A high fidelity light-shift gate for clock-state qubits

C. H. Baldwin, B. J. Bjork,∗ M. Foss-Feig,† J. P. Gaebler, D. Hayes,

M. G. Kokish, C. Langer, J. A. Sedlacek, D. Stack,‡ and G. VittoriniHoneywell | Quantum Solutions

To date, the highest fidelity quantum logic gates between two qubits have been achieved withvariations on the geometric-phase gate in trapped ions, with the two leading variants being theMølmer-Sørensen gate [1, 2] and the light-shift (LS) gate [3, 4]. Both of these approaches havetheir respective advantages and challenges. For example, the latter is technically simpler and isnatively insensitive to optical phases, but it has not been made to work directly on a clock-statequbit. We present a new technique for implementing the LS gate that combines the best featuresof these two approaches: By using a small (∼ MHz) detuning from a narrow (dipole-forbidden)optical transition, we are able to operate an LS gate directly on hyperfine clock states, achievinggate fidelities of 99.74(4)% using modest laser power at visible wavelengths. Current gate infidelitiesappear to be dominated by technical noise, and theoretical modeling suggests a path towards gatefidelity above 99.99%.

The promise of quantum computers to efficiently solvecertain classically intractable problems has spawned abroad experimental effort to realize scalable platformsfor quantum computation. At the very least, a goodplatform should offer high-fidelity state-initialization andreadout, a high-fidelity universal gate set, a large ratiobetween the qubit coherence time and the gate times,and a reasonable path toward scaling to large numbersof qubits. Trapped ions appear very capable of meetingall of the above requirements, with state of the art exper-iments demonstrating combined state preparation andmeasurement fidelities above 99.9% [5], two-qubit gatefidelities of 99.9% [2, 4], and laser-based single-qubit gatefidelities above 99.99% [6] (microwave-based single qubitgates have achieved even higher fidelities [5]). Althoughthese gate fidelities satisfy the fault-tolerance thresholdsof some error correcting codes, the technical overheadto achieve them and the qubit overhead of performingerror correction with them is still very high; scalabletrapped-ion quantum computation likely requires furtherimprovements of gate fidelity, reduced experimental over-head, or most likely both.

The best two-qubit gate fidelities reported for trappedions have been achieved using the geometric phase gate.A spin-dependent force (SDF) is engineered, and drivesexcursions of a chosen collective mode of motion of atwo-ion crystal. This motion imparts spin-dependent ge-ometric phases to the two-qubit wave function, and forsuitably chosen parameters results in a maximally en-tangling gate Uzz = diag(1, i, i, 1). There are severaldifferent microscopic realizations of the spin-dependentforce underlying this gate, but the two-most successfulapproaches to date have been (1) The Mølmer-Sørensen(MS) gate, in which blue- and red-sideband transitionscorresponding to a chosen collective mode of motion aredriven simultaneously, and (2) The light-shift (LS) gate,in which a traveling wave generates a spatially and tem-porally modulated light shift of the qubit frequency (seeFig. 1). Both gates can in principle be achieved on an op-

tical qubit or between qubit states within the electronicground-state manifold. The state-of-the-art fidelities re-ported in Refs. [2, 4] were both achieved in the lattercontext, generating couplings within the qubit manifoldby coupling off-resonantly to a dipole-allowed S → Pelectronic transition.

The MS gate and LS gate both have benefits and drawbacks. For example, the former requires high-frequency(∼ GHz) laser modulation, and one must be careful toremove the gate dependence on optical phases [7], but itcan be employed directly with qubits encoded in hyper-fine clock states, which have very long coherence times.The LS gate requires only low-frequency (∼ MHz) lasermodulation and is more natively insensitive to opticalphases, but is incompatible with a clock-state qubit whenoperated off a dipole transition [8]. Here, we demonstratethat a high-fidelity LS gate can in fact be operated on aclock-state qubit by utilizing a long-lived (D rather thanP ) excited state. We build on an approach originallyconceived in Ref. [9]. In that work the authors consid-ered a scheme involving large (∼ GHz) detunings fromthe D state, which in turn required extremely high laserpower. Here we present a related gate scheme that worksat much smaller (∼ MHz) detunings, and can be operatedat essentially the same laser power used in more conven-tional approaches. It combines many of the best featuresof the MS and LS gates while enabling a variety of tech-nical simplifications, such as operating at much friendlier(visible rather than UV) wavelengths, for which power ismore readily available, trap charging is mitigated [10],and integration of on-chip light delivery is easier [11].

