A high-fidelity method for a single-step N -bit Toffoli gate in trapped ions
Juan Diego Arias Espinoza, Koen Groenland, Matteo Mazzanti, Kareljan Schoutens, Rene Gerritsma
AA High-Fidelity Method for a Single-Step N -bit Toffoli Gate in Trapped Ions Juan Diego Arias Espinoza, Koen Groenland, Matteo Mazzanti, Kareljan Schoutens,
2, 3 and Rene Gerritsma Van der Waals-Zeeman Institute, Institute of Physics,University of Amsterdam, 1098 XH Amsterdam, the Netherlands QuSoft Science Park 123, 1098 XG Amsterdam, the Netherlands Institute for Theoretical Physics, Institute of Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, the Netherlands (Dated: October 19, 2020)Conditional multi-qubit gates are a key component for elaborate quantum algorithms. In a recentwork, Rasmussen et al. (Phys. Rev. A 101, 022308) proposed an efficient single-step method for aprototypical multi-qubit gate, a Toffoli gate, based on a combination of Ising interactions betweencontrol qubits and an appropriate driving field on a target qubit. Trapped ions are a natural platformto implement this method, since Ising interactions mediated by phonons have been demonstratedin increasingly large ion crystals. However, the simultaneous application of these interactions andthe driving field required for the gate results in undesired entanglement between the qubits and themotion of the ions, reducing the gate fidelity. In this work, we propose a solution based on adiabaticswitching of these phonon mediated Ising interactions. We study the effects of imperfect groundstate cooling, and use spin-echo techniques to undo unwanted phase accumulation in the achievablefidelities. For gates coupling to all axial modes of a linear crystal, we calculate high fidelities ( > N -qubit rotations with N = 3-7 ions cooled to their ground state of motion and a gate timebelow 1 ms. The high fidelities obtained also for large crystals could make the gate competitivewith gate-decomposed, multi-step variants of the N -qubit Toffoli gate, at the expense of requiringground state cooling of the ion crystal. I. INTRODUCTION
Quantum computers promise dramatic speedups in avariety of disciplines [1–5], but remain challenging toscale up in practice. A major obstacle to executing elab-orate quantum algorithms, is the need for gates that actconditionally on a large number of qubits. The prototyp-ical example of such a gate is the N -qubit Toffoli gate,which flips a single ‘target’ qubit if and only if all N − | (cid:105) . Even though quan-tum devices with over 50 qubits have been reported [6, 7],the largest Toffoli gate ever performed is, to our bestknowledge, the case N = 4 [8]. This gap is surpris-ing, because Toffoli gates (or equivalents) are essentialingredients of many basic computation steps, such as el-ementary arithmetic [9–11], error correction [12], and theGrover diffusion operator [13].Two different strategies exist to implement Toffoligates. The first consists of decomposing a single N -qubitToffoli gate into a circuit consisting of one- and two-qubitgates [14–16] or multiqubit gates, such as the Mølmer-Sørensen gate in trapped ions [17–19]. The second ap-proach is to perform the gate in a single step using inter-actions that are native to the specific platform [20–23]. Inparticular, a recent proposal [23] has demonstrated thatby exploiting systems with an all-to-all Ising interactionin combination with a drive field on a single target qubitan i -Toffoli gate can be implemented. This gate differsonly from the regular Toffoli by a phase + i on the targetqubit.Trapped ions are a natural candidate to implementthis proposal, as intrinsic Ising interactions have beendemonstrated in increasingly large ion crystals [6, 17, 24–26]. Moreover, quantum operations have beendemonstrated [27, 28] with fidelities higher than 99.9%.Ising interactions generally arise from qubit-phononcouplings ˆ H q-ph generated from state-dependent laser-induced forces on the ions. Combining this mechanismwith the driving field ˆ H drive required for an i -Toffoli gateposes a problem, as both process do not commute i.e.