Robust Trapped-Ion Quantum Logic Gates by Microwave Dynamical Decoupling

A. Bermudez,1 P. O. Schmidt,2 M. B. Plenio,1 and A. Retzker1

1Institut fur Theoretische Physik, Albert-Einstein Allee 11, Universitat Ulm, 89069 Ulm, Germany2QUEST Institute for Experimental Quantum Metrology,

Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany

We introduce a hybrid scheme that combines laser-driven phonon-mediated quantum logic gates in trappedions with the benefits of microwave dynamical decoupling. We demonstrate theoretically that a strong drivingof the qubit decouples it from the external magnetic noise, and thus enhances the fidelity of two-qubit quantumgates. Moreover, the scheme does not require ground-state cooling, is inherently robust to undesired ac-Starkshifts, and simplifies previous gate schemes thus decreasing the effort in their realization.

Introduction.– Cold atomic ions trapped by electrody-namic fields are among the most promising technologies forquantum-information processing [1] (and references therein).The exquisite control and isolation of their electronic statesmakes them ideal qubits where to store, manipulate, and re-trieve the information over long periods of time. This is par-ticularly true for the so-called clock states, which evade thedominant source of decoherence in trapped-ion experiments:magnetic noise [2]. Additionally, a quantum processor mustbe able to perform logic operations between distant qubits.Collective vibrations of a ground-state cooled ion crystal pro-vide such a mechanism: they serve as a quantum bus that me-diates an effective coupling between the qubits [3, 4]. Be-sides, this idea can be generalized to schemes for warmerions [5–7]. Recently, there has been a growing effort to re-alize microwave-based quantum-information processing withtrapped ions, owing to the excellent control over the phase andamplitude of microwaves compared to laser fields. Originally,by the use of magnetic gradients [8] and more recently vianear-field techniques [10]. Some of the above schemes canbe applied to the aforementioned clock states, yielding an ef-ficient toolbox for quantum-information processing [10, 11].

However, not all types of trapped ions have clock statesavailable, and not all of the quantum-gate schemes work forthis kind of states. For instance, this occurs for the geometri-cal phase gate [6, 7], and also for some of the schemes basedon microwaves [8, 9]. Besides, ions with a vanishing nuclearangular momentum (e.g. 40Ca+,138Ba+) do not posses clockstates either. In these cases, magnetic-field fluctuations limitthe coherence times to the millisecond range, and thus imposea fundamental limit to the number of quantum-logic opera-tions that can be performed within the coherence time [12].There are two different approaches to this problem: one mayi) optimize the gate speed [13], or alternatively ii) look forschemes that are intrinsically robust to the magnetic-fieldnoise, such as encoding in decoherence-free subspaces [14].We follow the second route in this work by using a continu-ous version of dynamical decoupling.

While dynamical decoupling is well developed [15], theoptimal combination of the decoupling scheme with gate op-erations [16] might require a considerable additional effort.Henceforth, simpler decoupling schemes are a subject of re-cent interest [9, 17]. We hereby present a scheme well suitedto trapped-ion experiments with three important properties: i)generality, ii) simplicity, and iii) robustness. This scheme is

general enough to be applied to any hyperfine or Zeeman ionqubit. It is simple since it combines two standard tools oftrapped-ion technology, namely a carrier and a red-sidebandexcitation. In particular, it relies on the strong driving ofthe carrier transition, which may be realized by a microwavesource. We show that this driving is responsible for the ro-bustness of the gate at different levels. On top of decouplingthe qubits from the magnetic-field noise, it also minimizes theerrors due to the thermal motion of the ions, and to undesiredac-Stark shifts. In contrast to the light-shift gates proposedin [18], we note that the introduction of an additional sourceto drive the carrier, which is independent with respect to thelaser beams, allows us to attain regimes where both the per-formance and the speed of the gate can be increased simul-taneously. Furthermore, we focus on the interactions medi-ated by transverse phonons, which are i) less sensitive to ion-heating/thermal motion, ii) easier to operate within the Lamb-Dicke regime, and therefore more convenient for quantum in-formation or simulation [19, 20].

