Pis’ma v ZhETF, vol. 116, iss. 8, pp. 570 - 571

Numerical simulation of the performance of single qubit gates

for trapped ions

L.A.Akopyan⁺¹⁾, O.Lakhmanskaya⁺, S.Yu.Zarutskiy⁺, N.D.Korolev⁺, O.Guseva⁺, K.Lakhmanskiy^+∗

⁺Russian Quantum Center, Skolkovo, 143025 Moscow, Russia

^∗Higher School of Systems Engineering MIPT, 141701 Dolgoprudny, Russia

Submitted 2 September 2022

Resubmitted 2 September 2022

Accepted 15

September 2022

DOI: 10.31857/S1234567822200113, EDN: kpoidh

Quantum computing with trapped ions has shown

ion p. Full Hamiltonian allows to account for the two ef-

significant progress over the last couple of decades [1-

fects responsible for gate errors: entanglement between

5]. The main advantages are the highest-fidelity quan-

the qubit states and the phonon modes and phonon

tum computing gates, long coherence times, inherent

mode heating which leads to the finite occupation of

uni-formality and all-to-all connectivity [6-11]. Nowa-

the phonon modes.

days the attention has shifted from miniature architec-

The results are obtained for optical Ca qubits and

tures towards more practical implementations requir-

for the two types of gates, G₁ and G₁₅ for which the

ing to scale up the computer performance [12-19]. For

π/2 pulse is performed during 1 s and 15 s, respectively

large ion crystals system performance is known to de-

[7, 2]. Figure 1 shows the fidelity of single-qubit R_φ(θ)

grade [1]. Therefore, it is crucial to understand scaling of

gate for 3 ions for gates G₁ and G₁₅ with different lev-

the finite errors in quantum gates with the system size

els of corrections to the Rabi frequency obtained in the

due to noise, decoherence, and control imperfections. In

Lamb-Dicke approximations (for details see [21]):

particular, we numerically studied the performance of





global single-qubit gates depending on the Lamb-Dicke

∑∑

Ω_p = Ω1 -

η^2pjk(n_kj + 1/2).

(2)

parameter, gate time, the number of ions, and the initial

j k=1

phonon mode occupation numbers.

The dynamics of single-qubit gates in trapped-ion

The corrected Rabi frequencies Ω account only for

systems is typically described using Lamb-Dicke ap-

the finite occupation of the phonon modes. According to

proximation meaning the exclusion of the phonon modes

Fig. 1, the minimum infidelity of 10^-4 is achieved for the

[2, 7, 20]. In this work we performed numerical simula-

gate G₁₅, when the corrections for all the modes are in-

tion of the action of the single qubit gate R_φ(θ) using

cluded. Additional simulations indeed have shown that

full Hamiltonian of the system (1):

the phonon mode occupation leads to a more rapid de-

)

crease in the fidelity. For the fast G₁ gate the situation

∑

∑∑ (

ω_q

Ĥ_N =

ℏ

σ^pz +

ℏω_jk

â^†jkâ_jk +

is worse: the oscillations of fidelity can not be removed

p=1

j k=1

by correcting Rabi frequency. These oscillations come

∑ [

]

from the entanglement between the phonon modes and

ℏ

Ω e-i(ωt-kRp+φ)σ^+p + h.c. .

(1)

the qubit states. Comparison of the two different types

p=1

of fidelities (obtained with and without tracing of the

Here indexes p and j refer to the ion index in the chain

phonon modes) and calculations of the entanglement en-

and to the choise of the Cartesian axis (x, y, z) respec-

tropy also show the existence of entanglement between

tively, ω_q is the qubit transition frequency, ω is the laser

phonons and ions. The amplitude of the fidelity oscilla-

frequency, ω_jk is the frequency of the normal mode k

tions scales with Ω and increases with the phonon mode

along axis j, â^†jk/â_jk are creation/annihilation opera-

occupation number. The largest contribution to the infi-

tors of the normal mode k along axis j, k is the wave

delity and to the amplitude of its oscillations comes from

vector of the laser, φ is the phase of the laser, and angle

the population of the center of mass (COM) mode. We

θ = τΩ of the gate is controlled via the gate time τ,

also studied the dependence of the fidelity on the Lamb-

Dicke parameter and on the number of ions (from 1 to

R_p is the quantised coordinate of the center of mass of

4) for the initial phonon mode state |100〉 correspond-

¹⁾e-mail: l.akopyan@rqc.ru

ing to a single phonon in a COM mode. We observe less

570

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Numerical simulation of the performance. . .

571

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This is an excerpt of the article “Numerical simula-

son, R. McConnell, D. Reens, G. N. West, J. M. Sage,

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and J. Chiaverini, Nature 586, 538 (2020).

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