[ ˆ H q-ph , ˆ H drive ] (cid:54) = 0. As a result, the qubit states andthe motion of the ions remain entangled at the end ofthe gate sequence, which leads to fidelity loss. This ef-fect could be mitigated by restricting the strength of thespin-phonon coupling such that the phonons are only vir-tually excited [26]. However, limiting the strength of theIsing interactions leads to undesirably long gate times.In this work, we show that this residual qubit-phononentanglement can be suppressed by adiabatic ramping ofˆ H q-ph . In this way, the i -Toffoli gate operates on thedressed eigenstates of ˆ H q-ph , that are adiabatically con-nected to the Fock eigenstates of the non-interacting sys-tem. The benefit of this approach is that the effectiveIsing interaction strength does not have to be limited tothe regime of virtual phonon excitation. We show thathigh-fidelity ¯ F > i -Toffoli gates shouldbe possible with up to 7 ions at gate times ∼ µ s.We start in Sec. II with the derivation of the modelfor a N -qubit i -Toffoli gate for a system of trapped ionsand introduce our proposal for adiabatic preparation ofdressed states. In Sec. III we analyze the results of nu-merical simulations for a linear 3 crystal and considerthe role of inhomogeneous Ising interactions mediated bymultiple phonon modes. We discuss the implementationof a method based on multi-frequency laser fields [29] toeliminate undesired phases originating from these inho- a r X i v : . [ qu a n t - ph ] O c t mogeneous interactions. Finally, in Sec. IV we calculatethe fidelities for 3-9 qubits gates and discuss sources oferrors and ways to mitigate them. We also consider theeffects of imperfect ground state cooling. II. MODEL OF A N -QUBIT TOFFOLI GATE INTRAPPED IONSA. Single step N -qubit i -Toffoli gate Briefly, the proposal [23] requires qubits coupled via anIsing interaction of the form ˆ H Ising = (cid:80) Nij J ( i,j ) ˆ σ ( i ) z ˆ σ ( j ) z with ˆ σ ( i ) r the Pauli matrix acting on ion i , and J ( i,j ) the strength of the interaction field [30]. Including adrive field of frequency ω g with strength g acting on thetarget qubit, ˆ H drive = g ˆ σ (t) x cos ( ω g t ), and the energy ofthe non-interacting qubits, ˆ H = ω / (cid:80) i ˆ σ ( i ) z , a simpleHamiltonian is obtained:ˆ H T = − ν (cid:88) i ˆ σ ( i ) z + (cid:88) i (cid:54) = j J ( i,j ) ˆ σ ( i ) z ˆ σ ( j ) z + g σ (t) x , (1)where we transformed into the interaction picture withrespect to ω g , using ˆ U = exp (cid:16) − i ω g t (cid:17) . We also de-fine ν = ω g − ω with ω the energy spacing betweenqubit states (or eigenstates of ˆ σ z ). These eigenstatesand their energies (Fig. 1) can be labeled as | x t , (cid:126)x c (cid:105) and E | x t ,(cid:126)x c (cid:105) with x t describing the state of the target qubitand (cid:126)x c is the string describing the state of the controlqubits. In particular, the two target states labelled as | , N c (cid:105) , | , N c (cid:105) , where N c correspond to the number ofcontrol qubits, correspond to those that are coupled bythe action of the Toffoli gate.The driving field frequency ( ω g ) is chosen such thatit resonantly couples these two states, i.e. ∆ Nc = E | , Nc (cid:105) − E | , Nc (cid:105) = ω g . According to Eq. 1 the energygap for any pair of states with equal control bits can bewritten as: ∆ (cid:126)x c = 4 N c (cid:88) i =1 J (t ,i ) ( − (cid:126)x i + ω , (2)where x i denotes the state of the i -th controlqubit. The resonant condition becomes then ν =4 (cid:80) N c i =1 J (t ,i ) ( − (cid:126)x i , which for the target states implies ν = − (cid:80) N c i =1 J (t ,i ) .Evolution under the Hamiltonian of Eq. 1 for a (gate)time τ g = π/g leads to the desired i -Toffoli gate. To pre-vent accumulation of unwanted dynamical phases duringthe gate, timing restrictions can be considered, or an echopulse can be applied. Both will be discussed later in thistext. FIG. 1. Energies of non-interacting eigenstates ( J = 0)and interacting (dressed) states ( J > | (cid:105) , | (cid:105) are highlighted. Because their energy gapis unique, an appropriate drive field can couple the statesresonantly. B. Implementation in trapped ions
