Fast entangling gates in long ion chains

Open Access

Fast entangling gates in long ion chains

Zain Mehdi, Alexander K. Ratcliffe, and Joseph J. Hope

Phys. Rev. Research 3, 013026 – Published 11 January 2021

Abstract

We present a model for implementing fast entangling gates ( $\sim 1 μ s$ ) with ultrafast pulses in arbitrarily long ion chains, that requires low numbers of pulses and can be implemented with laser repetition rates well within experimental capability. We demonstrate that we are able to optimize pulse sequences that have theoretical fidelities above $99.99 %$ in arbitrarily long ion chains, for laser repetition rates on the order of $100 - 300$ MHz. Notably, we find higher repetition rates are not required for gates in longer ion chains, which is in contrast to scaling analyses with other gate schemes. When pulse imperfections are considered in our calculations, we find that achievable gate fidelity is independent of the number of ions in the chain. We also show that pulse control requirements do not scale up with the number of ions. We find that population transfer efficiencies of above $99.9 %$ from individual ultrafast pulses is the threshold for realizing high-fidelity gates, which may be achievable in near-future experiments.

Received 13 April 2020
Accepted 18 December 2020

DOI:https://doi.org/10.1103/PhysRevResearch.3.013026

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum computation Quantum gates Quantum information with trapped ions

Quantum Information, Science & Technology

Authors & Affiliations

Zain Mehdi ^*, Alexander K. Ratcliffe , and Joseph J. Hope

Department of Quantum Science, Research School of Physics, Australian National University, Canberra, Australia

^*zain.mehdi@anu.edu.au

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Vol. 3, Iss. 1 — January - March 2021

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Figure 1
Simulated phase-space trajectories of the axial modes of a $N = 5$ ion chain during gate with 16 SDKs, for the two-qubit states $| ↑ ↑ 〉$ and $| ↑ ↓ 〉$ . In the gate scheme, individual SDKs are grouped together so that some kicks can be made larger than others. The accumulated entangling phase is proportional to differences in areas enclosed by the $| ↑ ↑ 〉$ and $| ↑ ↓ 〉$ trajectories, summed for each motional mode. Here, we represent each mode in position-velocity phase space, rotating at its normal frequency: (a) 1.9 MHz, (b) 3.3 MHz, (c) 4.6 MHz, (d) 5.6 MHz, and (e) 7.0 MHz. It is apparent that some of the trajectories do not close perfectly, which contributes to a nonzero infidelity of $\sim 5 \times 10^{- 5}$ .
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Figure 2
Infidelity of gates optimized for different numbers of ions in a trap, between two ions (a) on the edge of the ion chain and (b) in the middle of the chain. Each gate is optimized for a laser repetition rate of $300 MHz$ , and has a gate time between 0.75 and $1.2 μ s$ after optimization. The effect of imperfect SDKs due to pulse area errors is included in the optimization, with $ε$ giving the characteristic population transfer error from a single $π$ pulse. For error rates of $ε = 10^{- 5}$ or lower, fidelities above $99.99 %$ are achievable between neighboring ions in arbitrarily long ion chains.
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Figure 3
Infidelities of gates between two ions on the edge of a $N = 20$ ion chain, as a function of (a) repetition rate, and (b) pulse error. (a) Infidelities are shown assuming perfect pulses ( $ε = 0$ ) for gates optimized for different repetition rates. For repetition rates above about 300 MHz, fidelity does not significantly improve. (b) Infidelities of gates optimized for a 300-MHz repetition rate laser, with pulse imperfections (characterized by a typical population transfer error $ε$ ) included in the calculation. We find that achievable infidelity in long ion chains is approximately $1 - F \sim 10^{2} ε$ . In the limit of perfect pulses ( $ε \to 0$ ), the infidelity approaches the ideal value $1 - F_{0} \sim 10^{- 6}$ .
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Figure 4
Simplified diagram of the electronic structure in ${}^{40}{Ca}^{+}$ . The ultrafast pulses are resonant with the $S_{1 / 2} \to P_{3 / 2}$ (red, solid), and the shelved qubits are encoded in the metastable $D_{3 / 2}$ and $D_{5 / 2}$ levels (shaded). Dipole transitions from the $D$ to $P$ levels are indicated (blue, dashed). Transition wavelengths are indicated, taken from Ref. [27]. We note $D \leftrightarrow S$ dipole transitions are forbidden, and hence not indicated on this diagram.
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Figure 5
Motional infidelity (contribution of motional restoration errors to gate infidelity) as a function of the ions' temperature, for exemplary gates in ion chains of different length. The vertical line indicates the Doppler temperature $k_{B} T_{D} / ℏ \approx 7 \times 10^{7}$ Hz.
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