Resilient entanglement gates for trapped ions
A. E. Webb, S. C. Webster, S. Collingbourne, D. Bretaud, A. M. Lawrence, S. Weidt, F. Mintert, W. K. Hensinger
RResilient entangling gates for trapped ions
A. E. Webb, S. C. Webster, S. Collingbourne, D. Bretaud,
1, 2
A.M. Lawrence,
1, 2
S. Weidt, F. Mintert, and W. K. Hensinger ∗ Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, UK QOLS, Blackett Laboratory, Imperial College London, London, SW7 2BW, UK (Dated: September 18, 2018)Constructing a large scale ion trap quantum processor will require entangling gate operations thatare robust in the presence of noise and experimental imperfection. We experimentally demonstratehow a new type of Mølmer-Sørensen gate protects against infidelity caused by heating of the motionalmode used during the gate. Furthermore, we show how the same technique simultaneously providessignificant protection against slow fluctuations and mis-sets in the secular frequency. Since thisparameter sensitivity is worsened in cases where the ions are not ground state cooled, our methodprovides a path towards relaxing ion cooling requirements in practical realisations of quantumcomputing and simulation.
Building a quantum processor capable of solving someof the most complex real-world problems will requireboth a large number of qubits, and the ability to ac-curately perform gate operations on these qubits. Whilesuch gate operations have been demonstrated on pairs ofcarefully controlled ions with high fidelity [1, 2], main-taining such fidelities as systems scale towards a largequantum computer will require more robust operations.Many proposed trapped ion quantum processors will re-quire large numbers of ions to be trapped close to thesurface of a microfabricated chip [3–6], which can causeincreased gate infidelities due to heating and dephasingof the ions’ motion caused by voltage fluctuations in theelectrodes of the chip [7] - the heating rate scales unfor-givingly with distance as approximately d − [8]. In addi-tion, it is likely that there will be slowly changing varia-tions in experimental parameters, differing both from po-sition to position on the chip and drifting in time, whichwill be difficult to fully characterise and correct for duringthe operation of the processor. This problem is exacer-bated when the initial mean excitation of the motionalmode, ¯ n , is higher. This could, for example, occur as aresult of heating during shuttling processes which forma core part of a number of architectures for a large scalequantum computer [3, 5]. A quantum processor thus re-quires gate operations that provide low error rates notjust under ideal conditions but that are resilient enoughto be successfully implemented in realistic experimentalenvironments.The two-qubit Mølmer-Sørensen (MS) gate is one ofa class of trapped ion gates that operate by state-dependent coherent excitation and de-excitation of a mo-tional mode of a pair of ions during the gate operation[9–11], and this motional excitation can be representedas a circular path in phase space. Sørensen and Mølmerdiscussed how the effect of heating could be reduced byperforming multiple smaller circles in phase space [10].While this is effective at reducing the impact of heating,the gate time scales as the square root of the number ofloops. Hayes et al. experimentally demonstrated a sim- ilar technique as a method of reducing the effect of a‘symmetric’ detuning error, such as could be caused byan incorrectly measured trap frequency [12].Noise suppression can be effectively achieved with lessimpact on the gate time by tracing out more complicatedpaths in phase space [13–15]. Recent theoretical workproposed a method whereby the infidelity due to heatingcan be significantly reduced with smaller impact on gatetime than by just performing multiple smaller loops [13].Here, we experimentally demonstrate this effect, using apair of trapped Yb + ions. We then build upon thisresult to show that these same paths also dramaticallyincrease the resilience of the gate to errors caused bysymmetric detuning errors. We demonstrate how thisresilience becomes particularly significant in the case ofthe mode used for the gate not being cooled close to theground state.By off-resonantly driving the red and blue motionalsidebands of