To achieve the required Ising interaction in trappedions, a qubit state-dependent force is generated with twonon-copropagating bichromatic lasers with beatnote fre-quency µ , which excites phonons in the ion crystal. Foran homogenous laser field extending over the full ioncrystal, the laser-ion interaction Hamiltonian is ˆ H q-ph = (cid:80) i F i exp( i(cid:126)k · ˆ (cid:126)r ( i ) ) + h.c.. Here F i = (Ω / e − iµt ˆ σ ( i ) z isa state-dependent interaction [31] with Ω the interactionstrength, (cid:126)k the resulting wavevector of the interferinglaser fields, and ˆ (cid:126)r ( i ) the position operator of ion i . With (cid:126)k · ˆ (cid:126)r ( i ) = (cid:80) m η ( i ) m (ˆ a † m + ˆ a m ) the Hamiltonian can be writ-ten as:ˆ H q-ph = Ω2 (cid:88) i (cid:16) e i (cid:80) m η ( i ) m ( ˆ a † m +ˆ a m ) − iµt + h.c. (cid:17) ˆ σ ( i ) z , (3)where the creation and annihilation operators for the m -th phonon mode are denoted by ˆ a † m and ˆ a m . The Lamb-Dicke parameter η ( i ) m is scaled with the motion amplitudeof the i -th ion on the m -th phonon mode ( (cid:126)b ( i ) m ), i.e. η ( i ) m = (cid:126)b ( i ) m · (cid:126)k (cid:112) (cid:126) / (2 M ω m ) with M the ion mass and ω m thephonon mode frequency.Including again the drive field ( ˆ H drive ) and the energyof the non-interacting system ( ˆ H ), the total Hamiltonianin the interaction picture of ω g becomes:ˆ H T = − ν (cid:88) i ˆ σ ( i ) z + (cid:88) m ω m ˆ a † m ˆ a m + Ω2 (cid:88) i (cid:16) e i (cid:80) m η ( i ) m ( ˆ a † m +ˆ a m ) − iµt + h.c. (cid:17) ˆ σ ( i ) z + g σ (t) x , (4)which includes a new (second) term for the motionalenergy of the system. Now the eigenstates of the non-interacting system have the form | Ψ (cid:105) = | Φ (cid:105) ⊗ | x t , (cid:126)x c (cid:105) ,with | Φ (cid:105) = (cid:78) m | n m (cid:105) the motional wavefunction of thesystem in the Fock space of the m phonon modes ofthe crystal. For this system we define the target statesfor the i -Toffoli gate as the ones corresponding to anion crystal cooled to its ground state, that is the twotarget states are | Ψ (cid:105) = (cid:78) m | n m = 0 (cid:105) ⊗ | , (cid:126)x c (cid:105) and | Ψ (cid:105) = (cid:78) m | n m = 0 (cid:105) ⊗ | , (cid:126)x c (cid:105) [32].Next, we simplify this Hamiltonian by going into theinteraction picture of the phonon mode frequencies withthe transformation ˆ U = exp (cid:16) − it (cid:80) m ω m ˆ a † m ˆ a m (cid:17) :˜ H T = − ν (cid:88) i ˆ σ ( i ) z + Ω2 (cid:88) i (cid:16) e i (cid:80) m η ( i ) m ( ˆ a † m e iωmt +h.c. − iµt )+h.c (cid:17) ˆ σ ( i ) z + g σ (t) x , (5)where high frequency terms (2 ω g ) were ignored. Wenow consider a system within the Lamb-Dicke limit andtransform the Hamiltonian into a new interaction pic-ture [33] with respect to δ m = µ − ω m using ˆ U =exp( − it (cid:80) m δ m ˆ a † m ˆ a m ):˜ H T,mm = − ν (cid:88) i ˆ σ ( i ) z + i Ω2 (cid:88) m (cid:88) i (cid:0) ˆ a † m − ˆ a m (cid:1) η ( i ) m ˆ σ ( i ) z − (cid:88) m δ m ˆ a † m ˆ a m + g σ (t) x . (6)To recover a Hamiltonian having the desired Ising in-teraction as in Eq. 1, we apply a Lang-Firsov transfor-mation [34–36] to introduce a dressed-state picture ofqubits entangled with phonon modes of the crystal. Thetransformation, ˆ U I = exp (cid:104) − i (cid:80) i,m α ( i ) m (ˆ a † m + ˆ a m ) (cid:105) , with α ( i ) m = (Ω η ( i ) m / δ m )ˆ σ ( i ) z , has the form of a