Gate implementation.—We implement the gate ontwo 171Yb+ ions, with qubit state |↓ (↑)〉 encoded inhyperfine levels |F = 1(0),mF = 0〉 of the S1/2 man-ifold [12]. Two non-copropagating lasers with wave-vector difference ∆k parallel to the trap axis and dif-ference frequency µ are tuned ±µ/2 from resonance withthe |↑〉 (S1/2, |F = 1,mF = 0〉) → |e0〉 (D3/2, |1, 0〉) tran-sition (Fig. 1a). The directions and polarizations of

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these two beams (Fig. 1b) are chosen such that ∆mF =±1 transitions are allowed while ∆mF = 0,±2 arequadrupole-forbidden, thereby generating a coupling of|↑〉 → |e±〉 (D3/2, |1,±1〉) while avoiding coupling to the|e0〉 state [13]. Because the two gates lasers are detunedsymmetrically between the |e±〉 states, there is no time-averaged AC-Stark shift of the qubit-frequency, whichgreatly reduces the gate sensitivity to laser intensity fluc-tuations. Despite the uniform intensity, a spatially mod-ulated SDF is established by a polarization gradient, asshown in Fig. 1. We choose the beat note µ to be near-detuned from the axial stretch mode (µ = ωstr + δ), inwhich the two ions move out of phase along the trap axis(x-direction in Fig. 1b), with creation(annihilation) oper-ator a†(a). Adiabatically eliminating the excited states,moving to the interaction picture with respect to the ionmotion, dropping all terms rotating at frequencies fasterthan δ, and making the Lamb-Dicke approximation, weobtain the following Hamiltonian describing an SDF ap-plied to the two ions

H ≈ ηΩ∑j=1,2

(−1)j(1+σzj

)(aeiδt+iϕj + a†e−iδt−iϕj

). (1)

Here Ω ≈ g2/∆, with g the magnitude of the single-photon Rabi frequency for the |↑〉 → |e±〉 transitions(chosen to be equal for both lasers) and ∆/(2π) ≈ B ×1.4 MHz/Gauss the detuning of these levels from the av-erage frequency of the lasers, and η = |∆k|

√~/(4mωstr)

is the Lamb-Dicke parameter for the gate mode (with mthe ion mass). The ion dependent motional phases areϕj = ϕ+ |∆k|xj , with ϕ the optical (Raman) phase andxj the mean position of ion j along the trap axis, andthe factor of (−1)j reflects the antisymmetry of the gatemode. If the ions are spaced by an even integer multipleof π/|∆k| then the SDF is in phase on the two ions. The(antisymmetric) gate mode is therefore not driven whenthe ions are in the |↑↑〉 state. The |↓↓〉 state is triviallydecoupled by being far from resonance, so only the |↑↓〉and |↓↑〉 states are driven in this configuration, as shownin Fig. 1c. At a time 2π/δ the spin and motion decouple,imprinting the unitary U(Φ) = Rz(ε)diag(1, eiΦ, eiΦ, 1)on the spin wavefunction, where Φ = 2π(Ωη/δ)2. HereRz(ε) is a small unwanted spin rotation about the z axisdue to either AC-Stark shifts from the D2

3/2 manifold orfluctuations of the mean laser frequency. We implementa two-loop gate with a spin-echo to remove the effectof this small residual shift. Adjusting the laser powersuch that Φ = π/4 then yields the maximally entanglinggate Rx(π)U(π/4)R−x(π)U(π/4) = diag(1, i, i, 1). It isimportant to note that several approximations leadingto Eq. (1) are not that well justified. In particular, per-turbative adiabatic elimination of the excited states isa fairly crude approximation, with excited state popula-tions on the order of several percent being typical for theexperimental parameters used here. In the supplemen-tal material we give the Hamiltonian with the excited

ηΩ/δ

μ D13/2

S11/2

S01/2

|↑⟩

|↓⟩

435 nm

(a)

|e− ⟩

|e+ ⟩

|e0⟩

B(b)

(c)

intensity

σ +

|↓↑⟩

Re⟨ a⟩

Im⟨ a⟩

|↑↓⟩

x

y

z

(d) Gate sequence:waveμ

Loop 1 Loop 2

waveμ

R− x(π)Rx(π)time

A B

FIG. 1. (a) Level scheme used to implement a light-shift (LS)gate on a clock-state qubit. (b) Experimental setup: Due tothe orthogonal polarizations there is no intensity modulation,but there is a polarization gradient that induces a spatialmodulation of the AC-Stark shift experienced by an ion inthe |↑〉 state. (c) Gate operation: When the two ions areseparated by an even integer multiple of π/|∆k| only the |↑↓〉and |↓↑〉 states are driven, undergoing circular trajectories inphase space that return to their origin at a time 2π/δ. (d)We use a two-loop gate with spin echo in order to cancel anyresidual time-averaged AC-Stark shift during the gate.

states included, which is used for all numerical simula-tions. Importantly, we show that shaping the SDF pulsescan reduce residual excited state population at the endof the gate to significantly below the 10−4 level.