The system.– The gate scheme works for different ionspecies, however, we focus on 25Mg+ for concreteness [22].We consider two hyperfine states |F,MF〉 to form our qubit,|0〉= |3,3〉, |1〉= |2,2〉, whose energy splitting is ω0/2π = 1.8GHz [Fig. 1(a)]. The ions form a string along the z-axis of alinear Paul trap [21] with the following trapping frequenciesωx/2π,ωy/2π = 4 MHz ωz/2π = 1 MHz. We are inter-ested in the Hamiltonian for small transverse vibrations alongthe x-axis, expressed in terms of the phonon operators an,a†

n,

∆xi = ∑nMin√2mωn

(an +a†n), Hph = ∑n ωna†

nan. (1)

Here ωn,M jn stand for the phonon frequencies and ampli-tudes, and h = 1. As shown in Fig. 1(a), a pair of laser beamsin a Raman configuration (red arrows) induce a transition be-tween the hyperfine states via an auxiliary excited state. Bysetting their frequency beatnote ωL close to a red sidebandω0−ωn, such that the detuning δn is small with respect to thetrap frequency, one obtains the red-sideband excitation

Hr(t) = ∑in

Finσ+i ane−iδnt +H.c., Fin =

iΩL2 ηnMin (2)

which can be interpreted as a non-diagonal spin-boson in-teraction exchanging an excitation between the spins andphonons [Fig. 1(a)]. Here, ΩL is the two-photon Rabi fre-quency, ηn = kL/

√2mωn stands for the Lamb-Dicke param-

eter, where kL is the component of the difference of the two

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110.

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laser wavevectors along the x-axis, and σ+i = |1i〉〈0i|. This

expression is valid in the interaction picture for weak laserbeams ΩL δL, where the bare detuning δL = ωL−ω0 +ωx.Hence, we have set ΩL = 0.125|δL|/η , with δL = −0.2ωx,and the bare Lamb-Dicke parameter η = kL/

√2mωx ≈ 0.2.

Sideband gates and thermal noise.– For large detunings, thelasers only excite virtually the vibrational phonons, which canbe thus eliminated adiabatically from the dynamics. In fact, itis the process where a phonon is virtually created by an ion,and then reabsorbed by a distant one, which gives rise to theeffective spin interactions H→ Hph +Heff +Hres, where

Heff = ∑i j

Jeffi j σ

+i σ−j , Jeff

i j =−∑n

1δn

FinF∗jn. (3)

This phonon-mediated interaction gives rise to an indirectcoupling between the qubits that can be exploited to cre-ate logic gates U(t, t0) = e−iHeff(t−t0), or to perform a quan-tum simulation of the many-body Hamiltonian (3), whichcorresponds to a dipolar version of the so-called XY spinmodel [23]. In particular, for times

Ti j =(2n+1)π

2Jeffi j

, Jeffi j ≈

Ω2Lη2

8δ 2L

(ω2

z

ω2x

)(1+

δL

ωx

)ωx, (4)

the unitary evolution corresponds to the so-called SWAP gatewhich performs the logic operation |1i0 j〉 ↔ |0i1 j〉, leavingthe remaining states |0i0 j〉, |1i1 j〉 unchanged [24]. Let us alsonote that at half the swap periods, this gate generates entan-gled Bell states and together with single-qubit rotations be-comes universal for quantum computation. We focus belowon this entangling gate, and study how the fidelity of suchBell states can be protected against different types of noise.

There is an additional possibility neglected in the above ex-planation, namely, the process where a phonon is reabsorbedby the same ion. This leads to a residual spin-phonon couplingthat spoils the performance of the gate

Hres = ∑i

Di(t)σ zi , Di(t) = ∑

nmDinma†

mane−i(ωn−ωm)t , (5)

where Dinm = − 12FinF ∗

im(1δn

+ 1δm). Note that the strength

of this term is similar to the effective spin-spin couplings,O(Ω2

L/δn), forbidding a perturbative treatment. Accordingto this expression, the ion resonance frequency will fluctu-ate in time due to the collective motional dynamics. This ef-fect can be interpreted as a local thermal noise in the limitof many ions, where it leads to dephasing. In Fig. 1(c),the critical effect of this term on the SWAP gate can be ob-served. We show the results of a numerical simulation ofthe time evolution under the full spin-phonon HamiltonianHtot = Hph +Hr(t), and compare it to the effective idealizeddescription (3). We study the dynamics of the followinginitial state ρ0 = |1i0 j〉〈1i0 j| ⊗ ρth, where ρth is the Gibbsstate of the normal phonon modes corresponding to differ-ent temperatures, and thus different mean phonon numbersn ∈ 0,0.1,1,2,3,4. As evidenced in this figure, the perfor-mance of the gate is severely modified by the thermal phononensemble. In fact, the swapping probabilities P10(t),P01(t)

only approach the effective description (circles and squares)for ground-state cooled ions. As the mean phonon numberis increased, one observes that the coherent oscillations get astronger damping, and the generation of Bell states at half theSWAP periods (black arrows in Fig. 1(c)) deteriorates .