displacementoperator that displaces the state of the system in phasespace by a state dependent magnitude of α m, Ψ = (cid:80) i α ( i ) m .The result of the transformation is:˜ H T,I = ˆ U † I ˜ H T,sm ˆ U I = − ν (cid:88) i ˆ σ ( i ) z + (cid:88) i (cid:54) = j J ( i,j ) ˆ σ ( i ) z ˆ σ ( j ) z − (cid:88) m δ m ˆ a † m ˆ a m + ˜ g σ (t) x , (7) with J ( i,j ) = Ω (cid:80) m η ( i ) m η ( j ) m / δ m , a corrected drivestrength, ˜ g , and a transformed drive term, ˜ˆ σ (t) x =ˆ U † I ˆ σ (t) x ˆ U I . Because the drive and the Ising terms donot commute, this transformation introduces a term ∝ α (t) m ˆ σ (t) y which couples the drive to ion motion andcan cause a gate error ∝ α (t) m . For weak (virtual) phononexcitation, α Ψ (cid:28)
1, such that ˜ˆ σ (t) x ≈ ˆ σ (t) x , this error issmall. However, this regime corresponds to very slowgates and we are here interested instead in the regimein which the corrections to ˆ σ (t) x have to be taken intoaccount, i.e. α Ψ (cid:39) g = g/λ Ψ (cid:48) , Ψ c accountsfor the non-unitary overlap of the motional part of the(dressed) eigenstates of Eq. 7. These states are displacedFock states, i.e. | Φ (cid:105) I = (cid:81) m ˆ D ( α m, Ψ ) | n m (cid:105) , which can beproduced adiabatically from the Fock states of the non-interacting system. The correction factor λ Ψ (cid:48) , Ψ c is equalto the overlap between the displaced states of any pairof states | Ψ (cid:48) (cid:105) , | Ψ (cid:105) . The overlap is dependent on their ini-tial phonon occupation number | n m (cid:105) and can be writtenas [37]: λ Ψ (cid:48) , Ψ c = (cid:89) m (cid:104) n (cid:48) m | ˆ D † ( α m, Ψ (cid:48) ) ˆ D ( α m, Ψ ) | n m (cid:105) = (cid:89) m e − β m / β | ∆ n | m m (cid:18) n m ! n (cid:48) m ! (cid:19) sign(∆ n m ) / L | ∆ n m | n m (cid:0) β m (cid:1) , (8)where ∆ n m = n (cid:48) m − n m and β m = α m, Ψ (cid:48) − α m, Ψ , L ( γ ) n ( β )is the associated Laguerre polynomial. Note that thedrive strength needed for implementing the correct gatedepends therefore explicitely on the motional input state.For the target states in their ground states of motion, | Ψ (cid:105) , | Ψ (cid:105) , the overlap simplifies to λ Ψ , Ψ c = (cid:81) m e − β m / with β m = Ω η (t) m /δ m and where L (cid:0) β m (cid:1) = 1. C. Adiabatic Preparation of States
To guarantee a complete inversion of the target qubit,the system has to be prepared in a pure dressed eigen-state | Ψ (cid:105) I of the interacting system such that the drivestrength can be exactly corrected using Eq. 8. In thecase of a sudden quench (diabatic activation) of Eq. 7,a superposition of dressed eigenstates will result. Incontrast, by adiabatic switching (see Appendix A) thequbit-phonon interaction, ˆ H q-ph , and thus ˆ H Ising , pure(dressed) eigenstates are obtained for which an appropri-ate drive strength can be chosen.It also makes our gate robust against residual phonon-qubit entanglement which in turn makes it less sensitiveto timing errors. For quenched gates, this residual en-tanglement occurs if the total gate time t T (cid:54) = 2 k π/δ m ( k ∈ N ), as in this case the evolution of the states do (b)(a) (c) FIG. 2. Time evolution of states under the action of ˜ H T,sm (a) Phase space trajectories (zoomed in) of motional wavefunctionduring evolution with ˆ U T . Note that the adiabatic ramp ensures that dynamics take place along the momentum axis in thisframe, as explained more in detail in Appendix A. (b) Real and (c) imaginary part of process unitary matrix for the motionalground state ( | n = 0 (cid:105) ) subspace. (d) Evolution in the Bloch sphere of the two resonant states, and (e) the projections along x( −· ), y ( −− ) and z ( − ) of the trajectory of initial state | (cid:105) . Time is indicated with the color intensity from light ( t = 0) todark ( t = τ g ) in (d). The gate parameters are δ CM / π = 20 kHz, J/ π = 2 kHz (Ω / π = 126 .
