A Coherent Quantum Annealer with Rydberg Atoms
Alexander W. Glaetzle, Rick M. W. van Bijnen, Peter Zoller, Wolfgang Lechner
AA Coherent Quantum Annealer with Rydberg Atoms
A. W. Glaetzle,
1, 2, ∗ R. M. W. van Bijnen,
1, 2, ∗ P. Zoller,
1, 2 and W. Lechner
1, 2, † Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria (Dated: November 9, 2016)There is a significant ongoing effort in realizing quantum annealing with different physical plat-forms. The challenge is to achieve a fully programmable quantum device featuring coherent adiabaticquantum dynamics. Here we show that combining the well-developed quantum simulation toolboxfor Rydberg atoms with the recently proposed Lechner-Hauke-Zoller (LHZ) architecture allows oneto build a prototype for a coherent adiabatic quantum computer with all-to-all Ising interactionsand, therefore, a novel platform for quantum annealing. In LHZ a infinite-range spin-glass is mappedonto the low energy subspace of a spin-1/2 lattice gauge model with quasi-local 4-body parity con-straints. This spin model can be emulated in a natural way with Rubidium and Cesium atomsin a bipartite optical lattice involving laser-dressed Rydberg-Rydberg interactions, which are sev-eral orders of magnitude larger than the relevant decoherence rates. This makes the exploration ofcoherent quantum enhanced optimization protocols accessible with state-of-the-art atomic physicsexperiments.
Quantum annealing is a quantum computing paradigmwith the aim to solve generic optimization problems [1–4], where the cost function corresponds to the energy ofan infinite-range Ising spin glass [5]. Finding the optimalsolution of the problem is thus equivalent to determiningthe ground state of the spin glass. In quantum anneal-ing, this task is accomplished by adiabatic passage of asystem of N spins in the instantaneous ground state of aHamiltonian (denoted logical spin model ) of the form˜ H t = ˜ A t N (cid:88) ν =1 ˜ a ν ˜ σ ( ν ) x + ˜ B t (cid:34) N (cid:88) ν =1 ˜ h ν ˜ σ ( ν ) z + N (cid:88) ν<µ ˜ J µν ˜ σ ( ν ) z ˜ σ ( µ ) z (cid:35) . (1)Here ˜ σ { x,y,z } are Pauli spin operators, and schedulingfunctions ˜ A t and ˜ B t deform ˜ H t from a trivial initialHamiltonian with ( ˜ A , ˜ B ) = (1 ,
0) and transverse localfields ˜ a ν , into the spin glass Hamiltonian with ( ˜ A , ˜ B ) =(0 , h ν and infinite-range interactions ˜ J νµ [5]. Notethat implementing Eq. (A1) requires individually pro-grammable long-range interactions ˜ J µν , which is in con-tradiction to polynomially decaying interactions in coldatoms and molecule setups.Adopting the LHZ architecture [6], the infinite-rangespin glass is translated to a lattice spin model, wherenew physical spins ˆ σ ( i ) z represent the relative orientationof two logical spins ˜ σ ( ν ) z ˜ σ ( µ ) z of Eq. (A1). If two logicalspins are aligned in parallel, i.e. |↑↑(cid:105) or |↓↓(cid:105) , then thecorresponding physical spin is in state | + (cid:105) , while if thelogical spins are aligned anti-parallel, i.e. |↑↓(cid:105) or |↓↑(cid:105) ,then the physical spin is in state | - (cid:105) . The major advan-tage of this approach is that the interaction energy of apair of logical spins can now be implemented with a localfield acting on a single physical spin.A general optimization problem in the LHZ architec- (i) Optimization problem UV-laser (ii) Spin glass formulation (iii) LHZ (a)(d) (b) (c) DMD