a pair of ions coupled to a common motionalmode the MS Hamiltonian, H MS = (cid:126) δ S x (ˆ a † e iδt + ˆ ae − iδt ) , (1)can be realised, where ˆ S x = ˆ σ x + ˆ σ x is the sum of theˆ σ x matrices for the two ions, ˆ a † and ˆ a are the motionalmode raising and lower operators respectively, and δ isthe magnitude of the detuning from the red and bluesidebands. When the driving fields are applied for a time τ = 2 π/δ , the pair of ions undergo the unitary transfor-mation U MS = exp (cid:104) i π σ x ˆ σ x (cid:105) (2)which can produce a maximally entangled state from aninitial product state.In practice, there are a number of mechanisms bywhich the fidelity of the MS gate will be reduced fromone. Here we consider two sources of infidelity: a dephas-ing of the gate caused by heating of the motional modeduring the gate process, and an incorrect phase pickup a r X i v : . [ qu a n t - ph ] S e p and residual entanglement of the qubits with the mo-tional mode caused by a symmetric detuning frequencyerror. This symmetric detuning occurs as a result of themagnitude of the gate detuning δ being set incorrectlyby the same amount for both ions, ∆, for instance dueto a drift in trap frequency.If the total infidelity is small, the fidelity of the gatecan be expressed as a sum of independent infidelitiesF = 1 − (E h + E ∆ + E oth ), where E h is the infidelitydue to heating, E ∆ the infidelity due to symmetric de-tuning error, and E oth is the sum of any other infidelities,which we henceforth do not consider. Since the infideli-ties are small we will consider the infidelities only to theleading order in either heating rate or detuning error.The errors due to non-zero heating rate and symmetricdetuning error are thenE h = π ˙¯ nδ , E ∆ = (cid:18)
34 + ¯ n (cid:19) π (cid:18) ∆ δ (cid:19) (3)respectively, where ˙¯ n is the heating rate and ∆ is the errorin gate detuning (see Supplemental Material [16]). Notethat the error due to incorrect detuning is also dependenton the mean excitation of the motional mode at the startof the gate ¯ n (we assume the phonon distribution to bethermal) - if the ion is ‘hotter’ (larger ¯ n ) at the start ofthe gate operation, it becomes much more sensitive toparameter errors.Haddadfarshi et al. introduced a method to reduce theeffect of heating and dephasing of the motional modeon the gate fidelity [13]. They considered a multi-tonegeneralisation of the MS gate (MTMS), where insteadof driving each ion sideband with a single field, MTMSgates use N fields or tones to drive each sideband atdetunings δ j = jδ with { j = 1 , ...N } as shown in figure1(a), and each tone’s strength is given by coefficients c j .The Hamiltonian thus becomes H MS = (cid:126) δ ˆ S x N (cid:88) j =1 c j (ˆ a † e ijδt + ˆ ae − ijδt ) . (4)The condition to produce a maximally entangling uni-tary constrains the values of the coefficients c j to be (cid:80) Nj =1 c j j = , which corresponds to a standard singletone MS gate having a coefficient c = .The effect of any MS type Hamiltonian is to excite themotion of different spin components, causing them to se-lectively acquire a phase, before (ideally) returning themto their initial motional state. This excitation can beconsidered as a time varying displacement in a rotatingphase-space, and the effect of heating depends quadrat-ically on the magnitude of this displacement during thegate. Haddadfarshi et al. found that the best reductionin the effect of heating of the mode over the course of thegate is found when the average phase space displacementis zero, and average squared phase space displacement is FIG. 1. (a) Energy level diagram showing multi-tone gatefields detuned from the correct gate detuning δ by ∆ r and ∆ b for the red and blue sidebands respectively. MTMS gates pro-vide protection against errors of the form ∆ r = ∆ b = ∆. (b)Phase space trajectories for one (red), two (blue), and three(green) tone gates. Unlike the single tone case, the averagedisplacement (cid:104) α ( t ) (cid:105) = 0 for two or more tones. In addition,as the number of tones increases, (cid:104)| α ( t ) | (cid:105) becomes smaller.The effect of this reduction in squared displacement is to re-duce adverse effect of heating on gate fidelity. (c) For a singletone gate, a symmetric detuning error ( δ/ ∆ = 0 .
