Frequency-robust Mølmer-Sørensen gates via balanced contributions of multiple motional modes Brandon P. Ruzic1Matthew N. H. Chow1 2 3yAshlyn D. Burch1Daniel Lobser1

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Frequency-robust Mølmer-Sørensen gates via balanced contributions of multiple

motional modes

Brandon P. Ruzic,1, ∗Matthew N. H. Chow,1, 2, 3, †Ashlyn D. Burch,1Daniel Lobser,1

Melissa C. Revelle,1Joshua M. Wilson,1Christopher G. Yale,1and Susan M. Clark1

1Sandia National Laboratories, Albuquerque, New Mexico 87185, USA

2Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87106, USA

3Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131, USA

(Dated: October 6, 2022)

In this work, we design and implement frequency-robust Mølmer-Sørensen gates on a linear chain

of trapped ions, using Gaussian amplitude modulation and a constant laser frequency. We select

this frequency to balance the entanglement accumulation of all motional modes during the gate

to produce a strong robustness to frequency error, even for long ion chains. We demonstrate this

technique on a three-ion chain, achieving < 1% reduction from peak ﬁdelity over a 20 kHz range of

frequency oﬀset, and we analyze the performance of this gate design through numerical simulations

on chains of two to 33 ions.

I. INTRODUCTION

Linear chains of trapped ions are one of the leading

platforms for quantum computation in the near term.

The application of Mølmer-Sørensen (MS) gates [1] on

these systems has achieved some of the highest two-qubit

entanglement ﬁdelities to date, reaching above 99.9%

while targeting the axial motional modes of a two-ion

chain [2, 3]. To implement powerful quantum algorithms,

like digital quantum simulation [4] and quantum error

correction [5–7], one must extend these high-ﬁdelity gates

to systems of many physical qubits by, for example, in-

creasing the length of the chain and individually address-

ing each ion [8, 9]. In this approach, the MS gates provide

all-to-all connectivity between ion pairs, but the gate ﬁ-

delity can suﬀer due to the residual spin-motion entangle-

ment after the gate in the increased number of spectator

motional modes [1].

There have been many successful demonstrations of

high-ﬁdelity MS gates by modulating the amplitude [8–

14], frequency [15, 16], amplitude and frequency [17, 18],

or phase [19–21] of the laser beams. These approaches

have achieved 97% to 99.5% ﬁdelity when targeting the

radial modes of a two-ion chain, for which the tighter con-

ﬁnement than in the axial direction allows better cooling,

less heating, and faster gates. The modulation techniques

improve gate performance by eliminating the residual

spin-motion entanglement for ideal experimental condi-

tions and by adding robustness to this quantity in the

presence of motional frequency error. For example, sim-

ulations of frequency-modulated gates maintain a 99%

ﬁdelity with a motional frequency error of ±1.5 kHz for

a two-ion chain [15], and optimizing over a distribution

of gate parameters improves this level of robustness to at

least ±5kHz [22].

∗bruzic@sandia.gov

†mnchow@sandia.gov

Nevertheless, motional frequency error remains an im-

portant error source in MS gates and their applications.

Modulated MS gates attempt to minimize the sensitiv-

ity of the residual spin-motion entanglement to frequency

error, and as a result, the amount of spin entanglement

accumulated during the gate also gains robustness to this

error. However, signiﬁcant errors in the amount of ac-

cumulated spin entanglement can remain and create a

purely coherent rotation error in spin space, which is es-

pecially damaging to the performance of quantum algo-

rithms that involve many gates [23]. This sensitivity to

rotation error was recently demonstrated by the repeated

application of MS gates with a frequency oﬀset on two-

ion and four-ion chains [16].

For longer chains, the sensitivity to frequency error

increases due to the higher density of motional modes.

Further, the majority of frequency-robust gate designs

become more diﬃcult to implement due to more strin-

gent experimental requirements, including the need to

account for all modes by linearly increasing the num-

ber of optimized pulse-shape parameters with the num-

ber of ions [15]. Robust gate designs exist that reduce

this requirement by only targeting closely spaced ions or

a reduced set of motional modes [18], but the experi-

mental requirements to implement these techniques can

still grow with longer chains. Modulated gates on longer

ion chains can require larger laser powers [16] and gen-

erally have a higher sensitivity to drift in the calibrated

model parameters (e.g. motional frequencies, ion separa-

tion, laser power, and gate duration) that are used during

the optimization of pulse-shape parameters [22].

In this paper, we develop and implement an MS gate

with an analytic pulse shape that does not require opti-

mizing a large set of pulse-shape parameters yet is still

broadly robust to motional frequency error, even for long

ion chains. We perform amplitude modulation during

our gate with a simple, Gaussian time dependence that

strongly suppresses residual displacement errors in all

modes, as long as the detuning from each mode remains

suﬃciently large. While many studies have demonstrated

arXiv:2210.02372v1 [quant-ph] 5 Oct 2022

error suppression using amplitude modulation, including

modulation that resembles a Gaussian [14, 17, 18], we

also select a speciﬁc, constant detuning that balances

the amount of entanglement accumulation during the

gate from all motional modes and provides robustness

to this source of coherent gate error. With the ability

to adjust the detuning without signiﬁcantly impacting

displacement errors, we are free to tune the laser fre-

quency to a point where the derivative in the entan-

glement accumulation with respect to frequency goes to

zero. This produces a gate that is ﬁrst-order insensitive

to frequency error, resulting in regions of broad robust-

ness to this error. Our protocol is simple to realize exper-

imentally, as we can optimize performance by calibrating

only two pulse-shape parameters: the constant detuning

and the peak Rabi rate. As a result, our gate design

has a low classical computational overhead, facilitating

its adoption on other trapped ion quantum processors

and making it suitable for systems suﬀering from mod-

erate amounts of drift. We demonstrate the frequency

robustness of our gate on a three-ion chain and analyze

this robustness in numerical simulations for chains of up

to 33 ions.