Figure 1. (a) The cost function of a general optimizationproblem in the form of a spin glass with infinite-range interac-tions ˜ J µν is encoded in the LHZ architecture in local fields J i .(b) Rubidium (blue) and Cesium (red) atoms are trapped ina square lattice geometry representing physical and ancillaspins, respectively, where the spin degree of freedom is en-coded in two long-lived hyperfine states | + (cid:105) and | - (cid:105) . The pro-grammable local fields J i are induced by AC stark shifts fromlaser coupling the | - s (cid:105) state to low lying 5 P states using a dig-ital mirror device. The four-body gauge constraints at eachplaquette (e.g. the black dotted square) are engineered usingoff-resonant laser coupling of the | + s (cid:105) , | + a (cid:105) states to Rydberg P -states | r (cid:105) , | r (cid:105) or | r C (cid:105) and require only uniform illumina-tion of the system with UV laser light. ture becomesˆ H t = A t K (cid:88) i a i ˆ σ ( i ) x + B t K (cid:88) i J i ˆ σ ( i ) z + C t (cid:88) (cid:3) ˆ H (cid:3) , (2)with new schedule functions A t , B t and C t and transversefields a i . Physical spins are arranged on a square lat-tice [see blue spheres in Fig. 1], where the index i labelsthe entries of the matrix ˜ J µν . The number of physicalspins K equals the number of connections in the original a r X i v : . [ qu a n t - ph ] N ov model, which is quadratically larger than in the originalproblem [7]. This enlarged state space contains stateswhere collections of physical spins encode conflicting rel-ative orientations of the logical spins. These states can belocally identified and energetically penalized by 4-bodyconstraints H (cid:3) at each plaquette (cid:3) of the square lattice,such that at the end of the sweep plaquettes either con-tain all an even [6], or all an odd [8] number of spins inthe | - (cid:105) state, thus realising an even or odd parity repre-sentation of Eq. (A2). This ensures that the final groundstate of the LHZ Hamiltonian (A2) corresponds to thefinal ground state of the logical Hamiltonian (A1), andthus to the optimal solution of the optimization prob-lem. Importantly, the optimization problem is now en-coded in local fields ˜ J µν → J i , corresponding to singleparticle energy shifts. We show that the above model fora programmable quantum annealer can be emulated inan atomic Rydberg setup, which builds on the remark-able recent advances towards realizing complex spin mod-els with cold atoms in lattices interacting via designedRydberg-Rydberg interactions [9–13]. I. FOUR-BODY PARITY CONSTRAINTS
The key challenge of implementing ˆ H (cid:3) is resolved withRydberg atoms by combining (i) the odd parity repre-sentation [8] of Eq. (A2) with (ii) enhanced Rydberg-Rydberg dressing [14] schemes in a two-species mix-ture [15, 16]. In the odd parity representation, the sum ofthe four spins at each plaquette is either 2 or -2. We in-troduce a single ancilla qubit τ (cid:3) at each plaquette, whichcan compensate the two associated energies and makeodd parity plaquette states degenerate ground states ofthe constraint Hamiltonian ˆ H (cid:3) = (∆ (cid:3) /
4) ˆ S (cid:3) , with sta-bilizer operators ˆ S (cid:3) = (cid:80) i ∈ (cid:3) ˆ σ ( i ) z + 2ˆ τ (cid:3) z , and energy gap∆ (cid:3) . This allows to implement the four-body gauge con-straints via appropriately designed two-body Ising inter-actions between physical and ancilla qubits.Here we consider a more general and robust form of ˆ H (cid:3) ,consisting of all combinations of two-body interactionsalong the edges and diagonals of the plaquette, as well aswith the ancilla spin [see Fig. 2(a)], of the formˆ H (cid:3) (∆ (cid:3) /
2) = (cid:88) i,j ∈ edges ˆ σ ( i ) z ˆ σ ( j ) z + β (cid:88) i,j ∈ diag . ˆ σ ( i ) z ˆ σ ( j ) z + α ˆ τ (cid:3) z (cid:88) i ∈ (cid:3) ˆ σ ( i ) z , (3)where α and β are relative interaction strengths com-pared to spin interactions along the plaquette edge. Theenergy spectrum E (cid:3) of a single plaquette Hamiltonianis shown in Fig. 2(b), as a function of the parameters α and β . Importantly, there exists a parameter regime0 < − β < α < β with 0 < β <
