Quantum simulations with complex geometries and synthetic gauge fields in a trapped ion chain
Tom Manovitz, Yotam Shapira, Nitzan Akerman, Ady Stern, Roee Ozeri
QQuantum simulations with complex geometries and synthetic gauge fields in a trappedion chain
Tom Manovitz, ∗ Yotam Shapira, † Nitzan Akerman, Ady Stern, and Roee Ozeri Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
In recent years, arrays of atomic ions in a linear RF trap have proven to be a particularlysuccessful platform for quantum simulation. However, a wide range of quantum models andphenomena have, so far, remained beyond the reach of such simulators. In this work we introducea technique that can substantially extend this reach using an external field gradient along theion chain and a global, uniform driving field. The technique can be used to generate bothstatic and time-varying synthetic gauge fields in a linear chain of trapped ions, and enablescontinuous simulation of a variety of coupling geometries and topologies, including periodic boundaryconditions and high dimensional Hamiltonians. We describe the technique, derive the correspondingeffective Hamiltonian, propose a number of variations, and discuss the possibility of scaling toquantum-advantage sized simulators. Additionally, we suggest several possible implementationsand briefly examine two: the Aharonov-Bohm ring and the frustrated triangular ladder.
INTRODUCTION
Quantum simulators are highly controlled quantummachines with which it is possible to engineer andstudy complex quantum states and dynamics. Suchmachines, when large and accurate enough, are expectedto elucidate the behaviour of quantum systems thatdefy analytical treatment and which are intractable forclassical numerical simulations [1]. With the increasingsizes and abilities of quantum simulators and computers[2–5], quantum advantage in the context of quantumsimulation may be within reach in the near future[6, 7]. Of the diverse physical platforms used forquantum simulation, atomic ion chains in linear RFtraps have proven particularly fertile by virtue of theirlong coherence times and high operation fidelity [8, 9].Using trapped ion quantum simulators, researchers havecreated and studied a wealth of quantum phenomenaby applying both analog [3, 10–21] and digital [22–26]simulation techniques.A principal feature of ion chain quantum simulatorsis the precisely controllable long-range coupling betweenion-qubits, driven by an external field and mediated bythe motional modes of the chain [27–29]. Representing aspin- particle by two electronic energy levels in each ion,a uniform external driving field can induce an effectivespin-spin interaction of the form [30]: H c = (cid:88) i Our main result is a simple recipe for generating aclass of Hamiltonians using trapped ions. The class isdescribed by the following formula: H = N − (cid:88) n =1 H n = N − (cid:88) n =1 Ω n e i ( φ n − δ n t ) N − n (cid:88) i =1 σ + i σ − i + n + h.c. (3)where σ + i ( σ − i ) denotes the raising (lowering) Paulioperator on ion i , and Ω n , φ n and δ n are tunableparameters corresponding to the coupling strengths,static phases and time-dependent phases, respectively, ofan n -neighbor hopping interaction. As we show below,the Hamiltonian in Eq. (3) can be used to implement spinHamiltonians on various geometries, and can furthermoremanifest static and time-dependent gauge fields.The method relies on the application of a staticexternal field gradient (e.g. a spatially varying magneticor light shift field) for shifting the atomic energylevels along the ion chain, applied together with acorresponding global uniform driving field. The fieldgradient collapses the translational symmetry of theion chain, effectively suppressing the standard couplingform of Eq. (2). However, because the gradientis spatially uniform, the driving field can be used toselectively reinstate the translational symmetry of thespin-spin interaction. This is done in a controlledmanner by bridging the resonance difference betweenequally-separated ion-pairs using the frequency differencebetween pairs of bichromatic fields. Furthermore,the breaking of spatial symmetry differentiates theinteraction of an ion with its neighbors to the left andright, which gives rise to a gauge-field-like phase and aneffective breaking of time-reversal symmetry.The tools needed to implement (3) are standardin trapped ion experiments: for every nonzero Ω n ,representing an n -neighbor interaction, a four-tone fieldis added. The corresponding Ω n , φ n and δ n are set by thefield’s amplitudes, phases and frequencies. The drivingfield is activated using a single uniform-intensity beam.The magnetic field gradient can be modest, on the orderof 10 G/cm. Additionally, the interaction in Eq. (3)is excitation-number-preserving implying robustness toglobal dephasing noise.A wide