We show below how to protect the coherent spin dynamicsfrom this thermal dephasing by driving the carrier transitionbetween the internal levels according to

Hd(t) = 12 ∑i Ωdeiφdσ

+i (ei(ω0−ωd)t + ei(ω0+ωd)t)+H.c., (6)

where Ωd is the driving Rabi frequency, and φd its phase. Forour particular qubit choice, this driving can be performed witha microwave source tuned in resonance with the carrier tran-sition ωd ≈ ω0, [blue arrow in Fig. 1(a)] and we consider astrong-driving regime: Ω2

L/|δn| < 2δn Ωd ω0. As willbecome clear later on, the same driving is not only responsi-ble for the thermal-noise decoupling, but also minimizes theundesired errors due to Stark shifts and magnetic noise.

Robust gates against thermal noise.– The dynamics of thedephasing term in Eq. (5) is characterized by the differencebetween normal-mode frequencies. For transverse modes, thisis bounded by |ωn−ωm| ≤ 2(ω2

z /ωx) ωz, and the thermaleffects can be considered as some low-frequency noise. Un-fortunately, this noise is still much faster than the typical gatetimes, and simple spin-echo techniques [26] would not sufficeto get rid of the thermal gate errors. A neat account of the de-coupling by the strong driving of the carrier transition is that,by moving to the dressed-state basis |±i〉 = (|1i〉± |0i〉)/

√2,

the dephasing term (5) rotates very fast even when involvingtwo vibrational modes that are close in frequency. Note thatwe have set the driving phase to φd = 0, but the scheme worksequally well for any stable phase. In the dressed-state basis,the residual thermal error would be

Hres(t) = ∑inm

Dinm|+i〉〈−i|a†manei(Ωd−(ωn−ωm))t +H.c. (7)

For a strong driving Ω2L/δn < 2δn Ωd, this term evolves

very fast and can be neglected in a rotating wave approxima-tion Hres ≈ 0, increasing the gate fidelity for warmer ions.

This argument has to be readdressed for a combinationof the carrier and red-sideband interactions, since the resid-ual error terms are no longer described by Eq. (5). In or-der to show that a similar argument can still be applied,we have performed a polaron-type transformation an→ an +∑i F

∗inσ x

i /2δn [8, 20], that allows us to find the effective in-teraction and residual error terms to any desired order of per-turbation theory. One finds that the dynamics is accuratelydescribed by the effective Hamiltonian

Heff = ∑i j

Jeffi j σ

xi σ

xj +

12 ∑i Ωdσ x

i , Jeffi j = 1

4 Jeffi j , (8)

whereas the residual spin-phonon coupling is given by

Hres = ∑in

i2(Finan−F ∗

ina†n)(

coshΘiσyi − isinhΘiσ

zi)

(9)

where Θi = ∑m Fimam/2δm −H.c. Note that by moving tothe dressed-state basis, the residual error term to any order

3

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0.88

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σxj num

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σx j(t

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t(ms)0

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t)

P1,0(t) : n =

P0,1(t) : n =

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t(ms) 51 4

ba

Ω1 Ω2

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|2, 2

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···

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

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Ωd = 2.3ωz

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Ωd = 2ωz

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Ωd = 0

Ωd = 5.2ωz

F |Ψ

-

d

L

MW

MW

PU(tf ,

tf2 )

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Initialization Measurement

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F |Φ

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tf(ms)0 0.7

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U( tf2 , t0)

ttf/2t0 = 0

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Figure 1. Robust trapped-ion quantum logic gates: (a) Hyperfine structure of 25Mg+. Two laser beams Ω1,Ω2 drive the red-sidebandvia an off-resonant excited state, and a microwave Ωd directly couples to the transition. (b) Scheme of the robust two-qubit gate. The qubitsare initialized as follows: they are projected onto |0i〉 by optical pumping P , and then |ψ0〉 = |1i0 j〉 is obtained after a π-pulse from themicrowave Xπ

i = σ xi . This can be achieved with an off-resonant laser beam such that the ac-Stark shift effectively hides ion j from the MW,

or by employing magnetic field gradients. To account for the worst possible scenario, we consider different switching times of the laser andthe decoupling microwave, and also that the timing with the microwave is not perfect. We introduce an global π-pulse Zπ

i = σzi that refocuses

the oscillations due to the resonant microwave at t = t0, and follows from the energy shift of an off-resonant microwave which is sufficientlystrong. This spin-echo pulse corrects the possible difference of switching times. At this time, the two-qubit coupling is applied by switchingon the laser beams responsible for the red-sideband. Again, a spin-echo refocusing pulse is applied at t = tf/2 which shall correct for theimperfect synchronization of the experiment with the resonant microwave. After switching of the lasers at tf, the decoupling microwave is alsoswitched off, and the qubit state measured by state-dependent fluorescence techniques. (c) Dynamics of the swap probabilities P10(t),P01(t)for the effective gate (3) (squares and circles), and the exact spin-phonon Hamiltonian H = Hph +Hs for a two-ion crystal with differenttemperatures in the absence of microwave driving. The Hilbert phonon space is truncated to nmax = 20 excitations per mode. (d) Fidelityfor the generation of the Bell state |Ψ−〉 by the driven entangling gate as a function of the mean phonon number and setting nmax = 14. Asthe driving power is increased, the fidelity approaches unity. In the inset, we represent the gate error and compare it to the fault-tolerancethreshold εt ∼ 10−4−10−2. (e) Exponential decay of the coherences 〈σ x