05) results inboth incomplete loops in phase space, causing error due to theresidual entanglement between the spin and motional statesof the qubit, and incorrect phase accumulation. For the twoand three tone gates, the loops come much closer to comple-tion (visually indistinguishable from closed loops), effectivelyeliminating residual qubit-motion entanglement as a contri-bution to the infidelity, and although not visually obvious,the phase picked up also becomes closer to the ideal. Both ofthese effects lead to a reduction in sensitivity to symmetricdetuning errors. minimised over the course of the gate. This correspondsto parameters where (cid:80) Nk =1 c k k = 0, and (cid:80) Nk =1 | c k | k isminimised (see Supplemental Material [16] for more de-tail). The effect of using the MTMS gate on the phase-space trajectories can be seen in figure 1(b).For MTMS gates, the heating rate defining the infi-delity is modified by a factor given by (see SupplementalMaterial [16])˙¯ n MT = 8 N (cid:88) k =1 c k k + (cid:32) N (cid:88) k =1 c k k (cid:33) ˙¯ n. (5)The minimisation conditions mean that this can be un-derstood as a smaller effective heating rate. For N = { , , } , then ˙¯ n MT = { , / , / . } × ˙¯ n respectively.We show here that MTMS gates also protect againsterrors due to an incorrect symmetric detuning. In thiscase, the MTMS Hamiltonian is modified to become H MS = (cid:126) δ ˆ S x N (cid:88) j =1 c j (ˆ a † e i ( jδ +∆) t + ˆ ae − i ( jδ +∆) t ) (6)By expanding the fidelity of the gate in powers of thefractional symmetric detuning error, ∆ /δ , we obtain thesame set of constraints on the coefficients, c j , as wasobtained when minimising the effect of motional deco-herence, so the gate is protected against both sources ofinfidelity (see Supplemental Material [16]).The infidelity due to symmetric detuning error of theoptimised MTMS gates to leading order in ∆ /δ is givenbyE MT∆ (cid:39) π (cid:18) ∆ δ (cid:19) N (cid:88) j =1 c j j (7)= 136 π (cid:18) ∆ δ (cid:19) (cid:39) . π (cid:18) ∆ δ (cid:19) ( N = 2)(8)= 39 − √ π (cid:18) ∆ δ (cid:19) (cid:39) . π (cid:18) ∆ δ (cid:19) ( N = 3)(9)The sensitivity to ∆ is significantly reduced for twoand three tone gates compared with the standard MSgate (eq. 3). The infidelity is also independent of theinitial distribution of motional states, unlike eq. 3.This is because the effect of residual qubit-motionalentanglement on the fidelity is zero to this order of∆ /δ , the infidelity being completely due to the incorrectphase being acquired during the gate. This may be ofparticular interest when gates are combined with shut-tling operations which induce heating [3, 5]. Figure 1(c)shows gates with symmetric detuning error δ/ ∆ = 0 . Yb + ions [20]. The hyperfine ground stateis driven using microwave and radiofrequency radiation,and a magnetic field gradient of 23.6(3) T/m is gener-ated using permanent magnets to enable the requisitecoupling between the internal spin and collective mo-tional modes [21, 22]. The ions are decoupled frommagnetic field noise using a dressed state system [23–25]. A pair of microwave fields for each ion couple | S / , F = 0 (cid:105) ≡ | (cid:105) with | S / , F = 1 , m F = − (cid:105) ≡|− (cid:105) and | S / , F = 1 , m F = − (cid:105) ≡ | +1 (cid:105) and, in the interaction picture, this gives three well protected statesincluding ( | +1 (cid:105)−|− (cid:105) ) / √ ≡ | D (cid:105) . The pair of states | D (cid:105) and | S / , F = 1 , m F = 0 (cid:105) ≡ | (cid:48) (cid:105) gives a well protectedqubit with transition frequencies 11.0 MHz and 13.9 MHzfor each ion respectively and a coherence time of 500 ms.A maximally entangled Bell state is created and anal-ysed for standard single tone ( N = 1) and two tone( N = 2) MS gates, since moving from one to two tonesshould show the largest improvement in gate robustness.The single tone MS gate procedure is detailed in moredetail in [26]. The gate was performed on the stretchmode of the ions, with frequency ν/ π = 461 kHz whichgives an effective Lamb-Dicke parameter of η = 0 . τ anddetuning δ were compared. The single tone gate uses apair of gate fields per ion, each of carrier Rabi frequencyΩ / π = 36 kHz, and, since δ = 2 η Ω the detuning is δ/ π = 292 Hz and the gate time is τ = 2 π/δ = 3 .
42 ms.The two tone gate uses two pairs of gate fields per ion,with the Rabi frequencies of the two tones in each pairbeing Ω = − . at δ and Ω = 1 . at 2 δ ,corresponding to c = − .
