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arXiv:2002.03730 · Physical Review A

Properties of phonon modes of an ion-trap quantum computer in the Aubry phase

Justin Loye, José Lages, Dima L. Shepelyansky

Published Mar 30, 2020• 5 citations
arXivHTMLPDF

Abstract

We study analytically and numerically the properties of phonon modes in an ion quantum computer. The ion chain is placed in a harmonic trap with an additional periodic potential of which dimensionless amplitude $K$ determines three main phases available for quantum computations: at zero $K$ we have the case of Cirac-Zoller quantum computer, below a certain critical amplitude $K<{K}_{c}$ the ions are in the Kolmogorov-Arnold-Moser (KAM) phase, with delocalized phonon modes and free chain sliding, and above the critical amplitude $K>{K}_{c}$ ions are in the pinned Aubry phase with a finite frequency gap protecting quantum gates from temperature and other external fluctuations. For the Aubry phase, in contrast to the Cirac-Zoller and KAM phases, the phonon gap remains independent of the number of ions placed in the trap keeping a fixed ion density around the trap center. We show that in the Aubry phase the phonon modes are much better localized compared to the Cirac-Zoller and KAM cases. Thus in the Aubry phase the recoil pulses lead to local oscillations of ions, while in the other two phases they spread rapidly over the whole ion chains, making them rather sensible to external fluctuations. We argue that the properties of localized phonon modes and phonon gap in the Aubry phase provide advantages for the ion quantum computations in this phase with a large number of ions.

DOI
10.1103/physreva.101.032349

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We study analytically and numerically the properties of phonon modes in an ion quantum computer. The ion chain is placed in a harmonic trap with an additional periodic potential of which dimensionless amplitude KKK determines three main phases available for quantum computations: at zero KKK we have the case of Cirac-Zoller quantum computer, below a certain critical amplitude K<KcK<{K}_{c}K<Kc​ the ions are in the Kolmogorov-Arnold-Moser (KAM) phase, with delocalized phonon modes and free chain sliding, and above the critical amplitude K>KcK>{K}_{c}K>Kc​ ions are in the pinned Aubry phase with a finite frequency gap protecting quantum gates from temperature and other external fluctuations. For the Aubry phase, in contrast to the Cirac-Zoller and KAM phases, the phonon gap remains independent of the number of ions placed in the trap keeping a fixed ion density around the trap center. We show that in the Aubry phase the phonon modes are much better localized compared to the Cirac-Zoller and KAM cases. Thus in the Aubry phase the recoil pulses lead to local oscillations of ions, while in the other two phases they spread rapidly over the whole ion chains, making them rather sensible to external fluctuations. We argue that the properties of localized phonon modes and phonon gap in the Aubry phase provide advantages for the ion quantum computations in this phase with a large number of ions.
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 NNN 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.
The intrinsic decoherence from vibrational coupling of the ions in the Cirac-Zoller quantum computer [Phys. Rev. Lett. 74, 4091 (1995)] is considered. Starting from a state in which the vibrational modes are at a temperature TTT, and each ion is in a superposition of an excited and a ground state, an adiabatic approximation is used to find the inclusive probability P(t)P(t)P(t) for the ions to evolve as they would without the vibrations, and for the vibrational modes to evolve into any final state. An analytic form is found for P(t)P(t)P(t) at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0, and the decoherence time is found for all TTT. The decoherence is found to be quite small, even for 1000 ions.
A string of cold ions confined in a linear trap represents a man-made quantum object with a broad range of applications in atomic and molecular spectroscopy, such as high-precision measurement of atomic properties. An efficient isolation from the environment guarantees excellent coherent properties of such systems and makes them suitable for practical realization of the quantum information processing. The pioneering theoretical [Cirac and Zoller] and experimental [Wineland and Monroe] work in 1990s resulted in the explosive expansion of this field during the last decade. In this dissertation an alternative new method for controlling the quantized motional/vibrational states of ions in a trap is explored theoretically. It is proposed to create small anharmonicity in the trapping potential which would modify the spectrum of states and allow addressing the state-to-state transitions selectively. In this approach all ions remain in the ground electronic state and their motion is controlled adiabatically and coherently by applying the optimally shaped electric fields (RF). The optimal control theory, accurate numerical calculation of the energies and wavefunctions, and numerical propagation of wave packets are employed. Two sources of vibrational anharmonicity are studied: the intrinsic Coulomb anharmonicity due to the ion-ion interactions and the external anharmonicity of the trapping potential. It is shown that the magnitude of Coulomb anharmonicity is insufficient for the control. In contrast, anharmonicity of the trapping potential allows controlling the motion of ions very accurately. It is demonstrated that one ion in a slightly anharmonic trap can be easily controlled and used to represent one qubit. A multi-qubit system can be created by employing a long progression of states of a single ion, or by trapping multiple ions and controlling several normal vibration modes of the ion string. Up to four qubits are modeled in this work and accurate pulses are optimized for a set of universal quantum gates: NOT, conditional NOT (CNOT) and Hadamard transformation. The control field for Shor's algorithm (quantum algorithm for factorization onto prime numbers) is also obtained. It is demonstrated that a careful choice of system properties allows achieving very high accuracy of qubit transformations, up to 0.999.
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.