using a simple linear optical setup.
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.
We have theoretically and experimentally studied how a Mach-Zehnder interferometer with an additional mirror transforms a light beam composed of the second lowest transverse modes, HG10,{\mathrm{HG}}_{10},HG10​, HG01,{\mathrm{HG}}_{01},HG01​, LG01,{\mathrm{LG}}_{01},LG01​, and ${\mathrm{LG}}_{0\ensuremath{-}1}$ (HG denotes Hermite-Gaussian mode; LG denotes Laguerre-Gaussian mode). In certain conditions, the interferometer divides the incident beam into the HG10{\mathrm{HG}}_{10}HG10​ and HG01{\mathrm{HG}}_{01}HG01​ components as a transverse-mode beam splitter. We propose a practical device involving the two interferometers for quantum cryptography, in which a photon carries two bits corresponding to the polarization and the transverse mode.
We report an experimental implementation of the minimal Deutsch algorithm in an optical setting. In this version, a redundancy is removed from the most famous form of the algorithm. The original version involves manipulation of two qubits, while in its minimal version, only one qubit is used. Our qubit is encoded in the transversal spatial modes of a spontaneous parametric down-converted signal photon, with the aid of a double slit, with the idler photon playing a crucial role in creating a heralded single-photon source. A spatial light modulator (SLM) is programmed to physically generate one-bit functions necessary to implement the algorithm's minimal version, which shows that the SLM can be used in future implementations of quantum protocols.
We show that deterministic quantum computing with one qubit (DQC1) can be experimentally implemented with a spatial light modulator, using the polarization and the transverse spatial degrees of freedom of light. The scheme allows the computation of the trace of a high-dimension matrix, being limited by the resolution of the modulator panel and the technical imperfections. In order to illustrate the method, we compute the normalized trace of unitary matrices and implement the Deutsch-Jozsa algorithm. The largest matrix that can be manipulated with our setup is $1080\ifmmode\times\else\texttimes\fi{}1920$, which is able to represent a system with approximately 21 qubits.
The first section of the work investigates light modes as a means of implementing optical qubits and qudits. The modes considered are polarization mode, path mode, transverse spatial mode, frequency mode, temporal bin-mode and temporal mode. Subsequently, mathematical model of linear optical elements like beam splitters (BS) and phase shifters are deduced and their capability of representing any single qubit optical gate is exposed. Finally Knill, Laflamme and Milburn (KLM) method of using linear optical elements to promote nonlinear operations based on nonlinear (nondeterministic) sign-flip gate (NS) is explained, and designs of two qubits conditional sign flip gate (c-z1/16) and CNOT gate, both based on KLM method, are demonstrated. So, universal set of quantum gates based on linear optics is possible.