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arXiv:1209.1749 · Physical Review A

Double-slit implementation of the minimal Deutsch algorithm

Breno Marques, Mariana R. Barros, Wanderson M. Pimenta, M. A. D. Carvalho, José Ferraz, Raphael C. Drumond, Marcelo Terra Cunha, Sebastião Pádua

Published Sep 7, 2012• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.86.032306

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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.
In this work we implement the Deutsch-Jozsa algorithm with one qubit by removing a redundancy on it, which reduces the size of the register and simpli es the function evaluation. In order to do this, we used a pair of photons produced by down convertion, where one of them is used as triguer and the other passes trougth a double slit; as result its output is in a superposition of paths, i.e., one qubit. Using a spatial light modulator (SLM) that puts phases on the paths of the photons, and detecting in coincidence, we demonstrate the Deutsch-Jozsa algorithm with photons, and using only one qubit.
In this thesis we demonstrate experimentally the implementation of quantum processes and simulation of complex dynamics, through the manipulation of a quantum system. Pairs of entangled photons generated by spontaneous parametric down conversion (SPDC) were used as the quantum system. The entangled state in the transversal path has been manipulated using programmable spatial light modulators (SLM). A nonlinear crystal with phase matching type I and a continuouslaser beam with wavelength in the violet region were used for the generation of twin photons in entangled path states that allow us to vary the Hilbert space dimension. Three quantum protocols were experimentally studied in this thesis: minimum Deutsch algorithm, entanglement concentration and simulation of dissipative dynamics. In the Deutsch algorithm implementation were used twin photons generated by SPDC and a double slit to encode a spatial qubit were used. The second photon was used as a trigger to prepare a conditional state of the first. The SLM was used as the oracle mentioned in the algorithm, i. e., it is responsible for implementing the balancedand constant functions. The entanglement concentration protocol for two-qutrit states was also implemented. Partially entangled path states were generated using SPDC along with a SLM that modules amplitude only. A second SLM phase modulator inside a Sagnac interferometer was used to implement the protocol using the method known as Procustean. The third protocol demonstrated experimentally was the simulation of two dissipative dynamics in qudits: dephasing and amplitude. One of the photons passes through a noisy channel, implemented using a SLM, and the transmitted state is characterized allowing us to identify the noisy channel properties.
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.
This work has as a main goal the implementation of the experimental minimal quantum tomography in spatial variables using a spatial light modulator. To produce the two photons states entangled in spatial variables we use the properties of the photons produced by Spontaneous Parametric Down Conversion (SPDC) and double slits. A theoricaldescription was necessary for the understanding of the minimal tomography method. Four states which describe a tetrahedron in the Bloch sphere were chosen and associated to these states, were built four operators, making it a POVM set for tomography. For the experimental implementation of these operators, it took two calibrations of the spatial light modulator. Values of grayscales and polarization angles of input and output polarizers were determined in order to apply dierent phases and/or specific amplitude diferences between the beams that pass through each slit path. Thus, through the studyof minimal tomography and the knowledge of the operation of the modulator and their calibration, it was possible to experimentally perform the minimal quantum tomography for one and two qubits in spatial variables.
We study and experimentally implement a double-slit quantum eraser in the presence of a controlled decoherence mechanism. A two-photon state, produced in a spontaneous parametric down-conversion process, is prepared in a maximally entangled polarization state. A birefringent double slit is illuminated by one of the down-converted photons, and it acts as a single-photon two-qubits controlled-not gate that couples the polarization with the transversal momentum of these photons. The other photon, which acts as a which-path marker, is sent through a Mach-Zehnder-like interferometer. When the interferometer is partially unbalanced, it behaves as a controlled source of decoherence for polarization states of down-converted photons. We show the transition from wavelike to particle-like behavior of the signal photons crossing the double slit as a function of the decoherence parameter, which depends on the length path difference at the interferometer.