spectrum of quantum phenomena can beaccessed using this method. For example, by choosingΩ = Ω N − = Ω and φ = − φ N − = 2 π Φ /N (andnulling all other parameters) we arrive at a lattice ringHamiltonian with a hopping term: H r (Φ) = Ω N (cid:88) i =1 e πi Φ /N σ + i σ − i +1 + h.c. (4)where boundary conditions are periodic. The phaseΦ corresponds to the Aharonov-Bohm phase acquiredby an electric charge encircling a ring penetrated bya magnetic flux (note that a 1D spin system mayalways be described as a fermionic system, using theJordan-Wigner transformation) [61]. Accordingly, anexcitation will travel clockwise or counter-clockwise onthe ring, generating a persistent current [62]. While thismodel is easy to solve, it clearly showcases the main toolsof the proposed technique. We analyze the model in moredetail below. Figure (1) highlights our method’s main principles ofoperation. Taking the 1D ring in a N = 5 ion chainas an example, a pair of driving fields (a) bridge theenergy difference between the ion created by the externalgradient, i.e ∆ for neighboring ions and 4∆ for the edgeions (b). These driving fields form tailored couplingsbetween the ions, and are here used to generate a 5-sitering penetrated by a magnetic flux Φ (c). Accordingly,a simulation of the ion chain’s evolution shows anexcitation travelling around the ring (d).Using these principles, Hamiltonians of the formexpressed in Eq. (3) can be generated. Significantly, awide variety of coupling geometries are reachable. Weillustrate some possible coupling geometries in Figure2. Besides a ring (a), these include triangular ladders(b); 2d rectangular lattices (c) (which can be closed ontoa cylinder, not shown); a M¨obius-strip ladder (d); ahelical lattice on a cylinder (e); and a torus (f). Theselattices can be threaded by a variety of magnetic fluxes,as illustrated for the torus (f). While the 1D ring can bemapped to a system of free noninteracting fermions andis thus simply solvable, all other models shown here areexpected to show complex behavior and can be difficultto solve. PHYSICAL PICTURE Ions in a Paul trap are frequently modeled as two levelspins with a set of harmonic modes, where the formercorresponds to the ions’ electronic degrees of freedom,and the latter to the motional normal-modes of the ionchain. External electromagnetic fields can couple to spinand motional degrees of freedom and, with proper tuning,can be used to engineer effective interactions betweenthe spins of different ions via mediation by the motionalmodes.In the Mølmer-Sørensen (MS) interaction [27, 28], theexternal field is bichromatic and tuned to frequencies ω ± = ω ± ( ν + ξ ), with (cid:126) ω the single qubit energyseparation, ν the frequency of a normal mode of motion ofthe ion-chain, and ξ a constant detuning which togetherwith the field intensity determines the interaction rate.The tone ω + ( ω − ) mediates interactions via the blue(red) motional sideband, i.e it employs transitions whichexcite the ion’s electronic degree of freedom while adding(removing) a phonon of the motional normal-mode.While only two driving tones are used, this interactioncouples any two ions in the chain in four different“pathways”, as is shown in Fig. 1 of Ref. [28].When coupled to the center-of-mass (COM) mode ofthe ion chain, this driving field induces an effective σ x σ x interaction through a two photon process, which equallycouples all of the ions-pairs in the ion chain. Thisinteraction can be decomposed to two contributions: apair creation/annihilation term, σ + σ + + h.c , which drives a bdce f Figure 2. Implementation of various geometries usingappropriate hopping interactions. (a) nearest-neighbourand N − W -neighbor interactions generate a rectangular lattice whenused with spacer ions (see discussion below), and with theaddition of N/W interactions generate a rectangular lattice ona cylinder; (d) nearest-neighbour, N/ N − W interactions generate a helical lattice on a cylinder; (f) byadding N − N/W interactions to the helical lattice,the cylinder is closed onto a torus. Controlling the phases ofthese interactions results in synthetic gauge fields representingfluxes threading these geometries. For instance, in the torus(f) an external axial flux (green), or within the torus, as inan anapole moment (red), can be produced. a |↑↑(cid:105) ↔ |↓↓(cid:105) two photon transition, changing the systemenergy by ω + + ω − = 2 ω ; and an excitation hoppingterm σ + σ − + h.c , driving a |↓↑(cid:105) ↔ |↑↓(cid:105) two photontransition, which leaves the system’s energy unchanged,i.e ω ± − ω ± = 0.Here we are interested in eliminating the paircreation/annihilation term while retaining the hoppingterm, and furthermore shaping its coupling matrix.The first goal is achieved by detuning the drivingfield frequencies from resonance with the two-photontransition, i.e by modifying the bichromatic drivefrequencies to ω ± = ω + (cid:15) ± ( ν + ξ ), shifting the paircreation/annihilation term 2 (cid:15) off-resonance. The second a bc d Figure 3. Coupling selectivity using an external gradient.Here black lines represent equal-excitation energy levels ofthe ions, while arrows represent the two-photon hoppinginteractions between ions. (a) In the absence of an externalgradient, the hopping interaction is resonant for any ion pair.