Angle-robust Two-Qubit Gates in a Linear Ion Crystal Zhubing Jia1 2Shilin Huang1 3Mingyu Kang1 2Ke Sun1 2Robert F. Spivey1 3Jungsang Kim1 2 3 4and Kenneth R. Brown1 2 3 5y

2025-04-27 0 0 935.47KB 10 页 10玖币

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Angle-robust Two-Qubit Gates in a Linear Ion Crystal

Zhubing Jia,1, 2, ∗Shilin Huang,1, 3 Mingyu Kang,1, 2 Ke Sun,1, 2 Robert

F. Spivey,1, 3 Jungsang Kim,1, 2, 3, 4 and Kenneth R. Brown1, 2, 3, 5, †

1Duke Quantum Center, Duke University, Durham, NC 27701, USA

2Department of Physics, Duke University, Durham, North Carolina 27708, USA

3Department of Electrical and Computer Engineering,

Duke University, Durham, North Carolina 27708, USA

4IonQ, Inc., College Park, Maryland 20740, USA

5Department of Chemistry, Duke University, Durham, North Carolina 27708, USA

In trapped-ion quantum computers, two-qubit entangling gates are generated by applying spin-

dependent force which uses phonons to mediate interaction between the internal states of the ions.

To maintain high-ﬁdelity two-qubit gates under ﬂuctuating experimental parameters, robust pulse-

design methods are applied to remove the residual spin-motion entanglement in the presence of

motional mode frequency drifts. Here we propose an improved pulse-design method that also guar-

antees the robustness of the two-qubit rotation angle against uniform mode frequency drifts by

combining pulses with opposite sensitivity of the angle to mode frequency drifts. We experimentally

measure the performance of the designed gates and see an improvement on both gate ﬁdelity and

gate performance under uniform mode frequency oﬀsets.

I. INTRODUCTION

Trapped atomic-ion system is a leading platform for

quantum computation and simulation [1,2]. It has sev-

eral appealing features including long coherence time [3,

4], fast and eﬃcient state preparation and measure-

ment [5–7], high-ﬁdelity single-qubit [7–9] and two-qubit

gates [10–15]. As the most challenging component, the

two-qubit gates are commonly implemented by Mølmer-

Sørensen (MS) type interactions [16,17], which utilizes

the motional modes to mediate couplings between spins.

For exactly two-ion systems, two-qubit gate ﬁdelity ex-

ceeding 99.9% has been demonstrated [10,11,13–15]. For

longer ion chains with more complicated motional-mode

spectrum, several approaches including amplitude modu-

lation (AM) [18–23], frequency modulation (FM) [12,24–

26], phase modulation (PM) [27–29] and multi-tone driv-

ing ﬁelds [30–32] have been utilized to eﬃciently decouple

spins from all motional modes, achieving 98.5% ∼99.3%

ﬁdelity with 15 ions [33] and 97.5% ﬁdelity with 13

ions [34], 16 ions [35] and 25 ions [23].

As the circuit depth increases, the performance of MS

gate is sensitive to slow drifts of motional mode fre-

quencies, resulting in undesired residual spin-motion en-

tanglement and two-qubit overrotation errors. Previous

works [22,24,29,36] have focused on minimizing both

the residual spin-motion entanglement and its ﬁrst-order

response to unknown mode frequency drifts, which are

usually referred to as robust gates. Meanwhile, few works

have addressed the sensitivity of the two-qubit overrota-

tion errors. On the circuit level, the overrotation error

can be suppressed by local optimization [37–40]. On the

gate level, one can use single-qubit composite pulses to

∗zhubing.jia@duke.edu

†kenneth.r.brown@duke.edu

correct two-qubit overrotation errors [41,42], but due to

the long two-qubit gate time it is not experimentally ap-

plicable [39]. Refs. [25,43] applied machine learning and

numerical optimization techniques to reduce the overro-

tation errors due to mode frequency drifts, without be-

ing able to completely remove the ﬁrst-order sensitivity.

Ref. [22] proposed a scheme that eliminates such sensitiv-

ity to arbitrary order by applying diﬀerent pulses on the

two ions, but the scheme was not veriﬁed experimentally.

In this paper, we propose a method called A(ngle)-

robust that achieves similar goal by concatenating two

diﬀerent robust MS pulses. Inheriting the robustness of

negligible residual spin-motion entanglement, the two-

qubit rotation angle of an A-robust gate is ﬁrst-order in-

sensitive against uniform frequency drifts on all modes.

Our scheme is experimentally veriﬁed on a two-ion plat-

form and can in principle be realized in longer ion chains.

The paper is organized as follows. In Sec. II, we review

the general theory of robust MS gate and discuss how its

ﬁdelity can be aﬀected. In Sec. III, we introduce the an-

alytic construction of A-robust pulses in two-ion chains

and long ion chains and show the simulated behavior of

A-robust gates comparing with that of robust gates. In

Sec. IV we experimentally verify that the A-robust gate

outperforms the robust gate against mode frequency oﬀ-

sets in a two-ion chain. Finally, in Sec. Vwe summarize

our results and discuss further directions.

II. BACKGROUND

A. The Mølmer-Sørensen Gate

The Mølmer-Sørensen (MS) gate entangles the spin

states of two ions in an ion chain by applying a state-

dependent force. On each target ion j1and j2, we apply

driving ﬁelds with the same Rabi frequency Ω(t) and op-

posite detunings ±δ(t) with respect to carrier transition,

arXiv:2210.04814v1 [quant-ph] 10 Oct 2022

where δis close to resonance with the motional frequency.