This work is done on the Quantum Scientiﬁc Com-

puting Open User Testbed, QSCOUT. We use qubits

encoded in the hyperﬁne clock states of 171Yb+ions

trapped in a linear chain on a surface trap. Gates

are site-selectively driven with an optical Raman tran-

sition. Details of the apparatus are described in previous

work [24].

II. GATE DESIGN

A. MS Gate Model

We model the application of an MS gate on two ions

that are part of a linear chain of ions in a surface trap

using the Hamiltonian,

H(t) = −Ω(t)X

Sy,kakeiδkt+h.c., (1)

which is in a rotating frame with respect to the atomic

and trap degrees of freedom. The collective spin oper-

ator Sy,k has the form: Sy,k = (η1,kσy,1+η2,kσy,2)/2,

where σy,j is the yPauli spin operator for the j-th ion

targeted by the gate. The Lamb-Dicke parameter ηj,k

can diﬀer for each ion and each motional mode, and Ω(t)

is the Rabi rate of the carrier transition for both ions.

In this work, Ω(t)is a time dependent parameter of the

drive ﬁeld, while δkis eﬀectively held constant in time for

each mode. The operators a†

kand akare the raising and

lowering operators, respectively, for a harmonic oscillator

that represents the motional mode of the ion chain with

angular frequency νk. During the gate, a dual-tone laser

illuminates the ions with detunings ±δk=±(δc−νk)

from their blue and red motional sideband transitions,

respectively, where the parameter δcis the detuning of

the blue-detuned laser tone from the carrier transition.

For simplicity, we have made the Lamb-Dicke approx-

imation: eiη(ak+a†

k)≈1 + iη(ak+a†

k). We have also

neglected the carrier transition and the far-oﬀ-resonant

sideband transitions with detunings larger than |δc|.

Since the Hamiltonian H(t)acts on each motional

mode independently, we can write the propagator U(t)

as a product over motional modes:

U(t) = ΠkUk(t),(2)

and the exact analytic solution for Uk(t)is [1, 25],

Uk(t) = e−iBk(t)S2

y,k D(Sy,kαk(t)),

Bk(t) = i

2Zt

0dαk(t0)

dt0α∗

k(t0)−α(t0)dα∗

k(t0)

dt0dt0.(3)

The displacement operator D(Sy,k αk(t)) =

exp hSy,k(αk(t)a†

k−α∗

k(t)ak)iis conditioned on the

spin state of the targeted ions, and αk(t)describes the

phase-space trajectory of the ion chain. The phase

η1,kη2,kBk(t), which governs the amount of spin entan-

glement accrued during the gate, is real and positive

(negative) for clockwise (counter-clockwise) trajectories.

To gain an intuitive picture of the gate dynamics, we

express the phase-space trajectory of each motional mode

in terms of the parameters of H(t),

αk(t) = iZt

Ω(t0)e−iδkt0dt0,(4)

where t=τcorresponds to the end of the gate. From this

equation, we see that αk(τ)is proportional to the Fourier

transform of Ω(t)evaluated at δk, assuming that Ω(t)is

zero before (t < 0) and after (t>τ) the gate. This is

a key insight that will aid our choice of pulse shape for

frequency-robust gates, as discussed in section II C.

In this study, we focus on the robustness of MS-gate

performance to a frequency error δω that is applied to

both laser tones and moves them symmetrically with re-

spect to the carrier transition, resulting in new carrier

detunings: ±δ0

c=±(δc+δω)and sideband detunings:

±δ0

k=±(δk+δω), for the blue-detuned and red-detuned

tones, respectively. Equivalently, this frequency error can

be interpreted as a common change in the motional fre-

quency of each mode: ν0

k=νk−δω. Although other

error sources can aﬀect gate performance, such as laser

power ﬂuctuations and anomalous heating [26, 27], we

choose to focus on frequency error due to the high sensi-

tivity of gate performance to this error [25], especially in

the context of long ion chains with many closely spaced

motional modes.

B. Performance Metrics

We use the state ﬁdelity Fas the ﬁgure of merit for

gate performance, which can be computed by wavefunc-

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Frequency-robustMølmer-SørensengatesviabalancedcontributionsofmultiplemotionalmodesBrandonP.Ruzic,1,MatthewN.H.Chow,1,2,3,yAshlynD.Burch,1DanielLobser,1MelissaC.Revelle,1JoshuaM.Wilson,1ChristopherG.Yale,1andSusanM.Clark11SandiaNationalLaboratories,Albuquerque,NewMexico87185,USA2DepartmentofPhysics...

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Frequency-robust Mølmer-Sørensen gates via balanced contributions of multiple motional modes Brandon P. Ruzic1Matthew N. H. Chow1 2 3yAshlyn D. Burch1Daniel Lobser1

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