1, where theodd parity states are degenerate and have a lower energythan the even parity states. Since the precise value of thegap in Fig. 2 is not relevant as long as it exceeds all other + + (iii) even (ii) odd, wrong ancilla(i) odd (iv) even - ++ edge diagonal ancilla (a)(b) Figure 2. (a) Four-body interactions between physical spins(blue) of the same plaquette are constructed from two-bodyinteractions between physical spins of strength 1 along theedge of the plaquette (left), interactions of strength β alongthe diagonal (middle) and additional interactions of strength α between an ancilla qubit (red) located at the center of eachplaquette and the surrounding physical qubits (right). (b)Eigenenergies E (cid:3) of the Hamiltonian of Eq. (3), as a functionof the physical spin-ancilla interaction strength α for a partic-ular β (cid:46)
1. Odd parity eigenstates with the right (i) or wrong(ii) ancilla orientation have an energy ± α . The maximallypolarized states (iii) with all four physical spins up or downhave energy 4 + 2 β ∓ α , while the ‘spin-ice’ states (iv) areindependent of the ancilla interaction α and have constantenergies − β and − β . The thick blue line indicates thewindow of interest where the odd parity states are the groundstates of the plaquette Hamiltonian. energy scales, ˆ H (cid:3) is quite robust against small variationsin interaction strengths, and the parameters α, β neednot be fine-tuned. II. RYDBERG IMPLEMENTATION
In the Rydberg quantum annealer illustrated in Fig. 1,qubits are encoded in two long-lived hyperfine groundstates | + s (cid:105) , | - s (cid:105) of Rb and | - a (cid:105) , | + a (cid:105) of Cs, represent-ing physical and ancilla spins, respectively. These statesare trapped in a deep optical or magnetic square latticewith unity filling [17, 18] and frozen motion [14, 15, 19,20]. In particular, we choose the | F = 1 , m F = 1 (cid:105) and | F = 2 , m F = 2 (cid:105) hyperfine states of the 5 S / groundstate manifold of Rb and the | F = 4 , m F = 4 (cid:105) and | F = 3 , m F = 3 (cid:105) hyperfine states of the 6 S / groundstate manifold of Cs. Choosing mixtures of Rb andCs has the advantage that unwanted cross-talk will bestrongly suppressed compared to a single species imple-mentation. The first term of Eq. (A2) can be realizedwith a coherent driving field of amplitude a i coupling thetwo physical spins, written in a rotating frame. The sec-ond term is obtained using single-particle AC-Stark shiftsfrom off-resonant laser coupling of the | - s (cid:105) spin state tolow-lying | e (cid:105) = | P (cid:105) states using a digital micro-mirrordevice [21].To implement the two-body interactions of Eq. (3)we turn to the technique of Rydberg dressing [22–25],where off-resonant laser light weakly admixes some Ryd-berg character into a ground state level. Specifically,we propose to couple the | + s (cid:105) and | + a (cid:105) states of Rb andCs using single photon transitions to Rydberg P states[11, 12, 26, 27]. For large laser detunings the Rydbergdressing acts as a perturbation. Thus the states | + s (cid:105) , | + a (cid:105) predominantly retain their ground state character and re-main trapped.Interactions between two spins i and j arise as spatiallydependent light shifts U ( s ) ++ and U ( a ) ++ of the dressed pairstates | + s + s (cid:105) and | + s + a (cid:105) , respectively [28]. These pairstates are coupled via two photon excitations to doublyexcited Rydberg states. Due to dipole-dipole interactionsthe energies of the doubly excited Rydberg states varystrongly as a function of the relative position R ij of theatoms, where the level shifts exceed typical laser detun-ings even at micrometer distances. Figure 3 shows theRydberg pair energies of (a) the 2 × | P / (cid:105) and (b) the2 × | P / (cid:105) Rydberg states of Rb and (c) the mixed | Cs :33 P / , Rb :45 P / (cid:105) Rydberg state, which show po-tential wells as a function of the relative distance due tothe vicinity of a F¨orster resonance [14]. Tuning the (two-photon) detuning of the dressing laser close to a minimumof a potential well results in strongly enhanced ground-state light shifts peaked at the position of the potentialminima. The particular Rydberg states are chosen suchthat the dressed ground state potentials U ( s ) ++ and U ( a ) ++ plotted in Fig. 3(d) show peaks at distances a L / √ a L and √ a L , commensurate with the square lattice geom-etry. III. RESULTS