i (t)〉 due to the magnetic-field dephasing. (f) Error in the generationof the Bell state |Φ−〉 as predicted by the effective gate (8). The spin-phonon Hamiltonian incorporates the additional magnetic-field noiseH = Hph +Hs +Hd +Hn, together with a strong microwave driving Ωd/2π = 5.2 MHz. The error is presented as a function of the dephasingT2 times. In the inset, we represent the time evolution of the fidelity for the typical noise dephasing time T2 =5 ms.

of perturbation theory only involves transitions between thedressed eigenstates |+i〉 ↔ |−i〉, which are accompanied bythe transformation on the phonons encoded in the differentpowers Θi. Fortunately, all these transitions are inhibited dueto the large energy gap between the dressed states roughlyset by Ωd. More precisely, in the strong driving regimeΩ2

L/|δn| < 2δn Ωd, the leading order terms of the residualcoupling can be neglected in a rotating wave approximation.

We test the validity of this idea by computing numeri-cally the fidelity for the unitary generation of the Bell state|Ψ−i j〉 = (|1i0 j〉− i|0i1 j〉)/

√2. We integrate numerically the

complete Hamiltonian, Htot = Hph +Hr(t)+Hd(t), consider-ing the thermal motion of the trapped ions. In order to ob-serve neatly the gate effects, we perform a spin-echo sequencethat refocuses the fast oscillations due to the microwave driv-ing, but does not modify the spin-spin coupling. In Fig. 1(b),we represent schematically the necessary ingredients for thisquantum gate. Let us note that we are considering the worstscenario, where a number of spin-echo refocusing pulses cor-

rect for possible errors of synchronization. Nonetheless, westress that these pulses may not be required in the case of ac-curate synchronization. After the unitary evolution U(tf) =U(tf, 1

2 tf)(σ zi σ

zj )U( 1

2 tf,0), we calculate the fidelity of the pro-duced state F|Ψ−〉 = maxtf

〈Ψ−i j |TrphU†(tf)ρ0U(tf)|Ψ−i j〉

for different initial thermal states and driving strengths. InFig. 1(d), we represent the results obtained, where the afore-mentioned decoupling from the thermal noise can be clearlyidentified. The parameters used in these simulations are thesame as before, and we set the driving strength to Ωd/ωz ∈0,2,2.3,2.8,3.6,3.8,4.4,5.2. In this figure, we observe thatthe fidelity of the gate improves considerably with respectto the non-driven case when the driving power is increasedΩd 2δn. In fact, for driving strengths in the 4-5 MHz range,the gate error (see inset) lies below the fault-tolerance thresh-old ε ∼ 10−2−10−4 [27] for thermal states with mean phononnumbers n≤ 2. Note that a magnetic field B0 ≈ 4 mT needs tobe applied to ensure that the Zeeman splitting between mag-

4

netic states is much larger than the microwave Rabi frequencyto avoid driving unwanted transitions.

Resilience to magnetic-field noise and Stark shifts.– So far,our discussion has neglected the effects of the environmentalnoise. In standard traps, the leading source of noise is due toexternal fluctuating magnetic fields and limits the coherencetimes of magnetic-field sensitive states to about T2≈ 5ms [21],which is particularly important for multi-ion entangled statesexperiencing super-decoherence [12]. Considering that theabove effective interactions lie in the kHz regime, decoher-ence is likely to play a key role. In order to study its ef-fect, we use a phenomenological model for the global mag-netic noise, which involves a fluctuating resonance frequencyω0→ ω0 +F(t). Here, F(t) is a a stochastic Markov processthat evolves according to a Langevin equation [28]. When theprocess is stationary and gaussian [29], there is an exact ex-pression for its time evolution

Hn =12 ∑i F(t)σ z

i , F(t2) = F(t1)e−δ tτ +

[ cτ

2 (1− e−2δ tτ )] 1

2 n,(10)

where c is the diffusion constant, τ the correlation time ofthe noise, δ t = t2− t1, and n is a unit gaussian random vari-able. As shown below, the strong driving provides an efficientmechanism for decoupling from this magnetic noise, whichcan be considered as a continuous version [9, 31] of the so-called dynamical decoupling [15]. Note that, even if the par-ticular noise in Eq. (10) corresponds to states with oppositemagnetic moments, the decoupling would work equally wellfor any pair of Zeeman sublevels. In fact, any Zeeman termwill be inhibited by the large energy gap set by the driving.