144 and c = 0 . n ≈ . | (cid:48) (cid:48) (cid:105) and | DD (cid:105) ,and the coherence between these two states [26]. A maxi-mum likelihood method was used to determine these twovalues, as well as the errors in their measurement (seeSupplemental Material [16]).To demonstrate the effectiveness of the MTMS tech-nique for protection against heating, the heating rate wasartificially increased. Randomly generated noise with aflat amplitude spectrum over a bandwidth of 20 kHz cen-tred around the secular frequency, ν , was capacitivelycoupled onto an endcap DC trap electrode, and the heat-ing rate controlled by changing the amplitude of thisnoise. Heating rates with no added noise, and for twodifferent amplitudes of artificial noise, were measured byintroducing a varying time delay after sideband coolingand measuring the temperature of the ion using side-band spectroscopy. Figure 2 shows the gate fidelity asa function of these three heating rates for both singleand two tone gates.The solid lines are the results of anumerical simulation of the master equation with appro-priate Lindblad operators to model heating, the resultsof which show good agreement with the theoretical val-ues for fidelity given by equation S.7 of the SupplementalMaterial [16]. The dashed line is the result of a numericalsimulation for a faster single tone gate at a higher power, ( phonons / s ) N o r m a li s ed f i de li t y FIG. 2. Infidelities due to heating are reduced by movingfrom a single to a two tone MS gate, shown in red and bluerespectively. Both gates are of duration τ = 3 . as defined by the peak Rabi frequency used for the twotone gate, and demonstrates that two tone gates still ex-hibit lower error due to heating. No increase in fidelityis observed for the two tone gate at no induced heat-ing compared to the single tone gate due to the smallcontribution to overall infidelity from heating, smallerthan the measurement uncertainty. The measured infi-delity at no induced heating is thought to be largely aresult of dephasing and depolarising, and parameter mis-set primarily of the form ∆ b (cid:54) = ∆ r , where ∆ r and ∆ b arethe detuning errors on each sideband (see Figure 1(a)).Methods to protect against this error exist [28] and com-bining these with MTMS techniques may offer a solution.In order to demonstrate robustness to symmetric de-tuning errors, a symmetric detuning error of up to ± . δ was added to the nominal zero error detuning [30]. Re-sults are shown in figure 3, where again solid lines showthe result of numerical simulation. A clear consistencybetween simulation and experimental results can be seen,demonstrating strong protection against both heatingand detuning errors obtained by using a two tone ratherthan the standard single tone MS gate.Since the symmetric detuning error for multi-tonegates no longer exhibits any dependence on the initialmean excitation of the motional mode of the ions tofirst order in ∆ /δ , this also opens up the possibility ofperforming gates at higher ¯ n which can for example bereached by Doppler cooling. A multi-tone gate of fidelity0.851(9) has been demonstrated at an initial thermal - - ( Δ / δ ) N o r m a li s ed f i de li t y FIG. 3. The effect of symmetric detuning error is significantlyreduced by moving to two tones, shown here for a gate dura-tion of τ = 3 . n = 53(4)for a single (red) and two tone (blue) gate after ions have beenonly Doppler cooled. A significant improvement in contrastis seen for the two tone gate, since the dependence of gateinfidelity on ¯ n to first order in ∆ is eliminated by the use ofmultiple tones. state with ¯ n = 53(4), compared to a single tone fidelity of0.50(5), as shown in figure 4. The dominant infidelity ofthe MTMS gate is expected to be due to detuning errorsof the form ∆ r (cid:54) = ∆ b , which remain sensitive to ¯ n .We have shown that for a given gate time, the use ofa two tone MS gate instead of a standard single tone MSgate substantially reduces the effect of motional heating,as well as significantly lowering the sensitivity to sym-metric detuning errors. This comes at a cost in terms ofresources - the peak power required to drive the gate isthree times higher, while the average power is timeshigher. However, provided heating is a significant sourceof error, MTMS