The final spin-spin interactions between atoms i and j following from the light shifts areˆ H i = 14 (cid:88) i,j (cid:104) U ( s ) ++ ( R ij )ˆ σ ( i ) z ˆ σ ( j ) z + U ( a ) ++ ( R ij )ˆ σ ( i ) z ˆ τ ( j ) z (cid:105) , (4)apart from additional single-particle corrections to thelocal fields. The height of the two peaks of U ( s ) ++ at a L and √ a L for Rb-Rb [green line in Fig. 3(d)] andof U ( a ) ++ at a L / √ = Ω = 2 π ×
35 MHz, Ω C = 2 π ×
13 MHz, anddetunings ∆ = − π ×
618 MHz, ∆ = − π ×
373 MHzand ∆ C = 2 π ×
83 MHz which leads to light-shifts (a)(b)(c)(d) (i) (ii)
Figure 3. Rydberg-Rydberg interaction energies E around the(a) | P / , P / (cid:105) and (b) | P / , P / (cid:105) states of Rband (c) the mixed | P / , P / (cid:105) Rydberg state of
Csand Rb in a magnetic field of B z = 26 G along the z axis.Due to a close-by F¨orster resonance the pair potentials devel-ops a local minimum. The Rydberg states are chosen suchthat the position of the minima of these wells are commen-surate with the square lattice geometry. The intensity of theblue coloring indicates the overlap with the targeted Rydbergstates. Panel (d) shows the resulting interaction potential U ++ between two Rydberg-dressed (i) Rb-Cs (yellow) and (ii) Rb-Rb (green) ground state atoms. Exciting close to the minimaof the interaction potential (vertical dashed lines) yields dras-tically enhanced peaked-like interactions at a L / √ a L and √ a L thereby realizing the required plaquette interaction ofEq. (3) for α = 2 and β = 1. of U ( s ) ++ ( a L ) = U ( s ) ++ ( √ a L ) = − π ×
40 kHz for Rb-Rband U ( a ) ++ ( a L / √
2) = − π ×
80 kHz = 2 U ( s ) ++ ( a L ) for Rb-Cs (see supplementary information). We note that anexternal magnetic field and slight vertical offset of theCs atoms is used to obtain these numbers.All interactions are at least two orders of magnitudesmaller at lattice distances that do not belong to theplaquette, thus minimizing unwanted cross-talk. Thisallows to restrict the sum in Eq. (4) to atoms belong-ing to the same plaquette, thus realizing ˆ H (cid:3) of Eq. (3)for the optimal parameters α = 2 and β = 1. For the τ =50 τ =100 τ =150 (a) (b) Figure 4. (a) Illustration of the time-dependent spectrum forthe minimal instance shown in Fig. 1. (b) Histogram of thesuccess probability, i.e. the probability P to populate theground state at final time τ for different sweep times. above system parameters, we obtain a final energy gap∆ (cid:3) = − π ×
20 kHz [29]. Due to the finite lifetime ofthe Rydberg states, the dressed ground state interactionscome at a cost of an effective decoherence rate 1 /τ foreach qubit. However, since there is only a small Ryd-berg component admixed, the effective decay rate is alsoonly a correspondingly small fraction of the bare Ryd-berg decay rate. Ultimately, the figure of merit for fullycoherent operation of the quantum annealer is the ratioof the attained interaction strength versus the effectivedecay rate. In the enhanced dressing scheme this ra-tio becomes particularly favorable and is of the order of | ∆ (cid:3) | τ ≈ for the system parameters above (see sup-plementary information).Using the above potentials we demonstrate numeri-cally the feasibility of the Rydberg annealer for the min-imal instance (see Fig. 1) with 8 qubits and 3 ancil-las. Fig. 4 depicts the time dependent spectrum in re-duced units. The time-dependent spectrum for instanceof Hamiltonian Eq. (A2) for random | J i / ∆ (cid:3) | <
