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arXiv:0710.3950 · Physical Review A

Geometric phase gate on an optical transition for ion trap quantum computation

Kihwan Kim, Christian F. Roos, Leandro Aolita, Hartmut Häffner, Volckmar Nebendahl, R. Blatt

Published May 8, 2008• 37 citations
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Abstract

We propose a geometric phase gate of two ion qubits that are encoded in two levels linked by an optical dipole-forbidden transition. Compared to hyperfine geometric phase gates mediated by electric dipole transitions, the gate has many interesting properties, such as very low spontaneous emission rates, applicability to magnetic field insensitive states, and use of a co-propagating laser beam geometry. We estimate that current technology allows for infidelities of around ${10}^{\ensuremath{-}4}$.

DOI
10.1103/physreva.77.050303

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Abstract Universal logic gates for two quantum bits (qubits) form an essential ingredient of quantum computation. Dynamical gates have been proposed1, 2 in the context of trapped ions; however, geometric phase gates (which change only the phase of the physical qubits) offer potential practical advantages because they have higher intrinsic resistance to certain small errors and might enable faster gate implementation. Here we demonstrate a universal geometric π-phase gate between two beryllium ion-qubits, based on coherent displacements induced by an optical dipole force. The displacements depend on the internal atomic states; the motional state of the ions is unimportant provided that they remain in the regime in which the force can be considered constant over the extent of each ion's wave packet. By combining the gate with single-qubit rotations, we have prepared ions in an entangled Bell state with 97% fidelity—about six times better than in a previous experiment3 demonstrating a universal gate between two ion-qubits. The particular properties of the gate make it attractive for a multiplexed trap architecture4, 5 that would enable scaling to large numbers of ion-qubits.

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Physical Review Research•Jul 19, 2023•13 citations
We propose a geometric phase gate of two ion qubits that are encoded in two levels linked by an optical dipole-forbidden transition. Compared to hyperfine geometric phase gates mediated by electric dipole transitions, the gate has many interesting properties, such as very low spontaneous emission rates, applicability to magnetic field insensitive states, and use of a co-propagating laser beam geometry. We estimate that current technology allows for infidelities of around 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4.
We propose the implementation of a geometric-phase gate on magnetic-field-insensitive qubits with ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{z}$-dependent forces for trapped ion quantum computing. The force is exerted by two laser beams in a Raman configuration. Qubit-state dependency is achieved by a small frequency detuning from the virtually excited state. Ion species with excited states of long radiative lifetimes are used to reduce the chance of a spontaneous photon emission to less than ${10}^{\ensuremath{-}8}\phantom{\rule{0.3em}{0ex}}\mathrm{per}$ gate run. This eliminates the main source of gate infidelity in previous implementations. With this scheme it seems possible to reach the fault-tolerant threshold.
To date, the highest-fidelity quantum logic gates between two qubits have been achieved using the geometric-phase gate in trapped ions, with the two leading variants being the M\o{}lmer-S\o{}rensen gate and the light-shift (LS) gate. Both of these approaches have their respective advantages and challenges. For example, the latter is technically simpler and is natively insensitive to optical phases, but it has not been made to work directly on a clock-state qubit. We present a technique for implementing the LS gate that combines the best features of these two approaches: By using a small ($\ensuremath{\sim}\mathrm{MHz}$) detuning from a narrow (dipole-forbidden) optical transition, we are able to operate an LS gate directly on hyperfine clock states, achieving gate fidelities of 99.74(4)99.74(4)%99.74(4) using modest laser power at visible wavelengths. Current gate infidelities appear to be dominated by technical noise, and theoretical modeling suggests a path toward gate fidelity above 99.9999.99%99.99.
Abstract Universal logic gates for two quantum bits (qubits) form an essential ingredient of quantum computation. Dynamical gates have been proposed1, 2 in the context of trapped ions; however, geometric phase gates (which change only the phase of the physical qubits) offer potential practical advantages because they have higher intrinsic resistance to certain small errors and might enable faster gate implementation. Here we demonstrate a universal geometric π-phase gate between two beryllium ion-qubits, based on coherent displacements induced by an optical dipole force. The displacements depend on the internal atomic states; the motional state of the ions is unimportant provided that they remain in the regime in which the force can be considered constant over the extent of each ion's wave packet. By combining the gate with single-qubit rotations, we have prepared ions in an entangled Bell state with 97% fidelity—about six times better than in a previous experiment3 demonstrating a universal gate between two ion-qubits. The particular properties of the gate make it attractive for a multiplexed trap architecture4, 5 that would enable scaling to large numbers of ion-qubits.
We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers. Strong polarization gradients near the tweezer focus lead to qubit-state-dependent forces on the ion. We show that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal. The qubit-state-dependent forces generated by this effect are located on the plane perpendicular to the direction of propagation of the laser beams opening alternate ways of coupling to motional modes of an ion crystal. The proposed gate does not require ground-state cooling of the ions although the waist of the tightly focused beam needs to be comparable to its wavelength in order to achieve the needed field curvature. Furthermore, the gate can be implemented on both ground-state and magnetic-field-insensitive clock-state qubits without the need for counterpropagating laser fields. This simplifies the setup and eliminates errors due to phase instabilities between the gate laser beams. Finally, we show that imperfections in the gate execution, in particular, a 30-nm tweezer alignment error, lead to an infidelity of $\ensuremath{\sim}{10}^{\ensuremath{-}3}$. In the absence of experimental imperfections and within the limits of the gate model explored in this paper the fidelity is predicted to be $\ensuremath{\sim}0.999\phantom{\rule{0.16em}{0ex}}88$ when using a Laguerre-Gaussian beam to suppress photon scattering errors.