gates should prove a powerful tool, par-ticularly for large scale systems required for quantumcomputing with comparatively low ion heights and po-tentially noisier and less stable environments [31, 32].In addition, we note that MTMS gates also provideprotection against fluctuations in the trap frequencycaused by Kerr coupling of the stretch mode to the radialmode [33, 34], which is only Doppler cooled, thus alleviat-ing one of the main restrictions on using the stretch modefor two qubit gates. Finally, use of MTMS gates also actto reduce off-resonant excitation of the carrier caused bythe gate fields compared to a single tone gate, despitethe higher peak power. This is due to the lower initialRabi frequency and the sinusoidal variation in Rabi fre-quency acting as a natural pulse shaping to reduce thisexcitation, opening up the potential for performing fastergates.After the preparation of the manuscript, we have be-come aware of related work where similar methods wereused to make laser based entangling gates more robust[35].The authors thank Farhang Haddadfarshi for help-ful discussions. This work is supported by the U.K.Engineering and Physical Sciences Research Coun-cil [EP/G007276/1; the U.K. Quantum Technologyhub for Networked Quantum Information Technologies(EP/M013243/1) and the U.K. Quantum Technologyhub for Sensors and Metrology (EP/M013294/1)], theEuropean Commissions Seventh Framework Programme(FP7/2007-2013) under grant agreement no. 270843 In-tegrated Quantum Information Technology (iQIT), theArmy Research Laboratory under cooperative agreementno. W911NF-12-2-0072, the U.S. Army Research Officeunder contract no. 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SUPPLEMENTAL MATERIAL
Protection against motional decoherence
Here we briefly summarise the work presented in [13]to protect the fidelity of an MS gate from the effectsof decoherence of the motional mode due to heating ordephasing of the motional mode. The MS Hamiltonian(equation 1) can be generalised H MS = (cid:126) δ ˆ S x ( a † f ∗ ( t ) + af ( t )) , (S.1)which by using the Magnus expansion gives a unitarytransformation U MS ( t ) = exp( − i ( F ( t ) a + F ∗ ( t ) a † ) S x − G ( t ) S x )) (S.2)where F ( t ) = δ (cid:82) t dt (cid:48) f ( t (cid:48) ) and G ( t ) = − iδ (cid:82) t dt (cid:48) ( f ∗ ( t (cid:48) ) F ( t (cid:48) ) − f ( t (cid:48) ) F ∗ ( t (cid:48) )). The function F ( t ) gives, modulo a phase factor, the displacement ofthe motional state of selected spin states during the gateoperation and should be equal to zero at the gate time τ . G ( t ) gives the amount of phase accumulated duringthe gate, and should be equal to π/ f ( t ) over the course of the gate. Haddadfarshi et al.found that the effect of heating was most reduced by set-ting (cid:104) F ( t ) (cid:105) to zero, and minimising (cid:104)| F ( t ) | (cid:105) .To find optimised functions f ( t ) the function is param-eterised as a Fourier series of N different tones f ( t ) = N (cid:88) j =1 c j exp ( ijδt ) . (S.3)The condition for a maximally entangling gate givesthe constraint (cid:80) Nk =1 c k k = , and (cid:104) F ( t ) (cid:105) = 0 means (cid:80) Nk =1 c k k = 0. For a two tone N = 2 gate, up to aphase factor this fixes the values of the coefficients to c = − .
144 and c = 0 . N ≥
2, the minimisa-tion of (cid:104)| F ( t ) | (cid:105) corresponds to minimising (cid:80) Nk =1 | c k | k .The solution to this minimisation can be shown to be c j = jb − jλ (S.4)where b = − N (cid:88) j =1 j (1 − jλ ) − (S.5)and λ is the smallest root of the equation m (cid:88) j (1 − jλ ) − = 0 (S.6)Values of c j for N ≤ Infidelity due to heating
We study the effect of applying multiple tones to thecase where the motional decoherence is purely due toheating of the mode, and here we quantify the effecton fidelity of different numbers of tones for a givenheating rate. We define the fidelity of the gate F = (cid:104) Ψ max | ρ int | Ψ max (cid:105) , where | Ψ max (cid:105) is the appropriate max-imally entangled state, and ρ int is the reduced densitymatrix of the internal states of the ions.The infidelity caused by heating can be modelled usinga master equation with appropriate relaxation operators[10]. In the case where the phonon decoherence is solelydue to heating of the mode at a rate ˙¯ n it can be shownthat the gate fidelity is given by the standard result fora single loop MS gateF = 18 [3 + 4 exp ( − ˙¯ n MT τ /