Let us first focus on the evolution of the coherences 〈σ xi (t)〉

under the effect of noise H = Hn(t). For the initial state|ψ0〉 = |±i〉, one obtains 〈σ x

i (t)〉 = ±exp(−〈ϕ2〉st/2), whereϕ(t) =

∫dt ′F(t ′), and the autocorrelation function for this

type of noise is 〈ϕ2〉st = cτ2(t − τ(1− e−tτ)). In the limitof short correlation times τ t, one obtains an exponentialdamping of the coherences with a decay time T2 = 2/cτ2.In Fig. 1(e), we show the coherence decay obtained by thenumerical statistical average of the magnetic noise (circles),which displays a clear agreement with the above analyticalexpression (lines). Accordingly, one can fix the parametersof the model c,τ in order to reproduce the experimentally-observed decay in the millisecond regime, and then study itseffects on the performance of the quantum logic gate. In orderto single out the effects of the magnetic noise from the thermaleffects, we consider a ground-state cooled crystal with n≈ 0,but set Ωd/2π = 5.2MHz which ensures that the results canbe carried out to higher temperatures [Fig. 1(d)].

Since the magnetic-field noise is global, the unitary evo-lution within the zero-magnetization subspace is not altered,and the generation of the Bell state |Ψ−i j〉 is thus robust. How-ever, the entangling gate (8) should also lead to all four Bellstates. In particular, |Φ−i j〉 = (|1i1 j〉− i|0i0 j〉)/

√2 is gener-

ated from the initial state ρ0 = |1 j1 j〉〈1i1 j| ⊗ ρth, and thussuffers from the decoherence due to magnetic noise. We studythe unitary evolution of such state, and compute the fidelityF|Φ−〉(tf) = 〈Φ−i j |TrphU†(tf)ρ0U(tf)|Φ−i j〉 of the noisy dy-namics under H = Hph + Hr + Hd + Hn with respect to thedesired noiseless evolution given by Eq. (8). In the inset ofFig. 1(f), we represent the fidelity obtained by the average ofN = 5 ·103 samplings of the random noise such that the decaytime lies in a range T2 = 5ms. One readily observes that due tothe strong driving Ωd/2π = 5.2MHz, the fidelity approachesunity. In the main figure, we study the gate error for differentcoherence times 5µs≤ T2 ≤ 50µs. According to these results,the decoupling mechanism can work for a much stronger mag-netic noise. Alternatively, this result also tells us that the gatescheme can support smaller gate speeds, and thus we can workwith a smaller laser Rabi frequency ΩL. This has the impor-tant consequence that the thermal error studied above will beminimized even further.

As an additional advantage of our scheme, we note thatit also minimizes the effects of undesired Stark shifts. Inthe derivation of the red-sideband Hamiltonian (2), one ne-glects the carrier transition by imposing a weak Rabi fre-quency ΩL (ω0−ωL). Nonetheless, the energy levels willbe off-resonantly shifted due to an ac-Stark effect. In a realis-tic setup, one must consider the shifts caused by off-resonanttransitions to all possible states, which must be compensatedby carefully selecting the laser intensities, frequencies, andpolarizations [30]. In our scheme, however, the effect of thisenergy shift is minimized by the driving in the same mannerthat the magnetic-noise shifts have been cancelled. Let us fi-nally comment on the effect of phase fluctuations on the gateperformance. In analogy to the conditional phase gate [7], butin contrast to Mølmer-Sørensen gates, our gate does not de-pend on the laser phases. The final interaction (8) only relieson the stability of the microwave field, which can be stabilizedmuch more easily compared to the relative phase between twopulsed Raman laser beams.

Conclusions and outlook.– We have introduced a schemethat merges the notion of continuous dynamical decouplingwith warm quantum gates in trapped ions. The decoupling, ontop of reducing effects of external magnetic noise, is also re-sponsible for the gate robustness with respect to thermal fluc-tuations and ac-Stark shifts. Hence, the direction of combin-ing quantum gates and dynamical decoupling is very promis-ing. Moreover, the use of continuous dynamical decouplingas opposed to pulse sequences yields elegant schemes whichare easier to analyze theoretically and realize experimentally.We note that the same decoupling scheme could also work incombination with a Mølmer-Sørensen gate, thus protecting itfrom magnetic field fluctuations and Stark shifts.

Acknowledgements.– This work was supported by the EUSTREP projects HIP, PICC, the EU Integrating ProjectsAQUTE, QESSENCE, and by the Alexander von HumboldtFoundation.