To understand why the gate can be operated with highfidelity at modest laser powers, it is helpful to first con-sider the effect of the excited state linewidth on laser-based trapped ion gates in general (these considerationsapply to both the MS gate and the LS gate). Gate fi-delities are limited in part by technical issues, such asfluctuations of experimental parameters, which in gen-eral are more detrimental for longer gates, and by off-resonant couplings that are more detrimental for shortgates. It therefore makes sense to demand some fixedgate time tg that minimizes these errors, which fixes the

size of δ = 2π/tg√K (for a gate comprised of K succes-

sive phase space loops); optimal values of tg are typicallyof order 10’s of µs. Since δ must be proportional to theSDF in order to achieve the desired phase Φ, we haveg2/∆ ∝ t−1

g [14]. Remembering from Fermi’s golden rulethat g2 ∝ PΓ, with P the laser power and Γ the excitedstate linewidth, this implies a proportionality constraintP (Γ/∆) ∝ t−1

g . A more fundamental source of gate er-ror for laser-based gates is spontaneous emission from

3

the excited state: For an excited state decay rate Γ, thegate error incurred due to spontaneous emission obeysε ∝ Γ(g2/∆2)tg ∝ (Γ/∆). Thus demanding a fixed mini-mum infidelity of ε fixes the ratio Γ/∆. Substituting thisback into the gate-time constraint we find that the laserpower should satisfy [15]

P ∝ 1/(εtg), (2)

which is independent of the linewidth Γ. There is onecatch: since g2/∆ is fixed by the gate time, the excitedstate population obeys Pe ∝ (g/∆)2 ∝ 1/(tg∆). Asthe linewidth narrows, ∆ must decrease in proportionto maintain the gate time at fixed laser power, implyingthat the excited state population must grow. This behav-ior is the underlying reason why adiabatic elimination ofthe excited state is not a particularly good approximationgiven modest laser power, as mentioned earlier.

Experimental gate demonstration.—Two 171Yb+ ionsare loaded into a nearly-harmonic trap with frequen-cies (ωx, ωy, ωz) = 2π×(1.16, 2.57, 3.05) MHz by Dopplercooling in a 5.57 Gauss magnetic field. The qubitstates |↑ (↓)〉 are encoded in the 2S1/2 hyperfine lev-els |F = 1(0),mF = 0〉 of the 171Yb+ electronic groundstate, where F and mF are the total angular momen-tum and its projection along the quantization axis (seeFig. 1a). The Yb ions are initialized to |↓〉 via opticalpumping [16]. We then Raman sideband cool the axialstretch mode (which serves as our gate mode) to a mo-tional occupation of n ∼ 0.1 quanta, and all other modesto n < 0.2. Global single qubit operations are realizedvia microwave radiation resonant with the hyperfine fre-quency of approximately 12.6428 GHz, and an averagesingle-qubit gate error of 9(6)× 10−5 for a single 171Yb+

ion is measured by randomized benchmarking [17]. Two-qubit operations are realized via two beams propagatingat a relative angle of 90, such that the wave vector dif-ference ∆k is parallel to the axis of the two-ion crystal,and detuned from each other by µ = ωstr+δ. To calibratethe gate, we first measure the axial trap frequency andthen set µ based on the desired gate time. We use a two-loop gate with spin echo (Fig. 2b) for a total gate time oftgate = 2(tloop + tπ), and choose tloop = 45µs, which inturn fixes δ = 2π/tloop ≈ 2π×22.2 kHz. The gate fidelityis then optimized by tuning the laser power, which in gen-eral is about 50 mW per beam for a spot size of ≈ 35µmat the ion positions. For each loop, the laser intensity isramped on and off with a sin2(t) profile over a 2µs dura-tion [18]. After gate operations, the ions are illuminatedwith a laser beam resonant with the 52S1/2 → 52P1/2

transition, and the qubit state is inferred from the result-ing fluorescence [16]. For a detection duration of 400µs,we detect on average approximately 9 photons for eachion in |↑〉 and 0.1 photons for each ion in |↓〉, leading toa detection fidelity of ∼ 99%.