2) + exp ( − ˙2¯ n MT τ )] (S.7)but with the heating rate modified by a factor to givean effective heating rate ˙¯ n MT which depends on the tonecoefficients˙¯ n MT = 8 N (cid:88) k =1 c k k + (cid:32) N (cid:88) k =1 c k k (cid:33) ˙¯ n. (S.8)Note that the second term is proportional to (cid:104) F ( t ) (cid:105) andis thus zero for N ≥ (cid:104)| F ( t ) | (cid:105) , which is the quantity minimised for N ≥ N = { , , } , then ˙¯ n MT = { , / , / . } × ˙¯ n respectively.When the infidelity is small, equation S.7 can be sim-plified by expanding to first order in ˙¯ n :F (cid:39) − π ˙¯ n MT δ . (S.9) Protection against detuning error
Here we show how taking the same approach of param-eterising the MS Hamiltonian by using multiple harmonictones can be applied to the problem of detuning error.In the presence of an arbitrary detuning error ∆, andusing the Fourier parameterisation, the MS Hamiltonian(S.1) becomes H MS = (cid:126) δ ˆ S x ( a † e − i ∆ t f ∗ ( t ) + a e i ∆ t f ( t )) (S.10)which by use of the Magnus expansion leads to the fol-lowing expressions for the functions F ( t ) and G ( t ) at theupplemental Material – 2conclusion of the gate: F ( τ ) = N (cid:88) j =1 c j δi ( kδ + ∆) ( e i π ∆ δ −
1) (S.11) G ( τ ) = N (cid:88) j,k =1 c j c k δ i ( kδ + ∆) (cid:32) πδ δ jk − sin( π ∆ δ ) jδ + ∆ (cid:33) (S.12)This results in two different decohering effects when∆ (cid:54) = 0. The fact that F ( τ ) (cid:54) = 0 means there is residualqubit-photon entanglement at the end of the gate andthe fact that G ( τ ) (cid:54) = π means that the wrong amount ofphase is acquired during the gate operation. The fidelityof the gate operation can be expressed in terms of thevalues of F ( τ ) and G ( τ ) asF = 38 + 12 cos (4 G ∆ ( τ )) exp ( − n + 12 ) | F ( τ ) | )+ 18 exp ( − n + 12 ) | F ( τ ) | ) (S.13)where G ∆ ( τ ) = G ( τ ) − π is the deviation of G ( τ ) fromthe ideal case.By expanding F ( τ ) in ∆, constraints can be foundwhich eliminate the qubit-phonon coupling to a givenorder in ∆ m : (cid:40) N (cid:88) k =1 c k k j = 0 (cid:41) mj =1 (S.14)Eliminating qubit-phonon coupling to first order in ∆results in the same constraint as when (cid:104) F ( t ) (cid:105) is set tozero, and thus for the two tone case this results in thesame coefficients as Haddadfarshi et al. found as optimumfor protection against heating.We can see from equation S.13 that eliminating qubit-phonon coupling to first order in ∆ removes its effecton fidelity to second order in ∆, which also removes thedependence on fidelity of the initial temperature ¯ n . Thefidelity can then be approximated, to leading order in ∆,to be F (cid:39) − G ( τ ) (cid:39) − π (cid:18) ∆ δ (cid:19) N (cid:88) j =1 c j j . (S.15)This then means that for N ≥ N the same set of coefficients c j minimises both forms ofinfidelity under consideration.We can then compare the fidelity of the standard singletone MS gate, to the same order in ∆, to the optimised two and three tone MS gates F = 1 − E MT∆ , where:E
MT∆ (cid:39) (cid:18)
34 + ¯ n (cid:19) π (cid:18) ∆ δ (cid:19) ( N = 1)(S.16)= 136 π (cid:18) ∆ δ (cid:19) (cid:39) . π (cid:18) ∆ δ (cid:19) ( N = 2)(S.17)= 39 − √ π (cid:18) ∆ δ (cid:19) (cid:39) . π (cid:18) ∆ δ (cid:19) ( N = 3)(S.18) Mølmer-Sørensen gate using a magnetic fieldgradient
In the interaction picture, applying radiation fields de-tuned from the red and blue stretch mode sidebands ofa pair of ionic qubits trapped within a magnetic fieldgradient produces the following Hamiltonian: H MS = i (cid:126) η Ω S y ( a † e iδ t − ae − iδ t ) (S.19)where ˆ S y = ˆ σ y − ˆ σ y , δ is the detuning, η is the Lamb-Dicke parameter, Ω the carrier Rabi frequency, and wehave set the phases of the driving field to zero. Thisdiffers from the canonical form of the MS Hamiltonian ina number of respects. The first is that ˆ S y is the differencebetween the single ion spin operators, caused by the factwe are driving the stretch mode, and the second is thefactor of i and sign difference between the a and a † terms,which is due to the fact that the atom-photon coupling isproduced by a combination of long-wavelength radiationand static field gradient, rather than photon