(b) By imposing an external gradient, all hopping terms aremoved off resonance and therefore suppressed. (c-d) Couplingis then selectively reinstated by adding another drivingfrequency that bridges the gap induced by the gradient. Hereeither a nearest-neighbor (c) or next-nearest neighbor (d)coupling is produced through choice of driving frequencies. goal requires a more elaborate approach. As the σ + σ − + h.c term is mediated by an excitation/de-excitationpair of identical photons, it resonantly couples onlystates that are degenerate under H ; this implies acoupling between all equal-excitation states. However,the hopping term can also be controllably suppressedby lifting the equal-excitation degeneracy [63]. Acontrolled suppression of the interaction will then allowfor selectively reinstating resonant conditions through amodulation of the driving field.To do so, an external (e.g. magnetic) field gradientis added along the ion-chain, such that the transitionfrequency between adjacent ions differs by ∆. In orderto selectively couple ions which are n sites apart wedrive the ions with four frequencies, composed of thefrequency pairs ω b, ± = ω + ( ν + ξ b ) ± ∆ n and ω r, ± = ω − ( ν + ξ r ) ± ∆ n . The pair ω b, ± couples an n -site hopresonantly, mediated by the blue sideband. Similarly,the pair ω r, ± couples the same hop, mediated by thered sideband. As in the MS interaction, both sidebandsare employed in order to mitigate temperature-dependenteffects. In order to keep the pair creation/annihilationterm non-resonant we use ξ b = ξ + (cid:15) and ξ r = ξ − (cid:15) .Figure 3 illustrates the transition from all-to-allcoupling in the absence of a gradient field (a) to acomplete suppression of coupling due to the gradient (b)and the selective resurrection of coupling by introducingthe resonant sideband modulation (c,d). DERIVATION We outline the derivation of Eq. (3). We focusonly on a single H n term, and later comment on thegeneralization to a summation of these terms. In theabsence of an external driving field, the Hamiltonian of N trapped ions in a magnetic field gradient is given by: H = (cid:126) N (cid:88) k =1 ( ω + k ∆) ˆ σ zk + N (cid:88) l =1 (cid:126) ν l (cid:18) ˆ a l † ˆ a l + 12 (cid:19) . (5)Here (cid:126) ∆ is the transition energy difference betweenadjacent ions due to the field gradient, and ν l is thefrequency of the l -th normal-mode with the annihilationoperator a l . In our derivations below we will assumecoupling to a single motional mode, the COM mode (witha frequency ν = ν and a Lamb-Dicke parameter η = η ),which couples equally to all ions in the ion-chain. Thisassumption can be relaxed [33], as will be shown in laterdiscussions.For each n we apply a four-tone field composedof the frequency pairs ω b, ± = ω + ( ν + ξ b ) ± ∆ n and ω r, ± = ω − ( ν + ξ r ) ± ∆ n , Rabi frequenciesΩ b and Ω r , and phases φ b, ± = ± φ/ φ r, ± = ± ( φ + π ) / 2. In addition ξ b and ξ r are chosen suchthat pair creation/annihilation transitions between allion-pairs are detuned from resonance.Specifically, (cid:15) = ( ξ b − ξ r ) / η Ω r Ω b /ξ [28, 63]. Thus, the red and bluesideband pairs contribute to the evolution independently.We first focus on the interaction mediated by the bluesideband, which is due to the pair ω b, ± . In a framerotating with respect to H above, this interaction isdescribed by, V I = i (cid:126) η Ω b a † cos (cid:18) n ∆2 t + φ (cid:19) N (cid:88) k =1 σ + k e i ( k ∆ − ξ b ) t + h.c, (6)This expression is valid in terms of a rotating waveapproximation by assuming that Ω b /ν and Ω b /ω aresmall, and in leading order in η .The Hamiltonian in Eq. (6) cannot be solvedanalytically. However, its resulting evolution operator, U ( t ), can be approximated by using the leading termsin a Magnus expansion [64, 65], U = exp (cid:2) − i (cid:126) χ ( t ) (cid:3) =exp (cid:2) − i (cid:126) (cid:80) n χ n ( t ) (cid:3) . Since χ n ∝ (cid:16) η Ω b ξ (cid:17) n , we are satisfiedwith terminating the expansion at the second order, which (as we show below) provides the leading orderresonant terms.The evolution due to χ is stroboscopic with afundamental period T , i.e that χ ( kT ) = kT H eff , with k ∈ Z and H eff an effective time-independent Hamiltonian. Inthe limit T → χ = (cid:90) T dtV I ( t ) . (7)By choosing T ∆ = 4 πm and T ξ b = 2 πM b , with m, M b ∈ Z , we arrive at χ ( T ) = 