In the apparatus used for this work, we do nothave easy access to high-resolution individual address-

(a)

(b) Gate sequence:waveμ

Loop 1 Loop 2

waveμ

Rx(π)Rx(π)

(c)

FIG. 2. Experimental gate fidelity measurement using theprocedure described as SRB-lite in Ref. [6]. The width ofthe shaded regions reflects the fraction of random Clifford se-quences resulting in the corresponding average survival, andthe line is from an exponential fit (see supplemental mate-rial for details). Making conservative assumptions describedin the supplemental material, we estimate the average gatefidelity to be 99.74(4)%.

ing, making full randomized benchmarking challenging.However, arbitrary global rotations can be performed us-ing microwaves, and together with the LS gate give us ac-cess to the complete symmetric subspace of the two-qubitHilbert space spanned by |↓↓〉 , (|↓↑〉 + |↑↓〉)/

√2, |↑↑〉.

We therefore benchmark the gate fidelity experimentallyusing the subspace randomized benchmarking protocolof Ref. [6] (specifically the protocol described therein asSRB-lite) restricted to this symmetric subspace. Theprocedure consists of applying random sequences ofqutrit-Clifford gates with varying sequence lengths. Eachsequence is designed to return the qubits to a known statein the absence of errors, and from the decay of the re-turn probability with sequence length one can extractthe process fidelity across the symmetric subspace of thegates involved. The error of the LS-gate can be extractedfrom the Clifford-gate fidelity (under certain reasonableassumptions [6]) by simple counting arguments. Fromthe data in Fig. 2b, we estimate an LS-gate average fi-delity of 99.74(4)%.

Theoretical fidelity limit and experimental errors.—Non-technical errors associated with this gate are mostlysimilar to those encountered in operating either an LS orMS gate on a P -state transition, namely: (1) deviationsfrom the Lamb-Dicke regime, (2) spontaneous emission,and (3) failure of the various rotating-wave approxima-tions (i.e. the presence of off-resonant couplings). Errorsdue to (1) are essentially the same as for P -state gates,and can be suppressed to below 10−5. Errors due to (2)and (3) require additional scrutiny. In everything thatfollows, we define the gate error as ε = 1 − Favg, withFavg being the average fidelity.

Raman scattering is an important error mechanism for

4

90 91 92 93 94

10-7

10-6

10-5

10-4

0.001

0.010

0.100

time ( )μs

Popu

latio

n

1.5 × 10−5

7 × 10−7

FIG. 3. Contributions to the gate infidelity from off-resonantcouplings to: radial modes (blue), the axial c.o.m. mode(green), and the D-states (purple). All of these curveshave been computed numerically, and filtered with a moving-average to remove structure at the time scale of 1/∆ (forvisual clarity). The gate starts at time t = 0, but we onlyshow the final ∼ 5µs in order to resolve the effect of pulseshaping in reducing transient populations. Dashed lines aretheoretical estimates in the absence of pulse shaping; withpulse shaping (solid lines) the total errors from all such cou-plings can be brought well below 10−4, with the single largestcontributor being the c.o.m. coupling.

all optical gates, and while it can in principle be madeextremely small given enough laser power, in practice itis often quite important (contributing nearly half of the≈ 10−3 infidelity in Ref. [2]). Because the D-state gatemust operate at single photon detunings ∆ ∆hf toachieve reasonable gate speeds at feasible laser powers,any photon scattering event (Raman or Rayleigh) off theD-state unambiguously determines the qubit to be in the|↑〉 state, so we take the associated error to be simply thetotal scattering rate multiplied by the gate time. This

approach gives εDγ ≈ 2π(Γ/∆) ×√Kη . For 100 mW total

laser power focused to a 35µm spot size, we find that a100µs 2-loop gate incurs an error εDγ ≈ 6 × 10−5. Asfor any light shift gate, this error can in principle bemade smaller by using more laser power and detuningfurther. Ultimately the spontaneous emission limit is setby Raman scattering off the P states, which causes anerror εPγ that increases for larger ∆ assuming the gatetime is held constant. Optimizing over ∆, we find thata minimum spontaneous emission error εγ = εDγ + εPγ ≈7 × 10−6 is achievable in principle, though we note thatnearly 2 W of laser power would be required for the samelaser spot size.

There are two types of off-resonant couplings that limitthe gate fidelity in this scheme: two-photon couplingsthat oscillate near motional frequencies, and single-photon couplings that oscillate at or near the single-photon detuning ∆. The former are dominated by atime-dependent light shift (with no coupling to the mo-tional modes), with the next-leading contribution beingcoupling to the axial center of mass (c.o.m.) mode. The

latter are dominated by carrier transitions directly to the|e±〉 states, but there are also non-negligible contribu-tions from coupling to the blue sidebands of |e−〉 [19].For any of these couplings, we can estimate the inducedgate error as simply being equal to the population trans-fer into the associated state. For a square gate pulse,these populations can in turn be estimated analyticallyusing second-order perturbation theory, and are shownas dashed lines in Fig. 3. While some of these errors canbe quite large, they can be significantly suppressed byshaping the gate pulses. The solid lines in Fig. 3 showthe associated populations during the gate, along withtheir final values, when the gate beams are turned onand off using a sin2(t) profile with a duration of 2µs, asin the experiment. The theoretical gate error estimatesreported in the top of Tab. I are obtained by numeri-cal simulation of the gate Hamiltonian without makingthe Lamb-Dicke approximation, assuming ground-statecooling of all modes, and including all of the off-resonantcouplings described above; at our current laser power andbeam waists, we estimate that ε < 10−4 should be possi-ble, with significant further improvements possible withmore laser power.