momentumfrom laser light.By applying the radiation fields for a time τ = 2 π/δ toa pair of ions whose internal states and motional statesare separable, over the course of the gate the motionreturns to its initial state, while the internal states aretransformed according to U = exp (cid:20) i πη Ω δ ˆ σ y ˆ σ y (cid:21) . (S.20)Note that as we use the stretch mode we perform a phasegate in the σ y basis, rather than the σ x basis described inthe main text. The effect of this is to change the phase ofthe final Bell state produced during the gate. By settingthe detuning δ = 2 η Ω the required entangling two-qubitunitary U = exp (cid:2) i π ˆ σ y ˆ σ y (cid:3) is then obtained. Finally, wecan rewrite equation S.19 in terms of δ : H MS = i (cid:126) δ S y ( a † e iδt − ae − iδt ) . (S.21)upplemental Material – 3 Stark shifts
Due to the asymmetry of the dressed state system, thegate fields produce an a.c. Stark shift. For single tonegates this is corrected for by shifting the frequency of thegate fields. For two tone gates the beating due to therelatively small frequency separation δ between the twotones causes an effective sinusoidal amplitude modulationof the gate fields. Since the a.c. Stark shift depends onthis amplitude, it is also sinusoidally time-varying, andis compensated for by varying the detuning of the gatefields over the gate time. Fidelity measurement
We measure the state of the ions by carrying out a flu-orescence measurement and detecting the scattered lighton a PMT. We then threshold the photon counts fromthe PMT to obtain three quantities x , x and x , whichare, respectively, the number of times we observe that both ions are in the dark state; exactly one ion is inthe bright state; and both ions are in the bright state,where x + x + x = n . From these we can estimate thetrue probabilities for each of these outcomes, by way ofa maximum likelihood method [17].The probabilities to measure each of the three out-comes p (cid:48) , p (cid:48) and p (cid:48) differ from the true probabilities p , p and p due to errors in the preparation and statedetection of the ions. When calibrating the state de-tection measurements, we typically measure a combinedstate preparation/ detection fidelity of around 87% (notethat near unity detection fidelity could be obtained us-ing appropriate imaging optics and detectors [18, 19]).From these calibrations, we can extract a linear map p (cid:48) i = (cid:80) j P ( i | j ) p j , which relates the probabilities of themeasurement outcomes to the true probabilities. (Notethat we only need to use p and p , as p is constrainedby p + p + p = 1.) Since the measured counts x i are dis-tributed according to a multinomial probability distribu-tion, we can find most likely values for the true probabil-ities p i by maximising the following log-likelihood func-tion, f ( p , p ) = log (cid:18) ( n + 1)( n + 2) n ! p (cid:48) ( p , p ) x p (cid:48) ( p , p ) x (1 − p (cid:48) ( p , p ) − p (cid:48) ( p , p )) n − x − x x ! x !( n − x − x )! (cid:19) , (S.22)over the variables p and p .In order to measure the gate fidelity we use the par-ity oscillation method described in [26]. Here, the par-ity Π = p + p − p undergoes oscillations of the formΠ = A cos(2 φ + φ ) where φ is the phase of the analy-sis pulse, and φ the phase of the entangled state. Wemake N parity measurements Π i at varying analysis pulsephases φ i , from which we must extract a best fit valueof A in order to obtain the Bell state fidelity. This fitis achieved using a maximum likelihood method simi-lar to that described above. This time our measuredvalues are the counts the ‘even’ and ‘odd’ parity states x even = x + x and x odd = x where x even + x odd = n . We use the log-likelihood function f = N (cid:88) i =1 log (cid:32) ( n + 1) n ! p (cid:48) odd ( p i odd ) x i odd (1 − p (cid:48) odd ( p i odd )) n − x i odd x i odd !( n − x i odd )! (cid:33) , (S.23)where p i odd = (1 − Π i ) / N data points x i odd in the fit. We can then maximise f withrespect to the fit parameters A and φ , and use these tocalculate the fidelity.In order to account for any change in the phase of theBell state produced, the amplitude of the parity fit wasmultiplied by a a factor cos ∆ φ , where ∆ φ = φ m − φ , φ m is the phase of the Bell state as measured and φ0