1. Thesweep functions A t , B t and C t are simple linear functions.Note, that the efficiency can be considerably increased byadopting non-linear sweep functions. In Fig. 4 all ener-gies are given relative to the ground state energy. The pronounced minimal gap is an order of magnitude smallerthan the gap in the final state. Figure 4(b) shows thehistogram of the success probability P = | (cid:104) ψ ( τ ) | ψ gs (cid:105)| ,defined as the overlap of the final state ψ with the groundstate ψ gs , averaged over N r = 40 random instances fordifferent sweep times far below the decoherence times τ < τ /K . For the fastest switching time | ∆ (cid:3) | τ = 50 theaverage success probability is 75% and approaches unityfor slower sweeps. IV. CONCLUSIONS
The proposed implementation of a quantum annealerwith ultracold Rydberg atoms in optical lattices pro-vides a new platform for adiabatic quantum computing,featuring a highly controllable environment to explorethe many-body adiabatic passage, the role of entangle-ment and effects of decoherence during the annealingsweep. The large lifetimes of Rydberg dressed atoms en-able coherent quantum annealing as an alternative to thecurrent paradigm of quantum enhanced thermal anneal-ing [30]. We anticipate that due to the coherent evolutionthe number of spins in future experiments can readilybe extended well beyond the minimal example presentedhere, by using shorter annealing cycles with many rep-etitions [31], or by employing counter-diabatic drivingschemes that could greatly increase the attained fideli-ties [32]. Finally, our proposal allows to realize atomicquantum simulators of arbitrary infinite-range Ising spinglass models (see e.g. Refs. in [33]), and the combi-nation of multi-color Rydberg-dressed interactions withtwo-species mixtures has applications in realizing Z lat-tice gauge theories beyond the present example [34].We acknowledge discussions with C. Gross,H. C. N¨agerl, M. Saffman and J. Zeiher. This work issupported by the Austrian Science Fund SFB FoQuS(FWF Project No. F4016-N23), the European ResearchCouncil (ERC) Synergy Grant UQUAM and the EUH2020 FET Proactive project RySQ. WL acknowledgesfunding from the Hauser-Raspe Foundation. Appendix A: Numerical example using LHZ
A general quantum annealing problem with an infinite-range spin glass Hamiltonian consisting of N logical spins ˜ σ with K connections has the form ˜ H (log) t = ˜ A t N (cid:88) ν =1 ˜ a ν ˜ σ ( ν ) x + ˜ B t N (cid:88) ν<µ ˜ J µν ˜ σ ( ν ) z ˜ σ ( µ ) z (A1)with scheduling functions ˜ A t and ˜ B t , local transverse fields ˜ a ν and programmable infinite-range interactions ˜ J µν . - Figure 5. The minimal instance shown in Fig. 1 in the main text corresponds to a fully connected spin glass with N = 4 spins.In the LHZ architecture, the entries of the interaction matrix ˜ J µν in Eq. (A1) translate to local fields J i . Here, the index i labellocal fields ordered from bottom to top. The bottom row of local fields are fixed to large positive numbers J = J = 5 | ∆ (cid:3) | .The minimal instance discussed here consists of N = 8 spins with 3 ancilla spins accounting for the three 4-body plaquetteconstraints (red dots). Using the LHZ architecture it can be mapped on a spin modelˆ H (LHZ) t = A t K (cid:88) i a i ˆ σ ( i ) x + B t K (cid:88) i J (cid:48) i ˆ σ ( i ) z + C t (cid:88) (cid:3) ∆ (cid:3) (cid:89) i ∈ (cid:3) ˆ σ ( i ) z , (A2)with K physical spins ˆ σ , arranged on a square lattice (green circles in Fig. 5), and problem independent 4-bodyinteractions between spins belonging to the same plaquette (cid:3) of the square lattice (red dots in Fig. 5). Here, a i aretransverse