Mechanism Gate error (×10−4)

Spontaneous emission 0.6Off-resonant + L.D. errors 0.2Total 0.8Min 0.27Leakage ∼ 3Laser phase noise ∼ 3Gate mode heating < 4c.o.m. heating 1Microwaves 2Total ε <∼ 12

TABLE I. Top section: Non-technical contributions to thetotal gate error (see text for definitions) for the parametersand pulse-shaping used in the experimental gate demonstra-tion. Total indicates the expected gate infidelity due to non-technical errors for the experimentally available laser power(100 mW), while Min refers to the theoretical lower limitlimit, which requires 2 W of laser power. Bottom section:estimated contribution of various sources of technical noise tothe total experimentally measured gate infidelity.

The experimental gate error of 2.6(4)×10−3 is consid-erably larger than the theoretical limit. In the bottomsection of Tab. I we list technical sources of error thatwe are aware of, which account for approximately half ofthe experimentally measured gate error (we believe thatSRB-lite likely underestimates the actual gate fidelity,as the return probability in Fig. 2 gets pulled down byheating during long gate sequences, causing the fit tooverestimate the initial per-gate error). Contributions tothe error described as “leakage” come from both sponta-neous emission and any residual population remaining inthe D states at the end of the gate; causes of this residual

5

population are due in large part to a small coupling of|↑〉 → |e0〉 that exists for imperfect polarization of thebeam pointing along the quantization axis. Laser phasenoise contributes to the gate error primarily by inducingfluctuations in the time-averaged AC-Stark shift duringthe gate, which the spin-echo fails to perfectly null. Amore detailed discussion of these error sources and esti-mates can be found in the supplemental material.

Outlook.—The relatively low technical overhead in im-plementing a D-state LS gate provides a promising pathtowards high-fidelity gates in a scalable architecture. Inparticular, we believe that modest improvements to bothtrap noise and laser line width should enable two-qubitgates operating at fidelities approaching or even exceed-ing 99.99% fidelity. Ideally, one would like to implement auniversal gate set while maintaining the technical simpli-fications of this two-qubit gate (i.e. without introducingUV lasers or GHz frequency modulation). In principle,one could combine global microwave rotations with lo-cal z-rotations induced by the gate lasers to implementarbitrary single-qubit rotations, though understandingthe feasibility and achievable fidelity with this approachwould need further study. Alternatively, at the addedcost of modulating the gate lasers at the hyperfine split-ting one could readily implement high-fidelity all opticalsingle-qubit gates.

Acknowledgements.—We thank Patty Lee and Hart-mut Haffner for helpful discussions.

∗ bryce.bjork@honeywell.com† michael.feig@honeywell.com‡ daniel.stack@honeywell.com

[1] A. Sørensen and K. Mølmer, Phys. Rev. Lett. 82, 1971(1999).

[2] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler,A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried,and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).

[3] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Bar-rett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer,T. Rosenband, and D. J. Wineland, Nature 422, 412(2003).

[4] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol,and D. M. Lucas, Phys. Rev. Lett. 117, 060504 (2016).

[5] T. P. Harty, D. T. C. Allcock, C. J. Ballance, L. Guidoni,H. A. Janacek, N. M. Linke, D. N. Stacey, and D. M.Lucas, Phys. Rev. Lett. 113, 220501 (2014).

[6] C. H. Baldwin, B. J. Bjork, J. P. Gaebler, D. Hayes, andD. Stack, arXiv[quant-ph]:1911.00085 (2019).

[7] P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan,L.-M. Duan, and C. Monroe, Journal of Optics B: Quan-tum and Semiclassical Optics 7, S371 (2005).

[8] C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. De-Marco, A. Ben-Kish, R. B. Blakestad, J. Britton, D. B.Hume, W. M. Itano, D. Leibfried, R. Reichle, T. Rosen-band, T. Schaetz, P. O. Schmidt, and D. J. Wineland,Phys. Rev. Lett. 95, 060502 (2005).