0, trivially satisfying thestroboscopic condition.In the next order we decompose χ to a hopping termand a rotation around the z -axis. The expansion is thengiven as, χ = − i (cid:126) (cid:90) T dt (cid:90) t dt [ V I ( t ) , V I ( t )] = χ , h + χ ,z χ , h = iT (cid:126) η Ω b (cid:88) k ξ b − (cid:0) k + n (cid:1) σ + k + n σ − k e iφ + h.cχ ,z = iT (cid:126) η Ω b (cid:88) k ξ b − k ∆( ξ b − k ∆) − (cid:0) n ∆2 (cid:1) σ zk (cid:18) a † a + 12 (cid:19) . (8)which can be translated to an effective Hamiltonian, H eff = H h + H z + H ∇ h + H ∇ z , (9)with corrections that scale as (cid:16) N ∆ ξ b (cid:17) . The first twoterms of this Hamiltonian are homogeneous. The firstterm represents the desired hopping interaction (hencethe subscript h ), while the second is effectively equivalentto a temperature-dependent global magnetic field in the z -direction (hence the subscript z ), H h = (cid:126) Ω n,b (cid:88) k σ + k + n σ − k e iφ + h.cH z = 2 (cid:126) Ω n,b (cid:18) a † a + 12 (cid:19) (cid:88) k σ zk , (10)with Ω n,b = η Ω ξ b .Since H h is excitation preserving, by initializing thesystem to an eigenstate of (cid:80) k σ zk , i.e. to a state with awell defined number of excitations, H z is reduced to aglobal phase and can be ignored. For these initial statesa two-tone driving field suffices.Both H h and H z are proportional to Ω n,b , however theformer also depends on the phase φ . This enables the useof the red sideband pair, ω r, ± , in order to eliminate H z entirely while maintaining H h . To this end, we choose ξ r and Ω r such that Ω n,r = − Ω n,b , and φ r, ± = ± φ + π .Thus the combination of the two pairs yields H h → H h and H z → H ∇ h = (cid:126) Ω n ∆ ξ b (cid:88) k (cid:16) k + n (cid:17) σ + k + n σ − k e iφ + h.cH ∇ z = 2 (cid:126) Ω n (cid:18) a † a + 12 (cid:19) ∆ ξ b (cid:88) k kσ zk . (11)These terms are gradient inhomogeneous terms, i.ethey are not tranlationally invariant, and vanish in thelimit ∆ → 0. By setting ξ b (cid:29) N ∆, H ∇ h and H ∇ z become negligible and we obtain a homogeneous effectiveHamiltonian. Furthermore, by using the red sidebandpair as described above H ∇ h is eliminated entirely as well.The remaining H ∇ z is more difficult to eliminate inthe non-adiabatic regime. If the ion chain is cooledto the ground state, its contribution to the effectiveHamiltonian is simplified to H ∇ z → (cid:126) η Ω ξ b ∆ (cid:80) k kσ zk ,which has the same form as that of the external gradientfield in (5). Hence, by making a small correction to theaddressing field frequencies the ground-state contributionof H ∇ z is eliminated.Thus, at the appropriate limits, the four-tonefrequency drive yields the effective Hamiltonian, H eff = 2 (cid:126) Ω n N (cid:88) k =1 σ + k + n σ − k e iφ + h.c. (12)The interaction may be made time-dependent bydetuning the two-photon transition, ω b, ± → ω b, ± ± δ/ 2, with δ (cid:28) ∆ (similarly for ω r, ± ). We obtain anoff-resonant coupling, which manifests in a time variationof the hopping phase: σ + k + n σ − k e iφ → σ + k + n σ − k e i ( φ + δt ) .With this transformation we obtain the general form of H n from Eq. (3).We further note that more generally the Peierls phasecan be changed in time in whatever way one wishes bychanging the appropriate driving field phases, as long asall spectral component of the phase dynamics are muchsmaller than ∆. Constant detuning, implying a linearchange in time of the phase, is a specific instance ofsuch phase dynamics. As an example, by periodicallymodulating the phase, the presence of AC magneticfluxes can be realized.For any additional hopping term, H n (cid:48) , another set ofdrive parameters, { Ω (cid:48) , ξ (cid:48) , φ (cid:48) , δ (cid:48) } is added, but with thefurther requirement that for any m = 0 , , ..., N − || ξ (cid:48) − ξ | − m ∆ | (cid:29) max { Ω n , Ω (cid:48) n } , inorder to avoid any unintended cross-term resonances.This requirement should hold independently for any twoterms, and can most simply be fulfilled when | ξ − ξ (cid:48) | > | N ∆ | . SOME TARGET MODELS In this section we briefly explore two models thatcan be simulated with our method and suggest severaladditional models. In order to clearly display thesalient features of the technique, we first focus onthe analytically solvable 1D Aharonov-Bohm ring. Wethen discuss a triangular spin ladder model exhibitinggeometric frustration. Despite the simplicity of thismodel, it gives rise to a rich phase diagram andinteresting physical phenomena. The discrete 1D Aharonov-Bohm ring For N ions, turning on the n = 1 and n = N − φ = − φ N − = 2 π Φ /N simulatesa magnetic flux Φ penetrating the ring, as is shown inFig. 