Under Lamb-Dicke and rotating wave approximations,

the eﬀective Hamiltonian can be written as [16,17]

HMS(t) =Ω(t)

j=j1,j2X

ηkbk

jˆake−iθk(t)+ ˆa†

keiθk(t)σj

(1)

where ηkis the Lamb-Dicke parameter of mode k,bk

jis

the normalized coupling strength of ion jto mode k, and

θk(t) = ωkt−Zt

δ(t0)dt0(2)

is the integrated phase of the detuning between the fre-

quency ωkof the k-th mode and the driving ﬁeld at time

t. By applying Magnus expansion, at time τ, the MS

Hamiltonian generates a unitary evolution of the follow-

ing form [18,20]

UMS(τ) = exp(X

j=j1,j2X

khαk

j(τ)ˆa†

k−αk∗

j(τ)ˆakσj

+iΘ(τ)σj1

xσj2

x)(3)

where

αk

j(τ) = ηkbk

2Zτ

Ω(t)eiθk(t)dt (4)

is the displacement of mode kwhen spin jis in (+1)-

eigenstate of σj

x, and

Θ(τ) = 1

η2

kbk

j1bk

j2Zτ

dt Zt

dt0

×Ω(t)Ω(t0) sin (θk(t)−θk(t0)) (5)

is the angle of the two-qubit rotation with respect to the

axis σj1

xσj2

At the end of the gate, the ion spins are completely

disentangled with the motional modes, i.e., αk

j= 0 for

all ions jand modes k. For a maximally-entangled MS

gate, the rotation angle Θ should be equal to π/4. In

this work we refer to maximally-entangled MS gates as

XX(π/4) gates. In a multi-ion chain, these conditions

can be achieved by modulating either the Rabi frequency

Ω(t) or the detuning δ(t) of the state-dependent driving

forces in Eq. 1. The gate error E=Eα+EΘcan be deﬁned

as follows [25]:

Eα=X

kαk

j1

2+αk

j2

2,

EΘ=Θ−π

42.

(6)

B. Robust Pulses against Mode Frequency Drifts

In practice, the mode frequencies ωkmight drift to

ωk+kfor some small kdue to miscalibration and low-

frequency noise (comparing to gate speed). As a result,

both the residual displacement and the rotation angle

experience a ﬁrst-order response

αk

j→αk

j+∂αk

∂ωk

k,(7)

Θ→Θ + X

∂Θ

∂ωk

k,(8)

where

∂αk

∂ωk

=iηkbk

2Zτ

Ω(t)eiθk(t)tdt (9)

and

∂Θ

∂ωk

=η2

kbk

j1bk

2Zτ

dt Zt

dt0Ω(t)Ω(t0)

×(t−t0) cos(θk(t)−θk(t0)).(10)

To make the MS gate behavior insensitive to mode fre-

quency drifts, it is desirable to have pulses with vanishing

ﬁrst-order responses. Minimizing the residual entangle-

ment and its ﬁrst-order response to mode frequency drifts

are referred to as the robustness condition:

•Robustness: For all jand k, we require that

αk

j= 0,∂αk

∂ωk

= 0.(11)

Previous works have been focusing on achieving the

robustness condition by applying diﬀerent modulation

methods, including AM [22], FM [12,24], PM [27,29] and

multi-tone driving ﬁelds [31,44]. In particular, Ref. [24]

shows that by using a time-symmetric pulse and minimiz-

ing the absolute value of the time-averaged displacement

αk

j=1

τZτ

αk

j(t)dt, (12)

the optimized pulse satisﬁes the robustness condition.

Meanwhile, Ref. [22] points out that using AM to elimi-

nate the ﬁrst-order sensitivity of rotation angle to drifts

on each mode, i.e., ∂Θ/∂ωk= 0 for all k, can only be

achieved by applying diﬀerent pulses on each ion. If the

pulses on each ion are identical, it is impossible to achieve

∂Θ/∂ωk= 0 for all kby using AM [22]. Studies on mini-

mizing ∂Θ/∂ωkfor all kusing other modulation schemes

remain undeveloped.

III. ANGLE-ROBUST GATE BY

CONCATENATING PULSES

Although minimizing ∂Θ/∂ωkfor each ωkis hard to

realize, we notice that in experiments, small drifts on mo-

tional modes caused by rf or dc voltage ﬂuctuations has

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Angle-robustTwo-QubitGatesinaLinearIonCrystalZhubingJia,1,2,ShilinHuang,1,3MingyuKang,1,2KeSun,1,2RobertF.Spivey,1,3JungsangKim,1,2,3,4andKennethR.Brown1,2,3,5,y1DukeQuantumCenter,DukeUniversity,Durham,NC27701,USA2DepartmentofPhysics,DukeUniversity,Durham,NorthCarolina27708,USA3DepartmentofElectric...

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Angle-robust Two-Qubit Gates in a Linear Ion Crystal Zhubing Jia1 2Shilin Huang1 3Mingyu Kang1 2Ke Sun1 2Robert F. Spivey1 3Jungsang Kim1 2 3 4and Kenneth R. Brown1 2 3 5y

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