[9] L. Aolita, K. Kim, J. Benhelm, C. F. Roos, andH. Haffner, Phys. Rev. A 76, 040303 (2007).

[10] S. X. Wang, G. Hao Low, N. S. Lachenmyer, Y. Ge, P. F.Herskind, and I. L. Chuang, Journal of Applied Physics110, 104901 (2011).

[11] C. Sorace-Agaskar, S. Bramhavar, D. Kharas, W. Loh,P. W. Juodawlkis, J. Chiaverini, and J. M. Sage, Fron-tiers in Biological Detection: From Nanosensors to Sys-tems X 10510, 36 (2018).

[12] A similar approach may work for other electronically sim-ilar ions (i.e. with a low-lying D level).

[13] Note that the beam configuration chosen for this experi-ment and drawn in Fig. 1(b) is just one of many possibil-ities, e.g. the beams could be counter-propagating, and∆k could be oriented along a radial crystal mode.

[14] This entire discussion holds only for g <∼ ∆, which is theregime we are interested in. We note that a similar gatecan be operated even outside this regime, as shown inRef. [20].

[15] The required power also has a wavelength dependencethat has been ignored in this expression, and slightly fa-vors the longer wavelengths encountered in aD-state gaterelative to a P -state gate.

[16] S. Olmschenk, K. C. Younge, D. L. Moehring, D. N. Mat-sukevich, P. Maunz, and C. Monroe, Phys. Rev. A 76,052314 (2007).

[17] D. Hayes, D. Stack, B. J. Bjork, A. C. Potter, C. H.Baldwin, and R. P. Stutz, arXiv[quant-ph]:1912.13131(2019).

[18] The loop time is defined as the time between the middleof the pulse rise and the middle of the pulse fall. Whileδ = 2π/tloop is not exactly optimal for non-zero rise time,it is an excellent approximation and contributes no ap-preciable error for a two-loop gate.

[19] Near-ground-state cooling makes coupling to red side-bands of |e+〉 much less important. Note that heating ofspectator modes that occurs during longer gate sequencesin SRB may lead to enhanced off-resonant coupling er-rors, which increase with the mean occupation numberof those modes.

[20] K. Kim, C. F. Roos, L. Aolita, H. Haffner, V. Nebendahl,and R. Blatt, Phys. Rev. A 77, 050303 (2008).

SUPPLEMENTAL MATERIAL

Approximations leading to the gate Hamiltonian

In the manuscript, for simplicity we wrote the gateHamiltonian after a number of approximations had been

6

made, some of which are not necessarily that well jus-tified. Here we give the microscopic Hamiltonian usedfor our numerical simulations, and describe the approx-imations resulting in Eq. (1) in more detail. Ignoring

terms that rotate at optical and hyperfine (GHz) fre-quencies, but keeping all optical excited states and mo-tional modes, the Hamiltonian can be written H =Hphonon +Hion +Hlaser, where

Hphonon =∑α,ν

ωα,ν a†α,ν aα,ν (S1)

Hion = ∆∑j=1,2

(|e+,j〉 〈e+,j | − |e−,j〉 〈e−,j |

)(S2)

Hlaser =∑j=1,2

∑τ=±

((gτAe−i(kA·rj+µ

2 t+φA) + gτBe−i(kB ·rj−µ2 t+φB)

)|ej,τ 〉 〈↑j |+ h.c.

). (S3)

Here ωα,ν is the frequency of the αth normal mode ofmotion of the two ion crystal along the νth principalaxis of the trapping potential, and aα,ν(a†α,ν) is the an-nihilation(creation) operator for that mode. The opti-cal phases of lasers A and B are denoted φA and φB(see Fig. 1 of the manuscript for a definition of the twobeams), and we have chosen to work in a frame for which|↑〉 and |e0〉 have zero energy. The matrix element gτA(B)

is proportional to the electric field amplitude of laserA(B), and (because this is a quadrupole transition) de-pends on both the polarization and wave-vector of thatlaser. Note that we do ignore coupling to the |e0〉 statein Eq. (S3), and also to the F = 2 states of D3/2. Im-perfections of the former approximation are discussed inthe section titled “Estimates of experimental error”. Thelatter generates an AC-Stark shift of the qubit frequencyon the order of 1 kHz; for a two loop gate with spin echo,and considering the intensity stability of our lasers, wefind that this shift should not contribute an appreciableerror.

It is common to perturbatively eliminate the ex-cited states, under the assumption that g ∆.Assuming |gτA| = |gσB | ≡ g and using the factthat, for our chosen beam directions and polarizations,arg(g−A , g

+A , g

−B , g

+B) = 0, 0,−π/2, π/2, this procedure

yields H ≈ Hphonon +HLS , with

HLS =4g2∆

∆2 − µ2/4

∑j=1,2

|↑j〉 〈↑j | sin(∆k · rj + µt+ δφ

).

