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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Physical representation of universal quantum gates for a trapped ion

G. Drobný, Pavol Gustafik

Published Mar 7, 2001• 0 citations
Publisher

Abstract

We discuss possibilities of quantum manipulations of trapped ions from the perspective of quantum information processing. We show how from a restricted set of feasible microscopic interactions a physical implementation of an arbitrary unitary operator can be realized. An example is given for the discrete Fourier transform. The synthesis of operators enables to realize universal quantum gates to process unknown input qubits which can be encoded into vibrational levels of trapped ions. We discuss also problem of an arbitrary measurement which can be performed using quantum gates.

DOI
10.1117/12.417814

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We discuss possibilities of quantum manipulations of trapped ions from the perspective of quantum information processing. We show how from a restricted set of feasible microscopic interactions a physical implementation of an arbitrary unitary operator can be realized. An example is given for the discrete Fourier transform. The synthesis of operators enables to realize universal quantum gates to process unknown input qubits which can be encoded into vibrational levels of trapped ions. We discuss also problem of an arbitrary measurement which can be performed using quantum gates.
Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for realization of any unitary operator which acts on a (truncated) Hilbert space of a single bosonic mode. The algorithm itself is not unitary because it involves a conditional measurement. However, it does yield a constant probability of the conditional measurement which does not depend on the input state of the bosonic system. We consider a physical implementation of unitary transformations acting on one-dimensional vibrational states of a trapped ion. As an example we present an algorithm which realizes the discrete Fourier transform.
Quantum computers hold the promise to solve certain problems exponentially faster than their classical counterparts. Trapped atomic ions are among the physical systems in which building such a computing device seems viable. In this work we present a small-scale quantum information processor based on a string of 40 Ca + ions confined in a macroscopic linear Paul trap. We review our set of operations which includes non-coherent operations allowing us to realize arbitrary Markovian processes. In order to build a larger quantum information processor it is mandatory to reduce the error rate of the available operations which is only possible if the physics of the noise processes is well understood. We identify the dominant noise sources in our system and discuss their effects on different algorithms. Finally we demonstrate how our entire set of operations can be used to facilitate the implementation of algorithms by examples of the quantum Fourier transform and the quantum order finding algorithm.
In this tutorial we review the physical implementation of quantum computing using a system of cold trapped ions. We discuss systematically all the aspects for making the implementation possible. Firstly, we go through the loading and confining of atomic ions in the linear Paul trap, then we describe the collective vibrational motion of trapped ions. Further, we discuss interactions of the ions with a laser beam. We treat the interactions in the travelling-wave and standing-wave configuration for dipole and quadrupole transitions. We review different types of laser cooling techniques associated with trapped ions. We address Doppler cooling, sideband cooling in and beyond the Lamb-Dicke limit, sympathetic cooling and laser cooling using electromagnetically induced transparency. After that we discuss the problem of state detection using the electron shelving method. Then quantum gates are described. We introduce single-qubit rotations, two-qubit controlled-NOT and multi-qubit controlled-NOT gates. We also comment on more advanced multiple-qubit logic gates. We describe how quantum logic networks may be used for the synthesis of arbitrary pure quantum states. Finally, we discuss the speed of quantum gates and we also give some numerical estimations for them. A discussion of dynamics on off-resonance transitions associated with a qualitative estimation of the weak-coupling regime is included in Appendix A and of the Lamb-Dicke regime in Appendix B.
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Recent experiments [K. R. Brown et al., Nature (London) 471, 196 (2011); M. Harlander et al., Nature (London) 471, 200 (2011)] have demonstrated the coherent manipulations on the external vibrations of two ions, confined individually in the separated traps. Using these recently developed techniques, we propose here an approach to realize the coherent operations, e.g., the universal quantum gates, between the separated ion-trap qubits encoded by two internal atomic states of the ions. Our proposal operates beyond the usual Lamb-Dicke limits and could be applied to the scalable ion traps coupled by their vibrations.