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arXiv:0710.1204 · New Journal of Physics

Ion trap quantum gates with amplitude-modulated laser beams

Christian F Roos

Published Jan 14, 2008• 152 citations
arXivHTMLPDF

Abstract

In ion traps, entangling gate operations can be realized by a bichromatic pair of laser beams that collectively interact with the ions. In this paper, a new method of modelling the laser–ion interaction is introduced that turns out to be superior to standard techniques for the description of gate operations on optical qubits. The treatment allows for a comparison of the performance of gates based on σ z ⊗σ z and σ ϕ ⊗σ ϕ interactions on optical transitions where the bichromatic laser field can be realized by an amplitude-modulated laser resonant with the qubit transition. Shaping the amplitude of the bichromatic laser pulse is shown to make the gates more robust against experimental imperfections.

DOI
10.1088/1367-2630/10/1/013002

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•Jan 1, 2014•1 citation
In ion traps, entangling gate operations can be realized by a bichromatic pair of laser beams that collectively interact with the ions. In this paper, a new method of modelling the laser–ion interaction is introduced that turns out to be superior to standard techniques for the description of gate operations on optical qubits. The treatment allows for a comparison of the performance of gates based on σ z ⊗σ z and σ ϕ ⊗σ ϕ interactions on optical transitions where the bichromatic laser field can be realized by an amplitude-modulated laser resonant with the qubit transition. Shaping the amplitude of the bichromatic laser pulse is shown to make the gates more robust against experimental imperfections.
Two-qubit gates are a fundamental constituent of a quantum computer and typically its most challenging operation. In a trapped-ion quantum computer, this is typically implemented with laser beams which are modulated in amplitude, frequency, phase, or a combination of these. The required modulation becomes increasingly more complex as the quantum computer becomes larger, complicating the control hardware design. Here we develop a simple method to essentially remove the pulse-modulation complexity by engineering the normal modes of the ion chain. We experimentally test the feasibility of the method in a three-ion chain. This opens up the possibility to trade off complexity between the design of the trapping fields and the optical control system, which will help scale the ion-trap quantum computing platform.
Highly-localized light-matter interactions are necessary for scaling trapped-ion architectures. In hyperfine qubits, counterpropagating beams generate entangling gates by coupling with motion, but this effect is undesirable during single-qubit operations. For that reason, single-qubit gates are traditionally implemented with copropagating beams, and the coexistence of two beam geometries adds hardware and computational overhead. In an effort towards collective performance improvement with minimal overhead, we design and implement pulse-amplitude and dephasing robust dynamically corrected gates using Space Curve Quantum Control (SCQC) and compare them against the constant-amplitude gate implementation. We perform gate set tomography on a four-qubit trapped-ion register, and we discover more than 50% error reduction when robust pulses are used. We find that counterpropagating robust gates often outperform their copropagating counterparts and reach error rates as low as (3.59±1.25)⋅10−3(3.59 \pm 1.25)\cdot 10^{-3}(3.59±1.25)⋅10−3, using diamond distance as a metric. This value establishes a laser-driven-gate error reference and is merely an order of magnitude higher than the best reported microwave\textit{microwave}microwave gate on a single\textit{single}single ion. Additional experiments reveal that robust pulses can effectively suppress non-Markovian errors that grow during runtime. Our work challenges the widely accepted belief that copropagating gates should be preferred for their weak motional coupling and invites the adoption of high-performance robust pulses that suppress multiple noise sources of the trapped-ion error budget.
Almost every quantum circuit is built with two-qubit gates in the current stage, which are crucial to the quantum computing in any platform. The entangling gates based on Mølmer-Sørensen schemes are widely exploited in the trapped-ion system, with the utilization of the collective motional modes of ions and two laser-controlled internal states, which are served as qubits. The key to realize high-fidelity and robust gates is the minimization of the entanglement between the qubits and the motional modes under various sources of errors after the gate operation. In this work, we propose an efficient numerical method to search high-quality solutions for phase-modulated pulses. Instead of directly optimizing a cost function, which contains the fidelity and the robustness of the gates, we convert the problem to the combination of linear algebra and the solution to quadratic equations. Once a solution with the gate fidelity of 1 is found, the laser power can be further reduced while searching on the manifold where the fidelity remains 1. Our method largely overcomes the problem of the convergence and is shown to be effective up to 60 ions, which suffices the need of the gate design in current trapped-ion experiments.
A crucial building block for quantum information processing with trapped ions is a controlled-NOT quantum gate. In this Letter, two different sequences of laser pulses implementing such a gate operation are analyzed using quantum process tomography. Fidelities of up to 92.6(6)% are achieved for single-gate operations and up to 83.4(8)% for two concatenated gate operations. By process tomography we assess the performance of the gates for different experimental realizations and demonstrate the advantage of amplitude-shaped laser pulses over simple square pulses. We also investigate whether the performance of concatenated gates can be inferred from the analysis of the single gates.
We introduce two types of two-qubit controlled-phase gates, which operate with trapped ions in a linear Paul trap. Each ion is addressed individually with two sequential bichromatic laser pulses, the two frequency components of which are tuned on exact resonance with the first blue and red vibrational sidebands of the qubit transition frequency. For suitably chosen pulse areas and relative laser phases the dependence on the phonon number vanishes and the net effect is a controlled-phase gate between the electronic qubit states. In contrast to earlier gates, in which the pulses addressing the two ions are separated in time, we use partly overlapping pulses, which make the present gates faster, e.g., by about 26% for rectangular and over 50% for sinusoidal pulses.
Trapped atomic ion systems are currently the most advanced platform for quantum information processing. Their long coherence times, pristine state initialization and detection, and precisely controllable and versatile interactions make them excellent quantum systems for experiments in quantum computation and quantum simulation. One of the more promising schemes for quantum computing consists of performing single and multi-qubit quantum gates on qubits in a linear ion crystal. Some of the key challenges of scaling such a system are the individual addressing of arbitrary subsets of ions and controlling the growing complexity of motional mode interactions as the number of qubits increases or when the gates are performed faster. Traditional entangling quantum gates between ion qubits use laser pulses to couple the qubit states to the collective motion of the crystal, thereby generating a spin-spin interaction that can produce entanglement between selected qubits. The intrinsic limitations on the performance of gates using this method can be alleviated by applying optimally shaped pulses instead of pulses with constant amplitude. This thesis explains the theory behind this pulse shaping scheme and how it is implemented on a chain of Yb ions held in a linear radiofrequency `Paul' trap. Several experiments demonstrate the technique in chains of two, three, and five ions using various types of pulse shapes. A tightly focused individual addressing beam allows us to apply the entangling gates to a target pair of ions, and technical issues related to such tight focusing are discussed. Other advantages to the pulse shaping scheme include a robustness against detuning errors and the possibility of suppressing undesirable coupling due to optical spillover on neighboring ions. Combined with ion shuttling, we harness these features to perform sequential gates to different qubit pairs in order to create genuine tripartite entangled states and demonstrate the programmable quantum information processing capability of our system.