(S4)

Here δφ ≡ φA−φB is the phase of the beat note betweenthe two gate lasers. To arrive at Eq. (1) in the maintext, we rewrite the νth cartesian component of the posi-tion operators (decomposed in a basis determined by thecrystal principal axes) in terms of creation/annihilationoperators for the crystal normal modes along that direc-

tion as

rνj = 〈rνj 〉+∑α

ηα,νj (a†α,ν + aα,ν). (S5)

Here ηα,νj = ∆kνBα,νj

√~/(2mωα,ν), Bα,νj is the com-

ponent of the αth normal mode along the νth princi-pal axis onto the jth ion, and ωα,ν is the frequency ofthat mode. Now expanding the sin() function in Eq. (S4)to lowest order in the Lamb-Dicke parameters, using∆/(∆2 − µ2/4) ≈ 1/∆, and defining Ω = g2/∆, we find

HLS ≈ 2Ω∑j=1,2

(1 + σzj ) sin(µt+ ϕj) (S6)

+ 2Ω∑j=1,2

(1 + σzj ) cos(µt+ ϕj)∑α,ν

ηj,ν(aα,ν + a†α,ν).

The first term is a time-dependent AC-Stark shift thatnominally contributes a phase shift to the qubit of orderΩ/µ, which is typically on the order of 0.1 in the experi-ments reported here. However, as will be discussed belowthis phase shift is greatly reduced by pulse shaping, andonly enters quadratically into the total gate error. Thegate is operated by choosing µ near (∼ 20kHz) detunedfrom the axial stretch mode; ignoring all terms abovethat oscillate faster than δ leads to Eq. (1) in the maintext.

Numerical gate simulations

Because we symmetrically detune the two lasers withinthe excited state manifold, the detuning ∆ is controlledentirely by the magnetic field. For the reported exper-iments, we chose a field of 5.57Gauss (corresponding toa single-photon detuning ∆ ≈ 2π × 7.8MHz). For thisdetuning, achieving a two loop gate in tgate ≈ 100µsrequires |g| ≈ 2π × 0.76MHz; thus we expect excitedstate populations to be on the order of (|g|/∆)2 ∼ 1%,

7

making perturbative adiabatic elimination not very welljustified (at the level of precision we care about for high-fidelity gates). Therefore, we chose to directly simulatethe Hamiltonian as written in Eqs. (S1-S3), with theexcited-state structure fully intact.

The Hilbert space dimension is given by

D = Dion ×Dphonon = 42 ×6∏

α=1

(nmax,α + 1), (S7)

with nmax,α the Fock-space occupation cutoff for modeα. Keeping nmax = 10 for the gate mode (which wefind to be sufficient) and nmax = 1 for all other modesleads to D = 5600. Considering the poor conditioningof the Schrodinger equation (large separation in timescales) caused by keeping the excited states, we foundthat numerical simulations in this full space are feasiblebut very time consuming. Instead, we assume that errorsdue to the spectator modes are small and uncorrelatedto lowest order, justifying the following procedure: (1)calculate the gate error by including only the c.o.m. as aspectator mode and optimizing over the laser power, andthen (2) repeat the calculation for each of the four radialspectator modes at the same parameters, each time cal-culating their population at the final gate time, (3) addthose populations to the gate error estimate obtained in(1). We note that the radial-sideband contributions tothe gate error are relatively small compared to the con-tribution from the axial c.o.m., and in principle could begreatly reduced by moving the beams towards a counter-propagating geometry.

Estimates of experimental error

Leakage errors are quantified by preparing both ions inthe |↑〉 state and repeatedly applying the gate sequence,except that we apply only one gate beam or the otherto quantify leakage due to each beam independently. Inaddition to the leakage caused by spontaneous emission,we expect to accumulate some population in the D state(or into other hyperfine states if it relaxes) due to fail-ure of pulse shaping. We observe a leakage rate of about3× 10−4 per gate, only about one quarter of which is ex-plained by spontaneous scattering during the gate time.Subtracting out leakage due to spontaneous emission,we find a leakage rate for the two different beams ofRA ≈ 1.7 × 10−4 and RB ≈ 0.7 × 10−4. The discrep-ancy between the two beams is consistent with an ex-pected 1 × 10−4 leakage rate for beam A due to a weakdirect coupling from |↑〉 → |e0〉 (for beam B this couplingis nominally forbidden by both the polarization and k-vector choices, and is extremely small), which we inferfrom a measured Rabi frequency for that transition of2π× 6 kHz. The remaining leakage of slightly more than1× 10−4 per gate is not understood, but may be the re-

sult of either (1) fast laser phase noise (on the timescaleof ∆) that interferes with the efficacy of pulse shapingand (2) heating during the long sequences used to mea-sure this leakage, which increases coupling to sidebandsof the excited states (finite mode coherence times willprevent all population from being pulse shaped out ofthese excited-state sidebands).