local fields and ∆ (cid:3) is the four-body interaction strength (which, for simplicity, we assume to be equal for allspins and plaquettes, respectively), and C t is the scheduling function of the constraints. In the LHZ architecture theprogrammable interaction matrix ˜ J µν is translated to programmable single-particle energy shifts J (cid:48) i which correspondto the entries of the matrix ˜ J µν .In an odd parity representation the 4-body interactions are resolved by introducing an ancilla qubit ˆ τ (cid:3) in the middleof each plaquette with fine-tuned 2-body interactions of the formˆ H (odd) t = A t (cid:32) K (cid:88) i a i ˆ σ ( i ) x + (cid:88) (cid:3) a (cid:3) ˆ τ (cid:3) x (cid:33) + B t K (cid:88) i J i ˆ σ ( i ) z + C t (cid:88) (cid:3) ∆ (cid:3) (cid:32)(cid:88) i ∈ (cid:3) ˆ σ ( i ) z + 2ˆ τ (cid:3) z (cid:33) , (A3)where the local fields J i are the entries of the matrix ( − µ ( ν − µ ) ˜ J µν , and a (cid:3) is a transverse local field driving theancilla spins.In the following we illustrate the annealing sweep and the time-dependent spectrum of Eq. (A3) for the minimalinstance of 8 logical qubits (Rubidium atoms) and 3 ancilla qubits (Cesium atoms) which makes a total of 11 qubitsarranged on three plaquettes illustrated in Fig. 1 of the main text and Fig. 5 of the supplemental material. This setupcorresponds to 4 all-to-all connected logical qubits in Eq. (A1).The order of the indices of the local fields is from bottom to top (e.g. ˜ J → J ). In the minimal instance depictedin Fig. 5 these are the three plaquettes formed by physical qubits (1 , , , , , ,
7) and (6 , , , − µ ( ν − µ ) will flip the sign of the fourth local field, i.e. J (cid:48) = − J , such that theparity of all plaquettes is odd.We demonstrate the feasibility of quantum annealing for this small instance numerically calculating the successprobability P for constant ∆ (cid:3) and a i . This is the overlap of the wave function at final time T with the ground statewavefunction ψ gs P = | (cid:104) ψ ( T ) | ψ gs (cid:105)| . (A4)The statistics of P is obtained by solving the time-dependent Schr¨odinger equation with Hamiltonian (A3), usingthe Rydberg interaction potentials given in the main text and in Sec. B of the supplemental information, with exactdiagonalization and explicitly including the dynamics of the ancilla qubits. Statistics is taken from N i = 40 randominstances of an interaction matrix with J i / | ∆ (cid:3) | ∈ [ − . , . A t = T − t and B t = C t = t bothinterpolate linearly from 0 to T . Note, that this linear ramp is a pessimistic toy model and more sophisticated choicesof the schedule function could considerably increase the success probability. The bottom two qubits i = 1 and i = 2fixing the gauge constraints in the parity architecture are explicitly part of the dynamics and subject to strong localfield J = J = 5 | ∆ (cid:3) | . The total time is | ∆ (cid:3) | T = 50 , , | ∆ (cid:3) | τ ≈ . The results shown inFig. 3 in the main text show that for this system size the overlap with the ground state wave function is between 70%and 90% for the given parameters. Appendix B: Rydberg dressed interactions1. Rydberg-Rydberg potentials
In this section, we outline the calculation of the Rydberg-Rydberg interaction potentials that underly the effectiveinteractions described in the main text. Throughout the discussion we set (cid:126) = 1 for notational clarity.The Rydberg states | r (cid:105) = | n ( (cid:96)s ) jm j (cid:105) of the valence electron with spin s = 1 / n , orbital angular momentum (cid:96) , and total angularmomentum j with its projection m j along the quantization axis. The states | r (cid:105) are eigenstates of a single particleHamiltonian ˆ H A = (cid:88) r (cid:16) E (0) r + ∆ E m j (cid:17) | r (cid:105)(cid:104) r | . (B1)The bare atomic energies