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istry, ( North-Holland, Amsterdam, 1981).[29] D. T. Gillespie, Am. J. Phys. 64, 3 (1996).[30] D. J. Wineland, et al., Phil. Trans. R. Soc. A 361, 1349 (2003).[31] P. Facchi and S. Pascazio, Phys. Rev. Lett. 89, 080401 (2002);

K. M. Fonseca, et al., Phys. Rev. Lett. 95, 140502 (2005).[32] D. F. V. James, Appl. Phys. B 66, 181 (1998).[33] R. P. Feynman, Statistical Mechanics: A Set Of Lectures (Ben-

jamin/Cummings Publishing, Massachusetts, 1972).[34] J. Kanamori, J. Appl. Phys. 31, S14 (1960); R. J. Elliot, et al.,

J. Phys. C 4, L179 (1971).

6

Appendix A: Driven-sideband gates and magnetic noise

Hyperfine qubit.– We consider the isotope of magnesium25Mg, which after being ionized can be confined in a linearPaul trap [22]. The lowest energy levels correspond to the va-lence electron lying in the s or p orbitals, which have a tran-sition wavelength of 280 nm. The ground-state 2S1/2 is splitdue to hyperfine interactions into a couple of long-lived stateswith total angular momentum F = 2,3 and an energy differ-ence of ω0/2π ≈ 1.8 GHz [Fig. 1(a)]. Note that the transitionF = 2 → 3 is electric-dipole forbidden, and the decay rateΓ ≈ 10−14 Hz is so slow that one can neglect spontaneousdecay. Besides, by applying an additional magnetic field,the magnetic sublevels are Zeeman split, and one can iso-late two particular |F,MF〉 states to form our hyperfine qubit|0〉 = |3,3〉, |1〉 = |2,2〉. This two-level system can be coher-ently manipulated by either a laser in a two-photon Ramanconfiguration ΩL, or a direct microwave Ωd [Fig. 1(a)]. Notethat the external magnetic field must be strong enough so thatthe drivings do not excite the population of undesired Zeemansublevels. Considering the microwave and laser Rabi frequen-cies employed in this work, it suffices to set B ≈4 mT. Alter-natively, one can exploit the polarization of the microwave toselect only the desired transition.

Transverse phonons.– At low temperatures, the ion trap-ping forces balance the Coulomb repulsion, and the ions self-assemble in an ordered structure. The equilibrium positionsare given by the minima of the confining and Coulomb poten-tials, and follow from the following system of equations

z0i −∑

i 6= j

z0i − z0

j

|z0i − z0

j |3= 0, z0

i = z0i /lz, lz = (e2/mω

2z )

1/3 (A1)

where we assume that the transverse confinement is muchtighter than the axial one, so that the ions arrange along a one-dimensional string [32]. For the two-ion crystals discussed inthe main text, we have z0

i ∈ −0.63,0.63.We now describe the properties of the transverse collective

modes of the ion chain around these equilibrium positions.The small vibrations along the x-axis, ∆xi, are coupled via theCoulomb interaction, which in the harmonic approximation is

H = ∑i

(1

2mp2

i +12

mω2x ∆x2

i

)+

12 ∑

i jVi j∆xi∆x j, (A2)

where Vi j = mω2z (|z0

i j|−3 − δi j ∑l 6=i |z0li|−3), and z0

i j = z0i −

z0j . This quadratic Hamiltonian can be diagonalized, leading

Eq. (1), by means of an orthogonal transformation

∆xi = ∑n

1√2mωn

Min(a†n +an),

pi = ∑n

i√

mωn

2Min(a†

n−an)

(A3)

where h = 1, an,a†n are the phonon annihilation-creation oper-

ators, and Min/√

2mωn can be interpreted as the vibrationalamplitude at site r0

i of the n-th normal mode, characterized

by the frequency ωn [33]. Such amplitudes satisfy the follow-ing relations ∑n MinM jn = δi j, and ∑i j MinVi jM jm = Vnδnm,where δnm is the Kronecker delta, and the normal-mode fre-quencies are expressed as ωn = ωx(1+ ξVn)

1/2. Here, ξ =(ωz/ωx)

2 1 quantifies the anisotropy between the axialand transverse frequencies, and also the width of the phononbranch ωn ∈ [ωx(1− 2ξ ),ωx]. For the two-ion crystals usedin the main text ωx = 4ωz, the diagonalization of Vi j yieldsω1 = 0.968ωx cfor the zig-zag mode Mi1 = (1,−1)/

√2, and

ω2 = ωx for the center-off mass mode Mi2 = (1,1)/√

2.

