Generation of high-fidelity quantum control methods for multilevel systems

J. Randall, A. M. Lawrence, S. C. Webster, S. Weidt, N. V. Vitanov, and W. K. Hensinger

Phys. Rev. A 98, 043414 – Published 8 October 2018

Abstract

In recent decades there has been a rapid development of methods to experimentally control individual quantum systems. A broad range of quantum control methods has been developed for two-level systems; however, the complexity of multilevel quantum systems make the development of analogous control methods extremely challenging. Here we exploit the equivalence between multilevel systems with SU(2) symmetry and spin-1/2 systems to develop a technique for generating new robust, high-fidelity, multilevel control methods. As a demonstration of this technique, we develop adiabatic and composite multilevel quantum control methods and experimentally realize these methods using a ${}^{171}{Yb}^{+}$ ion system. We measure the average infidelity of the process in both cases to be around $10^{- 4}$ , demonstrating that this technique can be used to develop high-fidelity multilevel quantum control methods and can, for example, be applied to a wide range of quantum computing protocols, including implementations below the fault-tolerant threshold in trapped ions.

Received 7 February 2018

DOI:https://doi.org/10.1103/PhysRevA.98.043414

Physics Subject Headings (PhySH)

Quantum computation Quantum control Quantum information processing

Trapped ions

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

J. Randall^1,2, A. M. Lawrence^1,2, S. C. Webster¹, S. Weidt¹, N. V. Vitanov³, and W. K. Hensinger^1,*

¹Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom
²QOLS, Blackett Laboratory, Imperial College London, London, SW7 2BW, United Kingdom
³Department of Physics, St. Kliment Ohridski University of Sofia, 5 James Bourchier Blvd, 1164 Sofia, Bulgaria

^*w.k.hensinger@sussex.ac.uk

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Vol. 98, Iss. 4 — October 2018