E (0) r = − E Ryd / ( n − δ (cid:96),j ) are different for rubidium and cesium atoms, and are determinedby their respective quantum defects δ (cid:96),j , see e.g. Refs. [35, 36] and [37], respectively. We include level shifts∆ E m j = µ B g j B z m j due to a magnetic field B = B z ˆz which lifts the Zeeman degeneracy and sets the quantizationaxis along the z -axis. Here, µ B = 1 . h MHz/G is the Bohr magneton and g j is the Lande factor for the Rydberglevel.The Hamiltonian describing the internal-state dynamics of two particles separated by some distance R = ( R, ϑ, ϕ ),to good approximation, given by ˆ H = ˆ H (1) A ⊗ ˆ I + ˆ I ⊗ ˆ H (2) A + ˆ V dd ( R ) , (B2)with the first term of each operator product acting on the first particle, and the second term acting on the second.The single particle atomic Hamiltonians ˆ H (1) A and ˆ H (2) A can describe rubidium or cesium atoms. The third termcorresponds to the interactions coupling the internal states of the two particles. For the typical distances consideredin this work these interaction couplings are predominantly of dipole-dipole formˆ V dd ( R ) = d · d R − d · R )( d · R ) R = − (cid:114) π R (cid:88) µ,ν C , µ,ν ; µ + ν Y µ + ν ( ϑ, ϕ ) ∗ d (1) µ d (2) ν . (B3)Here, ˆ d and ˆ d are the dipole transition operators for atom 1 and atom 2, respectively, with spherical components d (1) µ and d (2) ν and ( R, ϑ, ϕ ) are the spherical components of the relative vector. With C and Y we denote Clebsch-Gordancoefficients and spherical harmonics, respectively.We proceed by selecting a large basis set of pair states | rr (cid:48) (cid:105) = | n ( (cid:96)s ) jm j (cid:105) ⊗ | n (cid:48) ( (cid:96) (cid:48) s (cid:48) ) j (cid:48) m (cid:48) j (cid:105) , which are productstates of eigenstates of the single particle Hamiltonians ˆ H (1) A and ˆ H (2) A , with energies E rr (cid:48) given by the sum of thecorresponding single-atom energies. In this basis, the pair Hamiltonian of Eq. (B2) turns into a large but sparsematrix, with off-diagonal matrix elements (cid:104) r i , r j | V dd ( R ) | r k , r l (cid:105) = R i,k R j, × ( − ) s − m i [ (cid:96) i ][ j i ][ (cid:96) k ][ j k ] (cid:26) (cid:96) i (cid:96) k j k j i s (cid:27) (cid:18) (cid:96) k (cid:96) i (cid:19) × ( − ) s − m j [ (cid:96) j ][ j j ][ (cid:96) l ][ j l ] (cid:26) (cid:96) j (cid:96) l j l j j s (cid:27) (cid:18) (cid:96) l (cid:96) j (cid:19) × (cid:34) − (cid:114) π (cid:88) µ,ν C , µ,ν ; µ + ν (cid:18) j k j i m k µ − m i (cid:19) (cid:18) j l j j m l ν − m j (cid:19) Y µ + ν ( ϑ, ϕ ) ∗ (cid:35) , (B4)with R i,k = (cid:104) r i || r || r k (cid:105) the radial integral and the abbreviation [ x ] = √ x + 1. The matrix is block diagonal, couplingonly (cid:96) to (cid:96) ± ϑ = π/ M = m + m can change by 0 or ± H numerically for a range of distances R for fixed ϑ = π/
2. The basis set is chosen sufficientlylarge, containing ∼ states, to ensure convergence of eigenstates and eigenvalues down to distances of R ∼ . µ m.The diagonalization procedure yields distance dependent molecular eigenenergies E µ ( R ) depending only on the radialdistance R , which are the interaction potentials plotted in Fig. 3(a)-(c) in the main text. Simultaneously thecorresponding molecular eigenstates | µ ( R ) (cid:105) are computed, | µ ( R ) (cid:105) = (cid:88) rr (cid:48) c ( µ ) rr (cid:48) ( R ) | rr (cid:48) (cid:105) , (B5)which are superpositions of pair product states | rr (cid:48) (cid:105) with coefficients c ( µ ) rr (cid:48) ( R ) depending on ( R, ϑ, ϕ ). The coloringof the curves in Fig. 3(a)-(c) of the main text is indicative of the overlap with the laser targeted Rydberg state, |(cid:104) r λ r λ | µ ( R ) (cid:105)| = | c ( µ ) rλ ,rλ ( R ) | , with λ , λ { ‘1’, ‘2’, ‘C’ } defining the particular Rydberg states [see e.g. Fig. 1in the main text].