491 kHz), g/ π = 1 kHz for agate time of τ g = π/g = 500 µ s. not describe closed trajectories in phase space. In con-trast, the adiabatic ramp assures that the system remainsin an eigenstate during the laser-ion interaction. There-fore, the exact timing is not crucial as long as the ramptime is long enough to assure adiabaticity. In practice,however, setting t T = 2 k π/δ m still proves to be usefulto reduce errors due to off-resonant drive field couplingbetween dressed states and to reduce errors caused bynon-adiabaticity.The gate sequence consists then in ramping up the in-teraction for a time t a and performing the i -Toffoli gate(Eq. 7) for a time τ g , and finally ramp down the interac-tion to transform the system back to the non-interactingor computational basis. This complete i -Toffoli processhas a total length t T = 2 t a + τ g and is described by:ˆ U i Tof = ˆ U deg ˆ U T ˆ U aeg , (9)where ˆ U a(d)eg is the unitary of the adiabatic activation(deactivation) of ˆ H Ising and ˆ U T = exp( − iτ g ˜ H T,I ). III. SIMULATIONS OF A N -QUBIT TOFFOLIGATE IN A LINEAR ION CRYSTALA. Single mode coupling The main features of our model can be first studied byconsidering an ideal system. This consists of a ground-state cooled linear ion crystal and an interaction lasercoupling only to the axial modes of the crystal, witha beatnote µ tuned close to the center-of-mass phononmode frequency ω CM of the crystal, i,e. δ CM (cid:28) δ m (cid:54) =CM .We assume that the coupling with the remaining phononmodes can be ignored, i.e. J ( i,j )CM (cid:29) (cid:80) m (cid:54) =CM J ( i,j ) m .This results in an homogeneous Ising coupling strength J ( i,j ) = Ω η / δ CM ≡ J and the simplified Hamilto-nian:˜ H T,sm = 2 N c J (cid:88) i ˆ σ ( i ) z + J (cid:88) i (cid:54) = j ˆ σ ( i ) z ˆ σ ( j ) z + ˜ g σ (t) x − δ CM ˆ a † CM ˆ a CM . (10)The resulting i -Toffoli process unitary for a 3-ion crys-tal is observed in Figs. 2(b) and 2(c). We have chosen aramp time ( t a ) that ensures the adiabaticity of the pro-cess, and the disappearance of dynamical phases. Thesephases have the form φ t T = exp (cid:0) − iE | x t ,(cid:126)x c (cid:105) ˜ t T (cid:1) , wherethe total effective process time is ˜ t T = 2˜ t a + τ g and ˜ t a iseffective ramp time (See Appendix A). Because the Isingcouplings are homogeneous in this particular case, thephases vanish if ˜ t T J = 2 k π ( k ∈ N ). For a modula-tion of the form Ω( t < t a ) = Ω sin ( πt/ (2 t a )) and theseparameters both criteria are fulfilled by setting t a = τ g .To illustrate the dynamics under the action of Eq.10, we have plotted the phase space (Fig. 2(a)) [38]and Bloch sphere trajectories (Fig. 2(d)) of the (target)dressed states | Ψ (cid:105) I = ˆ U aeg | n = 0 (cid:105) ⊗ | x t , (cid:126)x c (cid:105) . As expectedfor the two target states, the motional and electroniccomponent are transformed from one to the other, i.e.ˆ D ( α Ψ ) | n = 0 (cid:105) ↔ ˆ D ( α Ψ ) | n = 0 (cid:105) and | , (cid:105) ↔ | , (cid:105) .For the off-resonant states, closed trajectories are ob-tained indicating that motion is disentangled from theelectronic component of the states. Finally, in Fig. 2(e)we observe that the coupling of drive with the ion motion,leads to a small drive error reflected as small oscillationsof (cid:104) ˆ σ x (cid:105) . B. Multi-mode coupling
In experiments, due to the finite spacing betweenphonon frequencies, the laser field will couple to mul-tiple phonon modes, as described in Eq. 6. Althoughthe dynamics of the gate will still be dominated by thecoupling to the center-of-mass mode, the contributions ofnearby modes, (cid:80) m (cid:54) =CM J ( i,j ) m , will lead to two additionalsource of errors. The first are additional terms ∝ α (t) m ˆ σ (t) y which increase the drive error, and the second are state-dependent dynamical phases. The latter occur becausethe Ising interactions are inhomogeneous, J ( i,j ) (cid:54) = J ( i,k ) ,thus the state energies are not longer proportional to asingle value of J . As a consequence, no single gate timecan be chosen such that they vanish at the end of thegate (Fig. 3(a)).The first error can be minimized by using a linear crys-tals with odd number of ions and by addressing the cen-tral ion with the drive field. In this way, the largest con-tribution, coming from the next nearest phonon mode,disappears. To cancel the second error, dynamical phasesare removed with an additional “echo” step. Duringthis step, the sign of all coupling strengths is inverted J ( i,j ) → − J ( i,j ) for a duration t T . To realize this echo,we follow a recent proposal [29] in which a