1, with the Hamiltonian H r expressed in Eq. (4).As the Hamiltonian is excitation preserving, inside anexcitation eigenspace H r can be mapped directly via theJordan-Wigner transformation [66] to fermions on a ringlattice threaded by a magnetic field: H f = N (cid:88) i =1 e i π Φ /N ψ † i ψ i +1 + h.c., (13)with periodic (antiperiodic) boundary conditions for anodd (even) number of excitations. H f is the spinlessfermion 1D tight-binding model [67].The magnetic flux threading the ring may give rise toa persistent current due to the Aharonov-Bohm effect,which survives even in the presence of impurities inthe chain [68, 69]. In order to observe this effect, onecan prepare an initial state with a position occupationdistribution that will rotate around the ring withoutdiffusing. The singly-excited subspace of H f is spannedby plane-wave eigenstates | k (cid:105) = √ N (cid:80) n e ink/N ψ † n | (cid:105) with energies E k = 2 cos (cid:0) πN ( k + Φ) (cid:1) . While each ofthese waves uniformly occupies all ions along the chain,we can initialize the system in a wave packet statewith a position-dependent occupation: | ψ W.P ( t = 0) (cid:105) = (cid:0) | k (cid:105) + e iϕ | k − (cid:105) (cid:1) / √ n -th site is |(cid:104) n | ψ W.P (0) (cid:105)| ∝ (cid:18) πnN − ϕ (cid:19) . (14)Equation (14) shows that | ψ W.P (cid:105) is a wave-packet, with ϕ determining its position on the ring. The wave-packet’sevolution can be described by the evolution of ϕ . Thestate will evolve according to ϕ ( t ) = ϕ (0) + v (Φ) t , with v (Φ) = − 4Ω sin (cid:16) πN (cid:17) sin (cid:18) πN (cid:18) Φ + k − (cid:19)(cid:19) . (15) a bdc Figure 4. Time evolution in the Aharonov-Bohm ring.Turning on the n = 1 and n = N − N − φ − φ N − . This simulation of the timeevolution of a 5 ion chain for fluxes (a) Φ = 0, (b) Φ = π/ 2, and (c) Φ = 2 π , shows the flux-dependent propagationdynamics of a wave-packet. Local spin excitation probability P e = (1 + (cid:104) σ z (cid:105) ) is color coded. The excitation, which isstatic in the absence of flux (a), rotates around the ring ata constant velocity when flux is added. This rotation is amanifestation of persistent current in the magnetically drivenring. The ion dynamics are integrated directly from Eq. (6)with the appropriate choice of driving parameters as detailedin this manuscript. (d) Comparison of simulation results forion dynamics (blue) and Eq. (15) (red) for the flux-dependentangular velocity of an excitation. That is, the packet rotates around the ring at a constant,flux-dependent, angular velocity. As expected, at thelarge N limit, we obtain v (Φ) ∝ Φ + k .Figure 4(a)-(c) shows simulations of the evolution of | ψ w.p (cid:105) with k = 0, N = 5 and different values of Φ,integrated from Eq. (6) with the appropriate drivingfield parameters. Indeed the wave packet circles aroundthe ring with a flux-dependent velocity, exhibiting apersistent current. Figure 4(d) compares the observedangular velocity of the wave packet with Eq. (15),showing an excellent agreement.The Aharonov Bohm ring can be used to observe Blochoscillations. For particles in a 1D periodic structure,the addition of a constant uniform force generates anoscillatory motion rather than unidirectional acceleration[71]. In a 1D ring such a force can be created bythreading the ring with a time-dependent magnetic flux[68]. An excitation, rather than encircling the ringwith a constant acceleration, will oscillate locally. Theeffect can be naturally incorporated using our technique,taking advantage of the ability to generate a time-varying synthetic gauge field using off-resonant driving pairswithin the coupling scheme of the AB ring, as previouslydescribed. Triangular Spin Ladder The Aharonov-Bohm ring can be mapped onto a freefermion model via the Jordan-Wigner transformation,and is thus easily solvable. However, spin interactionsbeyond nearest-neighbor can only be mapped ontointeracting fermion models, and accordingly generatecomplex dynamics and phases which are oftenchallenging for classical computation techniques.As an example, we briefly discuss the triangular ladder,a simple Hamiltonian that can be easily implementedusing our technique, but which nonetheless manifestscomplex behavior.Activation of the n = 1 and n = 2 terms in Eq.(3) generates a nearest neighbor (nn) and next-nearestneighbor (nnn) interaction spin Hamiltonian: H tl = (cid:88) i σ + i (cid:0) J e iφ σ − i +1 + J e iφ σ − i +2 (cid:1) + h.c. (16)where boundary conditions are open. Such Hamiltonianscan be graphically represented by triangular ladders inwhich rungs and rails represent nn and nnn interactionsaccordingly, as pictured in Figure 5. Due to thecompetition between nn and nnn terms, geometricallyviewed as the competition of interactions inside eachtriangle, models of this sort are frustrated and thus giverise to a relatively rich phase diagram [72–74]. H tl isgauge invariant under the transformation φ → φ + ϕ , φ → φ + 2 ϕ for any ϕ ; this is equal to the gaugetransformation σ + k → e ikϕ σ + k .For trivial