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Quantum Dynamics of Cold Trapped Ions With Application to Quantum Computation

Daniel F. V. James

Published Feb 26, 1997• 329 citations

Abstract

. The theory of interactions between lasers and cold trapped ions as it pertains to the design of Cirac--Zoller quantum computers is discussed. The mean positions of the trapped ions, the eigenvalues and eigenmodes of the ions' oscillations, the magnitude of the Rabi frequencies for both allowed and forbidden internal transitions of the ions, and the validity criterion for the required Hamiltonian are calculated. Energy level data for a variety of ion species are also presented. PACS: 32.80.Qk; 42.50.Vk; 89.80.+h A quantum computer is a device in which data can be stored in a network of quantum mechanical two-level systems, such as spin-1/2 particles or two-level atoms. The quantum mechanical nature of such systems allows the possibility of a powerful new feature to be incorporated into data processing, namely, the capability of performing logical operations upon quantum mechanical superpositions of numbers. Thus in a conventional digital computer each data register is, throughout an...

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. The theory of interactions between lasers and cold trapped ions as it pertains to the design of Cirac--Zoller quantum computers is discussed. The mean positions of the trapped ions, the eigenvalues and eigenmodes of the ions' oscillations, the magnitude of the Rabi frequencies for both allowed and forbidden internal transitions of the ions, and the validity criterion for the required Hamiltonian are calculated. Energy level data for a variety of ion species are also presented. PACS: 32.80.Qk; 42.50.Vk; 89.80.+h A quantum computer is a device in which data can be stored in a network of quantum mechanical two-level systems, such as spin-1/2 particles or two-level atoms. The quantum mechanical nature of such systems allows the possibility of a powerful new feature to be incorporated into data processing, namely, the capability of performing logical operations upon quantum mechanical superpositions of numbers. Thus in a conventional digital computer each data register is, throughout an...

Using simple physical arguments we investigate the capabilities of a quantum computer based on cold trapped ions of the type recently proposed by Cirac and Zoller. From the limitations imposed on such a device by decoherence due to spontaneous decay, laser phase coherence times, ion heating and other possible sources of error, we derive bounds on the number of laser interactions and on the number of ions that may be used. As a quantitative measure of the possible performance of these devices, the largest number which may be factored using Shor`s quantum factoring algorithm is determined for a variety of species of ion.

Using simple physical arguments we investigate the capabilities of a quantum computer based on cold trapped ions of the type recently proposed by Cirac and Zoller. From the limitations imposed on such a device by decoherence due to spontaneous decay, laser phase coherence times, ion heating and other possible sources of error, we derive bounds on the number of laser interactions and on the number of ions that may be used. As a quantitative measure of the possible performance of these devices, the largest number which may be factored using Shor's quantum factoring algorithm is determined for a variety of species of ion.

The purpose of this article is to review two complementary schemes for quantum computation with trapped ions. We initially discuss the first proposal of quantum computations with cold trapped ions (J. I. Cirac and P. Zoller; Phys. Rev. Lett. 74, 4091, 1995) which requires, prior to any computation, laser cooling to the motional ground state. This proposal is closely related to the physics of generating and manipulating N-particles entangled states in both ion traps and high-Q cavities (cavity quantum electrodynamics). The second scheme is that of quantum computations with hot trapped ions (J.F. Poyatos, J.I. Cirac and P. Zoller; Phys. Rev. Lett. 81, 1322, 1998) which works at finite temperature and resembles physical ideas found in atom interferometry.

In this paper we describe one of the most promising platforms for the construction of a universal quantum computer, which consists of a chain of

N

ions trapped in a harmonic potential, whose internal states work out as qubits, and are coupled to collective vibrational modes of the chain. From such coupling, it is possible to build interactions between different ions of the chain, that is, qubit-qubit interactions that, together with individual operations on the ions, allow building a quantum computer as first proposed by Cirac and Zoller in the 1990s [Phys. Rev. Lett. 74, 4091 (1995)]. Here we discuss from the physics involved in trapping ions in electromagnetic potentials to the Hamiltonian engineering needed to generate a universal set of logic gates, fundamental for the execution of more complex quantum algorithms. Finally, we present the current state of the art of quantum computing in trapped ion systems, highlighting recent advances made by companies and government projects that use such architecture, such as IonQ and AQTION.

An experimental realization of quantum computers is composed of two or more calcium ions trapped in a magnetic quadripole. Information is transferred to and read from the ions by means of structured lasers that interact with the ions’ vibration pattern, causing changes of energy distribution in their electronic structure. Departing from an initial state when the ions are cooled, the use of lasers modifies the internal state of one ion that is entangled with the others, then changing the collective states. In such quantum computers, some of the physically possible electronic states are avoided or not taken into consideration, to force the system to work as a binary device. In this essay, we discuss the dynamics that the ions could spontaneously display and its possible implications for the principles of computational logics.