Interactions by virtual phonon exchange.– We considerthe interaction of two laser beams in the Raman configurationshown in Fig. 1(a). These two lasers couple the two hyper-fine states of our qubit via an auxiliary excited state. Whenthe transitions to the excited state are far off-resonance [21],the excited state is seldom populated and can be eliminatedfrom the dynamics. In particular, a detuning ∆/2π ≈ 9.2GHzexceeds both the individual Rabi frequencies Ω1,Ω2, and thedecay rate from the excited state Γ. Accordingly, the effec-tive coupling between the qubit states is expressed as HL =ΩL2 ∑i σ

+i ei(kL·ri−ωLt) + H.c., where ΩL = Ω∗1Ω2/2∆ is the

two-photon Rabi frequency of the transition, ωL = ω1−ω2 isthe beatnote of the laser beams, and kL = k1−k2 the effectivelaser wavevector. The laser wavevector is directed along the x-axis, and thus will couple to the transverse phonon modes. Bytuning the laser beatnote such that ωL = ω0−ωn + δn with adetuning that fulfills δn ωn, it is possible to derive the red-sideband Hamiltonian in Eq. (2) by making a Taylor expan-sion in the small Lamb-Dicke parameter ηn = kL/

√2mωn

1. Note that the derivation is only valid if the Rabi frequencysatisfies ΩL |ωL−ω0| in order to neglect the terms corre-sponding to the carrier transition [21]. The full Hamiltoniancan be expressed in a picture where the phonon frequency isreplaced by the detuning with respect to the particular side-band, namely

H = ∑n

δna†nan +∑

in(Finσ

+i an +F ∗

inσ+i a†

n). (A4)

This expression is now amenable to perform the adiabaticelimination of the phonons, and obtain the effective cou-pling between the qubits (3). In the limit of large detuning,Fin δn, only virtual phonon excitations can take place.There are two possible paths for the virtual phonon exchangebetween distant ions: i) Either the system virtually populatesthe manifold with one extra phonon, or ii) it goes through alower-energy manifold with one less phonon. Since these twoprocesses have an opposite detuning , their amplitudes can-cel (FinF ∗

jm)(1/δn + 1/δm)− (F ∗inF jm)(1/δn + 1/δm) ex-

cept when the exchanged phonon belongs to the same moden = m. In such a case, one has to take into account the bosonicnature of the phonons ana†

n = 1+a†nan, which spoils the inter-

ference between both exchange paths, and leads us to an ef-fective Hamiltonian where the phonons have been eliminated

Heff = ∑i j

Jeffi j σ

+i σ−j , Jeff

i j = ∑n

−1δn

FinF∗jn. (A5)

At this point, we should note that there was a missing pro-cess in the above argument, namely that the virtually-excited

7

phonon can be reabsorbed by the same ion. In this case, itdoes not lead to a phonon-mediated interaction, but rather to aStark shift of the qubit resonance frequency. Due to the alge-braic properties of the Pauli matrices, the above cancellationbetween the two paths does not take place in this case, and oneobtains an ac-Stark shift that depends on the phonons, namely

Hres = ∑inm

Dinma†manσ

zi , Dinm =− 1

2FinF ∗im(

1δn+ 1

δm). (A6)

This term can be exploited for quantum simulations of Ander-son localization of the phonons [23], but has a dramatic effecton the internal-state dynamics, since it couples the spins to thephonons with a strength similar to that of the desired gate.

Polaron-type transformation for a strong driving.– Weconsider a resonant microwave driving ωd = ω0 with Ωd ωd, so that one can neglect the counter-rotating terms inEq. (6) via a rotating-wave approximation. In the picture in-troduced above, the driven Hamiltonian becomes

H =∑n

δna†nan+∑

i(Ωdeiφdσ

+i +H.c.)+∑

in(Finσ

+i an+H.c.).

For the sake of simplicity, we set φd = 0 since it does notchange the essence of the decoupling mechanism describedbelow. By moving onto the dressed-state basis, |±i〉= (|1i〉±|0i〉)/

√2, the red-sideband term becomes

Hr(t) = ∑in

12Fin

(|−i〉〈+i|e−iΩdt −|+i〉〈−i|e+iΩdt

)ane−iδnt

+∑in

12Fin

(|+i〉〈+i|− |−i〉〈−i|

)ane−iδnt +H.c.

From this expression, one readily observes that the terms in-volving transitions between the dressed eigenstates rotate veryfast for Ωd δn, and their contribution to the effective inter-actions will be negligible. In order to give a stronger weightto the terms diagonal in the dressed-state basis, we performa spin-dependent displacement an → an − ∑ j F

∗jnσ x

j /2δn,which is formalized in terms of a Lang-Firsov-type transfor-mation [34] as follows

U = eS, S = ∑jn

F ∗jn2δn

σ xj a†

n−H.c., (A7)

which has also been used in the context of trapped ions [8, 20].This unitary offers an alternative mechanism to obtain the spininteractions by virtual phonon exchange, and also allow us tocalculate all the residual spin-phonon couplings to any desiredorder of the small parameter F jn/δn 1. In particular, con-sidering the algebraic properties of the spins and phonons, ittransforms the relevant operators as follows

UanU† = an−∑j

F ∗jn

2δnσ

xj ,

Uσyi U† = cosh(Θi)σ

yi − i sinh(Θi)σ

zi ,

(A8)

where Θi = ∑m Fimam/2δm−H.c. Therefore, it displaces thephonon operators, whereas it rotates the spins around the x-axis. From these transformations, one recovers the effective

Hamiltonian in Eq. (8), and the residual error term in Eq. (9),which we rewrite for reading convenience

Hres = ∑in

i2(Finan−F ∗

ina†n)(

coshΘiσyi − isinhΘiσ

zi).