combinationof multiple beatnotes coupling to all the phonon modesis used to generate couplings with arbitrary magnitudeand sign.In short, the method uses beatnotes with frequen-cies µ k that are harmonics of the interaction time ( t mb )between the crystal and a multi-beatnote laser field,i.e. µ k = 2 πk/t mb for k ∈ N . Their amplitudes Ω µ k (Figs. 3(b) and 3(c)) are calculated such that after a (a) (b)(c) FIG. 3. Multimode unitaries and spectrum of multiple beat-notes for “echo” step for phases cancellation. (a) i -Toffoliunitary for a 3 ion crystal considering all-mode couplingswithout and (inset) with “echo” step. Frequency and ampli-tude of beatnotes for (b) 3 and (c) 7 ions gate with detunings δ CM / π = −
20 kHz and δ CM / π = −
50 kHz respectively. Thephonon mode frequencies are indicated in dashed red lines.The parameters of (a) are ω CM / π = 1 MHz, δ CM / π = − g/ π = 1 kHz) and for (b,c) the interaction time is t mb µ s. time t mb the entanglement phases of each mode matchesa target value ϕ m , and both dynamical phases and theentanglement with the phonon modes disappear. Theentanglement phases are obtained by expressing the ma-trix of couplings for the echo step, ˜J i,j = − J ( i,j ) , in termsof the phonon modes ( (cid:126)b m ) and the target entanglementphase: ˜J (cid:117) N (cid:88) m =1 ϕ m (cid:126)b m ⊗ (cid:126)b m . (11)To reduce the number of beatnotes required, we chosean interaction t mb ∼ k π/ω CM for a small integer k ,that also satisfies t T = k t mb ( k ∈ N ). The “echo” isobtained by sequentially applying k multi-beatnote fieldpulses with the same modulation of the amplitudes Ω µ k as for the laser-ion coupling strength Ω (See AppendixB). IV. GATE FIDELITIES AND ERROR SOURCES
We have shown that an i -Toffoli gate ( ˆ U i Tof ) can beimplemented in a linear crystal of ions in realistic con-ditions where the effective Ising interaction is generatedby coupling to multiple phonon modes of the crystal. Inthis section, we will compare this gate against an ideal i -Toffoli gate ( ˆ U Ideal ) for different number of qubits andfind conditions for fast gates with high fidelities. Addi-tionally, we are interested in identifying and estimatingthe effect of other sources. (a)(b)
FIG. 4. Process error in function of Ising strength and gatetime assuming single-mode coupling. The results are for a i -Toffoli gate of 3, 5, 7, and 9 ions for detunings of (a) 50, and(b) 200 kHz. We also show the result (7 ∗ ) for a 7 ion crystalincluding an “echo” step. In this case, the total process timecorresponds to 2 t T . To characterize the gate, we use as figure-of-merit theaverage fidelity ¯ F [39]:¯ F ( ˆ U i Tof , ˆ U Ideal ) = (cid:80) j tr[ ˆ U Ideal U † j ˆ U † Ideal ˆ U i Tof ( U j )] + d d ( d + 1) (12)where ˆ U i Tof ( U j ) ≡ tr FS (cid:0) ˆ U i Tof [ ˆ P ⊗ U j ] ˜ U † i Tof (cid:1) , U j are gen-eralized Pauli matrices in the qubit Hilbert space withdimension d = 2 N , ˆ P = (cid:78) m | (cid:105)(cid:104) | m is a projector ontothe n m = 0 Fock subspace and tr FS is the partial traceof the phonons Fock space.We start again by assuming single-mode coupling andcalculate faster gates by increasing both Ω and g andsetting t a = τ g to avoid phases accumulation. By in-creasing the interaction strengths and reducing gate andramp times three types of gate errors will have to be ac-counted for: couplings between off-resonant states, driveerrors and non-adiabatic couplings during ramping of theIsing interaction. To mitigate the first one, we require J > g , therefore we keep the ratio
J/g = 2 for all thegates we will study. The last two errors can be mini-mized either by extending the duration of the adiabaticramp or increasing the detuning of the laser beatnote δ m ,both reducing the amplitudes α m, Ψ and thus the final er-ror. Because our goal is a faster gate, we have chosen forthe latter.Fidelities higher than 99% with gate times below 500 µ s are obtained when δ CM / π = 200 kHz (Fig. 4(c)) forgates with 3-9 qubits. As a consequence of the reductionof the ramp time with increasing J , the activation of theinteraction becomes less adiabatic and the crystal motionis excited. This leads to coupling of motional excitedstates in the form of | n > (cid:105)| , N c (cid:105) ↔ | n > (cid:105)| , N c (cid:105) during the drive step. The larger drops in the fidelity areobserved for particular interaction strengths, e.g. J/ π = 3 . δ t / π = 50 kHz, originate also from undesiredcouplings between states of the type | n = 0 (cid:105)⊗| , (cid:126)x c (cid:105) , | n = k (cid:105) ⊗ | , (cid:126)x c (cid:105) , which become degenerate when ∆ (cid:126)x c ∼ kδ CM .These errors affect more strongly gates with largeramount of qubits as the number of states and the occur-rence of degeneracies increases. Furthermore, the driveand non-adiabaticity errors also increase, as the displace-ment amplitude α m, Ψ ∝ N . However, by choosing appro-priate gate parameters, these undesired couplings can beavoided. A. Multi-mode coupling with residual crystalmotion
