Phase-Adaptive Dynamical Decoupling Methods for Robust Spin-Spin Dynamics in Trapped Ions

Lijuan Dong, Iñigo Arrazola, Xi Chen, and Jorge Casanova

Phys. Rev. Applied 15, 034055 – Published 18 March 2021

Abstract

Quantum platforms based on trapped ions are the main candidates to build a quantum hardware with computational capacities that largely surpass those of classical devices. Among the available control techniques in these setups, pulsed dynamical decoupling (pulsed DD) has been revealed as a useful method to process the information encoded in ion registers, whilst minimizing the environmental noise over them. In this work, we incorporate a pulsed DD technique that uses random pulse phases, or correlated pulse phases, to significantly enhance the robustness of entangling spin-spin dynamics in trapped ions. This procedure was originally conceived in the context of nuclear magnetic resonance for nuclear spin detection purposes, and here we demonstrate that the same principles apply for robust quantum-information processing in trapped-ion settings.

Received 14 September 2020
Revised 17 December 2020
Accepted 26 February 2021

DOI:https://doi.org/10.1103/PhysRevApplied.15.034055

Physics Subject Headings (PhySH)

Quantum control Quantum gates Quantum information with trapped ions Spin dynamics

Optically detected magnetic resonance

Quantum Information, Science & Technology

Authors & Affiliations

Lijuan Dong^1,2, Iñigo Arrazola ^2,*, Xi Chen ^1,2, and Jorge Casanova ^2,3

¹International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, Shanghai 200444, China
²Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, Bilbao 48080, Spain
³IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, Bilbao 48080, Spain

^*inigo.arrazola@ehu.eus

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Vol. 15, Iss. 3 — March 2021

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Figure 1
(a) XY8 pulse block with duration $8 τ$ . The Rabi frequency takes the value $Ω$ while applying the $π$ pulse, i.e., in the red region. The relative phase between the pulses changes according to $ϕ_{x} = 0 + Φ$ and $ϕ_{y} = π / 2 + Φ$ . (b) Modulation function $f_{z} (t)$ corresponding to a single XY8 block. (c) The basic block shown in (a) is repeated to construct a longer sequence. In this case, the same global phase is chosen for all blocks. (d) The global phases $Φ_{s}$ can be chosen to be different for each block $s$ .
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Figure 2
(a) Fidelity with respect to the state generated by $H = \sum_{i, j} J_{i j} σ_{i}^{z} σ_{j}^{z}$ at $t_{G} = π / 8 max (| J_{i j} |)$ as a function of the strength of the unwanted shift $ϵ$ and for $N = 2$ (solid curves) and $N = 6$ (dashed curves). The pulsed cases (red curves) are protected up to $ϵ \approx (2 π) \times 1$ kHz, while the fidelity for the pulse-free cases (blue curves) takes values below $99 %$ around $ϵ \approx (2 π) \times 5$ Hz; an improvement factor approximately equal to $200$ is obtained in the pulsed case. Due to the effect of realistic finite-width pulses, the pulsed case for $N = 6$ gets a fidelity of $F \approx 99.8 %$ fidelity for $ϵ \to 0$ . (b) $J_{i j}$ matrix elements for $N = 6, η = 0.018$ and $ν = (2 π) \times 220$ kHz. (c) Accumulated two-qubit phase versus time for $η = 0.0113$ , $ν = (2 π) \times 200$ kHz, and $Ω = (2 π) \times 40$ kHz. The pulse-free case (blue) reaches $θ = π / 4$ around $t_{G} \approx 7.1$ ms, while for the pulsed case (red) it takes $t_{FW} \approx 7.5$ ms. Inset: phase accumulation around $t \approx 4.05 - 4.1$ ms. It can seen that phase accumulation stops during the application of a $π$ pulse leading to a plateau effect, which lasts approximately $12.5 μ s$ . (d) Bell-state infidelity for $η = 0.0113$ , $ν = (2 π) \times 200$ kHz and for the XY8 pulse sequence with 16, 32, 48, 64, and 80 pulses. The cases with Rabi frequencies $Ω = (2 π) \times$ (20, 40, and 60) kHz are shown with blue, red, and green colors, respectively. Due to the effect of transversal terms in Eq. (11), the fidelity decreases when increasing the number of pulses or decreasing $Ω$ .
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Figure 3
XY8 pulse sequence with 80 pulses (i.e., ten XY8 blocks) and different choices for the value of the phases $Φ_{s}$ . (a) Standard choice with $Φ_{s} \equiv Φ$ . For each experimental run, $Φ_{s}$ is chosen to be the same. (b) Full random phases. (c),(d),(e) Correlated phases where $\sum_{s = 1}^{S} \exp (- i Φ_{s}) = 0$ with $S = 2, 5$ and $10$ , respectively. In all cases, the first global phase $Φ_{1}$ is chosen randomly for each experimental run. (f)–(j) Results of the described pulse sequences in terms of the Bell-state infidelity for different values of the static control error $δ Ω$ and qubit frequency shift $ϵ$ . The numbers in white represent how many more points with values above $99.9 %$ fidelity are with respect to (f). For example, $100 %$ would mean there are twice as many points that fulfil this condition compared with (f).
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Figure 4
Similar to Fig. 2, the fidelity with respect to the state generated by $H = \sum_{i, j} J_{i j} σ_{i}^{z} σ_{j}^{z}$ at $t_{G} = π / 8 max (| J_{i j} |)$ as a function of the strength of the unwanted shift $ϵ$ and for $N = 6$ is shown. Here, for the pulsed case (dashed red) we consider a Rabi frequency of $Ω = (2 π) \times 160$ kHz, which results in a better fidelity than the case with $Ω = (2 π) \times 60$ kHz in Fig. 2 at the limit $ϵ \to 0$ .
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