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Figure 1
Use of an effective two-level system to generate three-level control methods. (a) We wish to implement a control method which transforms an initial state ${|ψ_{1 / 2}〉}_{i}$ on the effective two-level Bloch sphere (which here we take to be $|↓〉$ ) into a final state ${|ψ_{1 / 2}〉}_{f} = e^{- i θ \hat{u} \cdot S} {|ψ_{1 / 2}〉}_{i}$ , where $\hat{u}$ is the axis of rotation and $θ$ is the angle (equivalent to ${|ψ_{j}〉}_{f} = e^{- i θ \hat{u} \cdot J} {|ψ_{j}〉}_{i}$ in the real multilevel system). (b) In the effective two-level system, most control methods are implemented by applying a single control field of Rabi frequency $Ω_{1/2} (t)$ , instantaneous detuning $δ_{1/2} (t)$ , and phase $χ (t)$ . (c) By inverting the Majorana decomposition, we derive the control fields that we must apply to our real physical system, namely, two fields of equal Rabi frequency $Ω (t)$ and equal and opposite detunings and phases $\pm δ (t)$ and $\pm χ (t)$ , respectively.
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Figure 2
Adiabatic transfer to the dark state of a dressed three-level system. (a) Energy eigenvalues and (b) eigenstates ${|ξ_{1}〉, |ξ_{2}〉}$ of $H_{1 / 2}$ as a function of $δ / Ω$ for $χ = 0$ . (b) Analytically calculated amplitudes of these eigenstates, all of which can be defined as real numbers in this case. An avoided crossing is present at $δ / Ω = 0$ , at which point the eigenstates are the dressed states $(|↓〉 \pm |↑〉) / \sqrt{2}$ , which are separated in energy by $ℏ Ω / \sqrt{2}$ . Therefore, by adiabatically varying the detuning and Rabi frequency, the population can be coherently transferred from ${|↓〉, |↑〉}$ to $(|↓〉 \pm |↑〉) / \sqrt{2}$ . (c–e) Demonstrating the method using a single ${}^{171}{Yb}^{+}$ ion. In the three-level system, the adiabatic procedure will transfer population from $|0〉$ to the dark state $|D〉 = (|+ 1〉 - |- 1〉) / \sqrt{2}$ . (c, d) The temporal profiles for the Rabi frequency $Ω$ (solid green line in c) and the instantaneous detuning $δ$ (solid red line in d), where the relevant parameters are given in the text. (e) Measured probability for the ion to be in the ${}^{171}{Yb}^{+}$ $F = 1$ state given by $P (F = 1) = 1 - P_{0}$ as a function of time. Each point is the average of 300 repetitions. The theoretical probability for the ion to be in $F = 1$ as a function of time (solid red line) is obtained from a numerical simulation of the system with no free parameters, which can be seen to agree well with the measured data.
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Figure 3
Robust population transfer to the dark state using the ${TBB}_{1}$ composite pulse sequence. (a) The ${TBB}_{1}$ composite pulse sequence represented on the effective two-level Bloch sphere. The sequence consists of four resonant pulses with varying pulse area and phase which can be written as a sequence of rotations on the Bloch sphere of the form $R (θ_{R}, ϕ_{R})$ . Each of these rotations is represented as a colored line on the Bloch sphere, in the order red, orange, green, blue (numbered to show ordering). Above is the trajectory in the case of zero Rabi frequency error and below for the $Δ Ω = - 2 π \times 10 kHz$ case. We implement both of these cases experimentally to demostrate the robustness of the method to pulse area errors. (b) The phase $χ$ as a function of time implementing the ${TBB}_{1}$ pulse sequence for a fixed Rabi frequency $Ω_{0} / 2 π = 40 kHz$ . An extra phase change of $- π / 2$ at the end ensures the population remains in $|D〉$ after the procedure. Therefore the total pulse sequence is $R (*, 0) \cdot R (π / 2, π / 2) \cdot R (π, 3.267) \cdot R (2 π, 0.376) \cdot R (π, 3.267)$ . (c) The measured population in $F = 1$ as a function of time with $Δ Ω = 0$ (a, upper sphere; c, light blue line or light gray when printed in grayscale) and for a Rabi frequency error of $Δ Ω = - 2 π \times 10 kHz$ (a, lower sphere; c, black line), showing that the sequence is robust to such errors. (d) Measured population in $F = 1$ as a function of pulse area for a square $π / 2$ pulse (black) and the ${TBB}_{1}$ pulse sequence (light blue; light gray when printed in grayscale), demonstrating that the ${TBB}_{1}$ sequence maintains the robustness to pulse area error of the original two-level ${BB}_{1}$ sequence. The pulse area is normalized such that the nominal pulse area for a $π / 2$ rotation is 1. The solid lines in panels (c) and (d) correspond to numerical simulations of the sequence with no free parameters.
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Figure 4
Measuring the fidelity. (a) The fidelity with which we produce the $|D〉$ state can be obtained by applying two fields resonant with the $|0〉 \leftrightarrow |+ 1〉$ and $|0〉 \leftrightarrow |- 1〉$ transitions with equal Rabi frequency $Ω$ , and varying the phase $χ$ of the two fields in equal and opposite directions. (b) The measured population in $|0〉$ as a function of $χ$ after a single adiabatic operation (black points), which can be fitted to the function $A_{0} + A cos (2 χ + ϕ_{0})$ (solid red curve) to extract the mapping fidelity using a maximum likelihood fitting method (Appendix pp1). Each point is the average of 200 repetitions. (c) The fidelity as a function of the number of applications of the adiabatic method (black) and resonant ${TBB}_{1}$ sequence (light blue; light gray when printed in grayscale). For the adiabatic method, the population transfer back to $|0〉$ begins immediately after it reaches $|D〉$ ( $t_{h} = 0$ ). A linear least-squares fit to the data gives an average infidelity per operation of $1.4 (4) \times 10^{- 4}$ for the adiabatic method and $1.1 (4) \times 10^{- 4}$ for the composite pulse sequence.
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