interaction phases φ = 2 φ andantiferromagnetic nnn interactions ( J > H tl represents the one-dimensional frustrated XY chainmodel for spin- . This model is a paradigmatic exampleof frustration [73]. It supports a variety of phases,notably including an exotic chiral-ordered phase [75–78]. The system phase depends on j = J /J , whichcan be fully controlled using the techniques outlinedin this manuscript. At j = , also known as theMajumdar-Ghosh point, the ground state is an exactlysolvable dimerized state [74]. The control and flexibilityof trapped ion systems may enable generation of theseunique phases and direct measurement of their orderparameters [79], their entanglement properties [80] andtheir excitation dynamics [16].By choosing nontrivial values for φ , φ an additionalsynthetic gauge field representing a staggered flux isadded to the Hamiltonian. Each triangle is pierced bya gauge-invariant magnetic flux ± Φ = φ − φ , with theflux alternating signs between neighbouring plaquettes.Similar triangular ladder models with synthetic flux fieldshave been suggested and implemented in neutral atomsystems [81–84]. Figure 5 illustrates the connectivity andstaggered flux for this model. Figure 5. The triangular spin ladder. By turning on bothnn and nnn couplings, an effective triangular or zigzag laddercoupling geometry is generated. The triangular layout caneasily lead to frustration, which can be understood as acompetition of the order imposed by the different couplingterms. Corresponding coupling phases φ and φ createa synthetic gauge field representing a magnetic flux thatalternates in sign between plaquettes, with a gauge-invariantphase Φ = φ − φ . Rectangular lattice In the 2D examples discussed in this manuscript sofar, lattices were either triangular or helical, and notrectangular. This is due to the fact that a strictly2D rectangular lattice cannot be reduced to the formgiven by Eq. (3); placing qubits on the lattice, thenearest-neighbor coupling scheme would create a linkbetween the last qubit in row k and the first qubit inrow k + 1, violating the lattice geometry.This can be remedied by interrupting thenearest-neighbor interaction chains, represented bythe rows of the lattice, through insertion of an auxiliarypassive ion. The auxiliary ion acts as a spacer, generatingan effective jump in the gradient for the active ions.This ion can be of a different isotope or species, but ismost easily chosen to be an identical ion that is eitherstrongly light shifted by an individual addressing beamor prepared in any state outside the qubit subspace.In the simplest example, a single spacer ion in themiddle of a chain of 2 N + 1 ions, along with interactionterms H and H N +1 , would generate the rectangular spinladder, shown in Figure 6. With the addition of morespacer ions, more rows could be added to this array,effectively creating a complete rectangular lattice.This lattice can then be curled into a cylinder with theactivation of an additional term. For N = h · ( w + 1)ions, where h, w ∈ Z , activating the nn, w + 1 and N − − ( w + 1) terms generates an h × w rectangular lattice a b Figure 6. Spacers and rectangular ladders. (a) Placingthe spins on a rectangular chain and activating the n =1 and n = m terms generates a rectangular lattice withadditional unwanted cross-row terms. (b) By adding spacerions, initialized in a state which is uncoupled to the spindynamics, and by replacing the n = m term with a n = m + 1term, the unwanted coupling is corrected, giving an exactrectangular coupling geometry. on a cylinder. The curled dimension can be threaded bya flux, determined by phases of the non-nn terms. Additional geometries There are a number of other geometries which canbe generated in a straightforward manner using ourtechnique. We briefly mention several more examples:the M¨obius ring, the cylindrical helix, and the torus, allillustrated in Fig. 2.Turning on the terms H and H N/ generates a 2 × N/ n − n , as is shown in Fig. 6a for n = 3. Activatingin addition the term H N − forms a M¨obius ring, shown inFig. 2d. Such a system may be used to study topologicaleffects in non-trivial geometries [85–88].For N = w · h where w, h ∈ Z , turning on theterms H and H w in Eq. (3) generates the cylindricalhelix in Fig. 2e, with w sites per loop and height h .Furthermore, adding terms Ω N − and Ω N − w inducesperiodic boundary conditions, resulting in the torus seenin Fig 2f. Setting the phases φ = − φ N − and φ w = − φ N − w gives rise to two independent fluxes penetratingthe torus, Φ = Lφ w (green arrow) and Φ = W φ − φ w (red arrow). Such a system may be used for the study ofthe quantum Hall effect in the thin torus limit [89, 90].In Appendix A we show a simple resource efficientimplementation of our method, that can be used for therealization of some of the models above.The ideas