From the single-mode coupling analysis we have iden-tified conditions for high fidelity gates for ion crystals intheir ground state. We can use this information to calcu-late high-fidelity gates for systems where all axial phononmodes participate. We will also take into account resid-ual ion motion such that average number of phonons inthe crystal ¯ n m is not zero. In particular, we consider thecases where ¯ n CM > n m (cid:54) = CM = 0.To illustrate, we choose gates with the largest detuning( δ CM / π = −
200 kHz) to minimize drive errors and selecttwo drive strength values ( g/ π = 1 .
0; 4 .
762 KHz) forwhich no large drop of fidelities were obtained in thesingle-mode model. As a result, we obtain multi-modecoupled gates with fidelities better than 99% for bothfast (Fig. 5(a)) and slow gates (Fig. 5(b)). Even in thepresence of residual motion up to ¯ n = 1, the fidelitiesalways exceed 90%.Importantly, the addition of the “echo” step leads tofidelities that, in most of the cases, are better than thosefor single-mode model. Clearly, this step also compen-sates phases due to Stark shifts originated by couplingsof states | , (cid:126)x c (cid:105) ↔ | , (cid:126)x c (cid:105) , which remained uncorrected inFig. 4.Moreover, in absence of these phases, higher fidelitiesare obtained for larger gates (compare with Fig. 4). Theincreasing gaps between states, ∆ (cid:126)x c , for larger systemswill reduce any type off-resonant couplings. In partic-ular, it reduces couplings with excited motional states∆ (cid:126)x c ∼ kδ CM , as the ratio ∆ Nc /δ m increases. Further-more, not only do these gaps increase, there are alsovastly more states with large gaps than with small gaps as N increases. Thus state-specific errors weigh less in thecalculation of the average fidelity for larger qubit gates. V. DISCUSSION AND CONCLUSIONS
We have presented a high-fidelity method to imple-ment a single-step i -Toffoli gate in trapped ions. Ourmethod allows operating in a regime of strong Ising in-teractions between qubits, necessary for fast gate oper-ations. Although the adiabatic ramping of these inter-actions extends the total length of the process, the long (a) (b) FIG. 5. Effect of average phonon number in process fi-delity for gate with multi-mode coupling. The Ising anddrive strengths ( J/ π = g/ π ) are (a) 4.762 kHz and (b)1 kHz. The detuning, δ CM / π , and the center-of-mass fre-quency, ω CM / π , are -200 kHz and 1 MHz respectively. coherence times offered by trapped ions [40] should al-low the experimental implementation of this gate withhigh fidelities. Furthermore, recent methods of shortcutto adiabaticity [41–43] may be applied to speed up theadiabatic preparation of states.We have shown that, when the Ising interactions aremediated by multiple phonon modes, the residual dynam-ical phases can be effectively removed by using an “echo”step exploiting a recent non-adiabatic method for multi-ple qubit entanglement [29]. A natural next step wouldbe to combine our model and this method to generatehomogeneous Ising interactions which should allows usto avoid the “echo” step.A feature of our method is that the appropriate drivestrength ˜ g depends on the initial phonon state. Purephonon input states can be assured by ground state cool-ing the ion crystal. The necessity of ground state coolingsets the implementation apart from a decomposition ine.g. Mølmer-Sørensen gates [18, 19] that are more robustwith respect to the phonon states [17, 44]. On the otherhand, reaching the ground state via sideband cooling isan established technique in trapped ions and is used ex-tensively.Taking these considerations into account, our singlestep implementation of the i -Toffoli gate offers a com-petitive advantage compared to the gate-based decompo-sition, in particular for large N when accumulated gateerrors start to dominate. ACKNOWLEDGMENTS
We thank Georg Jacob for providing code for themultiple beatnote calculations, Arghavan Safavi-Naini,Philippe Corboz and Thomas Feldker for fruitful discus-sions. This work was supported by the Netherlands Or-ganization for Scientific Research (Grant No. 680.91.120,R.G. and M.M.) and by the QM&QI grant of the Uni-versity of Amsterdam (K.G.).