Laser phase noise enters into the gate infidelity in sev-eral possible ways depending on how fast it is. Noisethat is fast compared to the inverse single-photon detun-ing will reduce the efficacy of pulse shaping and lead toleakage, as described above. Noise that is slow comparedto this time scale will primarily impact the gate via in-duced fluctuations in the time-averaged AC-Stark shift.We quantify this effect by performing a Ramsey spin-echoexperiment on the clock transition |↑〉 → |e0〉 using thegate lasers (using a clock transition enables us to isolatethe effect of laser phase noise from magnetic field noise)with the same duration as the gate time. The proba-bility to return to the incorrect state after the Ramseysequence can be related to laser phase noise by the filterfunction F(ω) for the spin-echo sequence as

εramsey =

∫ ∞0

Sφ(ω)F(ω)dω. (S8)

Here Sφ(ω) is the spectral density of phase fluctua-tions of the laser. During the gate, laser frequencyfluctuations δω(t) that are slow compared to 1/∆ re-sult in a fluctuating AC-Stark shift, with the relationδ∆LS(t) ≈ δω(t)

[4g2/∆2

]. Since the gate is performed

with a spin echo separating the same wait times as theRamsey sequence described above, gate errors due tothe AC-Stark shift fluctuations can be written in termsof the same filter function as above. The only differ-ence is that the spectral density of noise gets an overall

scale factor of[4g2/∆2

]2, which enables us to extract

an expected gate error due to laser phase fluctuations of

εφ = εramsey ×(4g2/∆2

)2 ≈ 3× 10−4.

The effect of gate mode heating is calculated using theexpression in Ref. [4] and an experimentally inferred up-per bound on the gate mode heating rate. Heating of thec.o.m. mode is also considered because this mode has arelatively large transient population during the gate (seeFig. 3 in the main text), and the heating rate is large.Assuming that absorption of a single phonon preventspulse shaping from removing this population, we obtainan error estimate εc.o.m. ∼ κc.o.m.tgate × Pc.o.m., whereκc.o.m. is the heating rate of the c.o.m. mode and Pc.o.m.

is its transient population. We measure κcom ≈ 3 × 103

quanta/s, and numerically calculate a transient popula-tion of Pcom ≈ 3×10−4, giving an error of about 1×10−4

for a 100µs gate.

Errors due to imperfect microwave pulses are esti-mated by single-qubit randomized benchmarking of themicrowave gates [6]. We further verify the contribution of

8

microwave gates to SRB by repeating the SRB sequenceswithout LS gates and choosing an appropriate final gate,which consists only of microwaves, to ideally return tothe initial state. We call this procedure SRB gap bench-marking and while it cannot be used to extract a fidelityof the microwave gates it is useful for interpreting theeffects the single-qubit gates on the return probabilityin SRB. Under a simple single-qubit depolarizing errormodel, we can calculate the expected survival in SRB gapbenchmarking from single-qubit randomized benchmark-ing data. For 10 and 65 qutrit-Clifford gates this yieldsan expected survival of 0.982 and 0.965 compared to themeasured return probability from SRB gap benchmark-ing of 0.984(4) and 0.955(9). Therefore, for SRB withLS gates we expect the single-qubit randomized bench-marking estimate to accurately estimate the effects of themicrowave gates.

Experimental fidelity estimation

The SRB procedure used in this paper is similar tothe SRB-lite procedure proposed in Ref. [6]. In SRB-lite,

leakage errors, which cause population to move outside ofthe symmetric subspace, are ignored. This is a good ap-proximation for the current work since state preparationerrors and leakage are believed to be small, and thereforedo not contribute much to the fidelity. Even with leak-age, the estimate from SRB-lite provides an estimate oferror rates across most of the symmetric subspace. Theone difference between SRB-lite and the procedure usedhere is that we fix the asymptote of the decay curve to1/3 when fitting instead of leaving it as a free parameter.This reduces the number of fit parameters and allows usto fit data with shorter sequence lengths. Leakage orSPAM errors may cause the asymptote to be less than1/3 but we believe these effects to be small. From nu-merical simulations we see fixing the asymptote to 1/3actually tends to return a lower fidelity estimate thanleaving it as a free parameter. We ran this SRB proce-dure with three sequence lengths [1, 3, 7] each with 33random sequences.