presented here may be taken even further byadopting the powerful neutral atoms quantum simulationconcept of synthetic dimensions [47, 91, 92]. In neutralatom quantum simulators, extraneous internal degreesof freedom of the atom are used to represent additionallattice sites. For instance, a 1D system of atoms can beused to represent a 2D lattice, where the supplementarydimension is embodied by additional internal states ofeach atom. In such a case, engineered spin-orbit couplingcan be used to drive a synthetic gauge field. In trappedions, the additional Zeeman or hyperfine states maypresent a similar possibility, further extending ion chainsimulations to an additional dimension. Spatially varying potentials In ion chains, it is possible to generate site-dependentenergy shifts, H v = (cid:126) (cid:80) k V k σ zk , by using individualaddressing beams, nonuniform global beams, magneticfields or other spatially varying fields. Under thecondition V k (cid:28) ∆, H v can be applied in parallel toour technique. As H v commutes with H , it can besimply added to the interaction picture Hamiltonian, V I → V I + H v . In this instance, H v can be interpreted asa spatially varying potential on the lattice. By choosingrandom site-dependent shifts, disorder is added to thesystem. Disorder can give rise to localization [93], whichcan thus be studied in the variety of contexts presentedin the paper. SCALING UP While implementation of the technique we proposein small scale simulators should be straightforward,approaching larger, quantum-advantage (NISQ) sizedsimulators [51] seems feasible yet more demanding. Herewe discuss possible challenges in the implementation ofthe technique in large quantum simulators.The technique, as presented, calls for constantdifferences in resonance frequencies between neighboringions. This can be exactly achieved with a spatiallyuniform gradient only to the extent that the ions arespaced equidistantly. However, many linear ion trapscurrently use a harmonic axial potential, in which ionsare not equidistant [94]. Higher order components of theexternal field can be engineered to meet this issue, albeitwith increasing experimental complexity. Nevertheless,several groups have constructed - or are in the process ofconstructing - anharmonic traps, with the stated purposeof trapping ions equidistantly [34, 95–97]. Anharmonictraps are likely to become more useful in future ion trapsimulators and computers, due to their advantage inmaintaining high inter-ion spacing, critical for preventingcross-talk in addressing and detection; their resistance totransitions from linear to zig-zag crystal configurations;and their suppression of inhomogeneous quadrupoleshifts. Our proposal is best suited for such traps. Another hurdle for large scale implementation couldbe the inverse relation between coupling strength andnumber of ions, keeping driving field intensity constant.We define adiabaticity parameters α = ξ/ (∆ N ) and β = ∆ / η Ω ξ . In the low α limit the inhomogeneouscontribution of the gradient field to the Hamiltonianbecomes prominent (although strongly suppressed bythe double sideband frequency configuration), and for α < / β , non-resonant Hamiltonian terms, which areideally completely suppressed, can play a significantrole in the dynamics. Keeping these adiabaticityparameters constant, the effective coupling strengthis η Ω ξ = η Ω √ αβN . Since we assume coupling onlyto the COM mode, the Lamb-Dicke parameter alsodecreases with ion number: η = η √ N , where η isthe single-ion Lamb-Dicke parameter. Hence, overall,the coupling strength decreases as N , as opposed tothe normal COM MS degradation of √ N . In essence,the additional penalty emerges from the requirement topreserve spectral spacing in presence of the gradient field.The degradation can be remedied by increasing the fieldintensity or by coupling to many modes rather than justthe COM, which is not only convenient but necessarywhen using radial modes in large simulators.The radial modes of linear ion traps bunch up whenadding more ions, and consequently the radial COMmode cannot be spectrally resolved. Using these modesas interaction mediators will thus inevitably generatecoupling to a multitude of modes. Even so, the useof radial modes can be advantageous and is compatiblewith our proposal, up to a modification of the effectivecoupling, in analogy with [30] (see Appendix B for moredetails).In contrast, the axial COM mode remains spectrallyseparated from all other modes independently of thenumber of ions, and thus is in this sense ideal for theproposed technique. Nevertheless, axial modes comewith their own disadvantages. Working with large ionchain requires very low axial trap frequencies in orderto avoid the crystal zig-zag transition, or buckling, inthe middle of the chain. Low axial frequencies leadto higher carrier coupling, temperatures, and heatingrates, limiting simulation fidelity. These problems canbe strongly mitigated by using anharmonic trappingpotentials, which are beneficial for our proposal as statedabove. SUMMARY In conclusion, we have introduced and explored atechnique that realizes a variety of spin Hamiltoniansin trapped ion chains. The range of implementable0Hamiltonians includes spin lattices with dimensionlarger than one, closed boundary conditions, rectangularand triangular lattices, and full control of nearestand next-nearest neighbor couplings. Furthermore,the technique provides a means to realize static andtime-varying synthetic gauge fields in ion chains. 