Appendix A: Modulation of Ising interaction
The adiabatic transformation between the non-interacting and dressed states basis is realized by slowlyincreasing (decreasing) the strength of the Ising interac-tion for a time t a (cid:29) /δ s . This is achieved by modu-lating the Rabi frequency of the laser-ion Hamiltonianˆ H q-ph , such that Ω( t ) = Ω sin (cid:0) π t/t a (cid:1) for t < t a andΩ( t (cid:48) ) = Ω cos (cid:0) π t (cid:48) /t a (cid:1) with t (cid:48) = t − t a − τ g for t > t a + τ g (Fig. 6). As a result we obtain the time-dependent Isingcouplings J ( t ) ∝ Ω( t ) . This modulation leads to apulse area equivalent to that of a square pulse of halfthe length, such that we define an effective ramp time as˜ t a = 0 . t a . FIG. 6. Strength of Hamiltonian terms during length i -Toffoli gate. The Ising interaction (blue) is increased beforeacting with the drive field (purple) and then lower down again.A “echo” step (red) can be applied at the end of the gate tocorrect for residual entanglement or dynamical phases As seen in Fig. 7(a)-7(b), the displacement in phasespace of the two target states are significantly reducedfor the adiabatically initialized system. This minimizeserrors due to the non-commutativity between the driveand Ising interaction fields and also the ones arising fromresidual phonon-qubit coupling. To approximate the uni-tary evolution of this adiabatic process we use a Trotter-Suzuki expansion [45]:ˆ U aeg = t a (cid:89) t =0 e − i ∆ t ˆ H Ising ( t ) e − i ∆ t ˜ H ˆ U deg = (cid:89) t = t a e − i ∆ t ˆ H Ising ( t ) e − i ∆ t ˜ H (A1)where, ˆ H Ising ( t ) = J ( t ) (cid:88) i (cid:54) = j ˆ σ ( i ) z ˆ σ ( j ) z (A2)˜ H = 2 N c J ( t a ) (cid:88) i ˆ σ ( i ) z − δ s ˆ a † s ˆ a s (A3)and ∆ t (cid:28) (1 /k ) δ s (cid:28) t a is the time-step of the expansionand k = t a δ t / π . (a) (b)(c) (d) FIG. 7. Evolution of target states under the application ofˆ H q-ph . Trajectories of | (cid:105) (red) and | (cid:105) (blue) wavepack-ages and evolution of momentum expectation value of | (cid:105) due to (a,c) a quench activation of 500 µ s and (b,d) an adia-batic modulation of ˆ H q-ph Appendix B: Elimination of residual entanglementand dynamical phases
Whenever the timing condition for the elimination ofdynamical phases, ˜ t T J = 2 k π , is not fulfilled, it is pos- sible to add an additional “echo” step to the process tocorrect for these errors (Fig. 6). In this step the sign ofthe interaction strength is also reversed, i.e. J → − J .For the single mode coupling model, this is obtained byinverting the sign of the detuning δ s → − δ s . In the caseof multi-mode coupling, we have used a combination ofmultiple beatnotes to generate an effective Ising interac-tion reversing the sign of the couplings during the gatestep. 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