To any order of perturbation theory, the residual spin-phononcouplings only involves terms that try to induce transitions be-tween the dressed eigenstates, namely σ

yi = i|+i〉〈−i|+H.c.,

and σzi = |+i〉〈−i|+H.c.. Therefore, in the limit of strong

microwave driving Ωd ΩL, the residual spin-phonon termbecomes rapidly rotating and can be neglected in a rotatingwave approximation unless there is some resonance with aprocess that does not conserve the number of phonons. Forthe parameters of interest to our setup, it suffices to considerthat Ωd 2δn to avoid such processes, and thus minimize theeffects of the residual spin-phonon coupling.

Magnetic noise model.– The internal level structure oftrapped ions can be perturbed by uncontrolled external elec-tric and magnetic fields. Since the Stark shifts are typicallysmall [21], one is only concerned with magnetic-field fluc-tuations of the resonance frequency ω0 → ω0 + ∂Bω0(B−B0) +

12 ∂B2ω0(B−B0)

2. For clock states, either there is nosuch magnetic-field dependence (e.g. |0i〉, |1i〉 have both zeromagnetic moment), or the linear Zeeman shift vanishes at acertain B0. In such cases, the internal state coherence timescan reach even minutes. On the other hand, when none ofthe above conditions holds, one refers to magnetic-field sen-sitive states whose typical coherence times are reduced bymagnetic-field fluctuations B(t) to T2 ∼1-10ms. In Eq. (10),we assume that we are dealing with magnetic-field sensitivestates such that F(t) = −gµBB(t) fluctuates, where µB is theBohr magneton and g the hyperfine g-factor. Low-frequencyfluctuations (i.e. fluctuations that only take place between dif-ferent experimental runs) can be easily prevented by usingspin-echo sequences [26]. On the other hand, the effects offast-frequency fluctuations cannot be refocused unless com-plicated dynamical-decoupling sequences are used. We areconcerned with this fast type of fluctuations, which lead to anexponential decay of the coherences 〈σ x

i (t)〉 as measured byRamsey interferometry [21].

In order to reproduce such an exponential decay, we usea paradigmatic model of noise where F(t) corresponds to astationary, Markovian, and gaussian random process. Sucha stochastic process is known as a Ornstein-Uhlenbeck pro-cess [28, 29], and is characterized by the following Langevinequation

dF(t)dt

=−F(t)τ

+√

cΓ(t), (A9)

where c is the diffusion constant, τ the correlation time,and Γ(t) is a gaussian white noise that fulfills 〈Γ〉st = 0,〈Γ(t)Γ(0)〉st = δ (t). This particular stochastic differentialequation can be integrated exactly yielding a gaussian randomprocess with the following mean and variance

〈F〉st = F(t0)e−(t−t0)/τ ,

〈F2〉st−〈F〉2st =cτ

2 (1− e−2(t−t0)/τ),(A10)

8

which show that the correlation time τ sets the time scale overwhich the process relaxes to the asymptotic values. Besides,the autocorrelation function 〈F(t)F(0)〉st =

cτ

2 e−t/τ showsthat τ also sets the time scale such that the values of pro-cess are correlated or not, where we have assumed a vanishingmean as customary. Of primary importance to the numericalsimulations is the update formula (10), which is valid for anarbitrary discretization t2 = t1 + δ t [29], and is thus ideallysuited to for the numerical integration of the dynamics.

Let us finally comment on the modification of the Lang-Firsov transformation in the presence of the magnetic noise

term (10), which would lead to extra residual spin-phononcouplings. However, these contributions are negligible forthe regime of importance to ion traps T2 ∼ 1-10ms. Thiscan be understood by considering the variance at long times,which leads to

√〈F2〉st =

√1/τT2. Since the magnetic-field

noise in ion traps corresponds to the limit τ t, we haveconsidered τ = 0.1T2 throughout the text, which leads to atypical noise strength in the 0.1-1kHz regime. Accordingly,Ωd

√〈F2〉st, and the residual spin-phonon couplings that

come from the Lang-Firsov transformation can be safely ne-glected.