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Holz¨apfel,P. Jurcevic, M. B. Plenio, M. Huber, C. Roos, R. Blatt, et al. , Physical Review X , 021012 (2018). APPENDIX A - RESOURCE-EFFICIENTIMPLEMENTATION In the given formulation, each H n in Eq. (3) requiresa bichromatic field with frequencies f n, ± = ω eg + ν + ξ n ± n ∆ / 2, where care has to be taken to avoidcross-term resonances. However, for a subset of targetHamiltonians, it is possible to use less frequencies. Fora pair with frequencies f + − f − = n ∆ generating H n ,the inclusion of a single extra frequency f such that f − f + = n ∆ would generate another two resonantcoupling terms: H n and H n = H n + n . Given theaccording field amplitudes Ω ± , Ω , the effective couplingsare proportional to the appropriate amplitude products:(Ω n ) ∼ Ω + Ω − , (Ω n ) ∼ Ω + Ω , and (Ω n ) ∼ Ω Ω − .We may choose n = n ; in such a case, two terms aregenerated: n and n (cid:48) = 2 n .This method is naturally applicable in some cases.For instance, the triangular ladder Hamiltonian can beapplied by choosing n = n = 1. The Aharonov-Bohmring can be applied by choosing n = 1 , n = N − n = N term. We can combine the two by choosing n = n = 1and adding a fourth frequency f = f + + ( N − n = 1, n (cid:48) = 2, n = N − n = N − 1; this would generate the triangularladder with closed boundary conditions, i.e. a closedtriangular band. a b Figure 7. Resource efficient implementation. In somecases the model Hamiltonian can be implemented with lessfrequencies than prescribed by the general formula. Herethe Aharonov-Bohm ring (a) and the triangular spin ladder(b), both analyzed above, are generated using three ratherthan four driving frequencies. Any frequency difference in thepolychromatic field that corresponds to a frequency differencebetween ions in the chain will generate another coupling term. APPENDIX B - DERIVATION OF EFFECTIVEHAMILTONIAN USING RADIAL MODES Here we derive the effective hopping Hamiltonian inpresence of the multitude of modes which are necessarilyin play when using the radial motional modes asinteraction mediators . We assume that the laser drivegenerates non-negligible coupling to a multitude of radialnormal-modes of motion. As in our derivations in themain text we first focus on the blue sideband drivingpair. The interaction Hamiltonian in Eq. (6) is amendedto: V I = i (cid:126) Ω b cos (cid:18) n ∆2 t + φ (cid:19) · (cid:88) j a † j (cid:88) k η j,k e − i ( ω b − ν j ) t σ + k + h.c, (17)where the Lamb-Dicke matrix η j,k represents theparticipation of ion k in motional mode j . The drivingfrequencies have been generalized to ω b, ± = ω + ω b ± n ∆2 .This also generalizes the definitions of α and β withthe modification ξ b → ξ b,j = ω b − ν j . We assume thatthe limits α (cid:29) β (cid:29) j . As such, the leading order contributionof the Magnus expansion, χ scales as ( αβN ) − / and istherefore negligible.We decompose the second order Magnus term to χ = χ ,j + χ ,j,j (cid:48) . The former term does not couple betweendifferent normal modes of motion and therefore is a trivial3generalization of Eq. (8), it is given by, χ ,j = χ ,j ;h + χ ,j ; z χ ,j ;h = iT (cid:126) Ω b (cid:88) k B k,k + n σ + k + n σ − k e iφ + h.c + O (cid:0) α − (cid:1) χ ,j ; z = 2 iT (cid:126) Ω b (cid:88) j,k B k,k + n σ zk (cid:18) a † j a j + 12 (cid:19) + O (cid:0) α − (cid:1) . (18)where we defined B i,k = (cid:80) j η j,i η j,k / ξ b,j . That is, weobtain the same effective Hamiltonians as in Eq. (10) butwith the normalized spin-spin coupling B i,k , which comesabout due to contributions from all of the normal-modes,in analogy to [30]. Since the radial modes are bunched, typically | ξ b,j − ξ b,k | (cid:28) | ν j − ν k | . Consequently the red sidebandcan be used in order to eliminate H z , as described in thesingle normal-mode case above.The term χ ,j,j (cid:48) couples between mode j and mode j (cid:48) .It only contains operators of the form σ zk a † j a j (cid:48) , and itsconjugate; therefore, this term generate no spin-hopping.By using the red sideband frequency pair, the leadingorder contribution of this term scales as α − and istherefore neligible in the large αα