2. Rydberg dressing potentials
Having obtained the molecular eigenstates | µ ( R ) (cid:105) and their energies E µ ( R ), we can now proceed to calculate thelight shifts of the ground state levels resulting from laser coupling to the excited state manifold.The laser couplings are characterized by a Rabi frequency, Ω λ , and detuning from a targeted Rydberg level, ∆ λ .The subscript λ = { ‘1’, ‘2’, ‘C’ } indexes the three distinct laser couplings discussed in the main text: • λ = ‘1’ pertains to the laser coupling the | + s (cid:105) = | F = 2 , m F = − (cid:105) hyperfine ground state of Rb to the | r (cid:105) = | P / , m J = − / (cid:105) Rydberg state, • λ = ‘2’ indexes the laser coupling of the | + s (cid:105) state to the | r (cid:105) = | P / , m J = − / (cid:105) Rydberg state, • λ = ‘C’ refers to the laser coupling of the | + a (cid:105) = | F = 4 , m F = − (cid:105) hyperfine ground state of an ancilla Cs atomto the | r C (cid:105) = | P / , m J = − / (cid:105) Rydberg state.All lasers propagate in the xy -plane, with linear polarization along the z -axis coinciding with the quantization axisand magnetic field direction. This geometry is chosen such that the total system and resulting interaction potentialsare rotationally symmetric along the z -axis, and in particular the energies of the plaquette configurations are invariantunder rotation and mirroring operations.In the Rydberg dressing limit, the laser coupling is far off-resonant with Ω λ (cid:28) | ∆ λ | , such that the effect of thelaser coupling is perturbative with an associated small parameter (cid:15) = Ω λ / | ∆ λ | . In the following we only considerthe laser-coupled ground states | + s (cid:105) , | + a (cid:105) , as the uncoupled ground states | - s (cid:105) , | - a (cid:105) play no role in the interactioncalculation. To describe the state of two laser coupled ground state particles, we work in a basis consisting of pairstates | g g (cid:105) , where both particles are in one of the two ground state | + s (cid:105) or | + a (cid:105) , i.e. g , g = + s , + a . This basisis extended with pair states | g r λ (cid:105) , | r λ g (cid:105) where one of the two particles is in the ground state while the other isexcited to the laser-targeted Rydberg state, with λ , λ { ‘1’, ‘2’, ‘C’ } . Due to appropriately chosen laser frequencyand polarization we only couple to these targeted Rydberg states. Finally, the basis also contains the molecular states | µ ( R ) (cid:105) we obtained numerically in the previous section.The ground pair state | g g (cid:105) is defined to have an energy 0. The atomic Rydberg states of particle 1 are definedin a rotating frame corresponding to the laser frequency ω λ , such that the near-resonant, laser-targeted Rydbergstate | r λ (cid:105) has an energy − ∆ λ . Similarly, the Rydberg states of particle 2 have their energy defined relative to theenergy − ∆ λ of the | r λ (cid:105) state in the rotating frame of the laser transition λ
2. The molecular Rydberg states | µ ( R ) (cid:105) therefore have an energy δ ( µ ) ( R ) = E µ ( R ) − E r λ r λ − ∆ λ − ∆ λ . (B6)Expressed in the basis described above, the two-particle Hamiltonian thus becomesˆ H = − ∆ λ | r λ g (cid:105) (cid:104) r λ g | − ∆ λ | g r λ (cid:105) (cid:104) g r λ | + (cid:88) µ δ ( µ ) ( R ) | µ ( R ) (cid:105) (cid:104) µ ( R ) | + ˆ H L ( R ) , (B7) Figure 6. Energy levels involved to obtain the dressed ground state potentials. The double ground state | g , g (cid:105) is laser-coupled to the | r λ , g (cid:105) or | g , r λ (cid:105) single-excited Rydberg state with Rabi frequencies Ω λ and Ω λ and energies ∆ λ and∆ λ , respectively. These state can be further excited to doubly-excited Rydberg states | ¯ µ ( R ) with position dependent Rabifrequencies Ω (¯ µ ) λ and Ω (¯ µ ) λ . Close to a molecular potential well the energy δ (¯ µ ) ( R ) can be much smaller than the single particledetunings ∆ λ resulting in an enhanced light shift of the dressed | g , g (cid:105) state. where the operator ˆ H L is the laser couplingˆ H L ( R ) = Ω λ | r λ g (cid:105) (cid:104) g g | + Ω λ | g r λ (cid:105) (cid:104) g g | + h . c . + (cid:88) µ (cid:34) Ω ( µ ) λ ( R )2 | µ ( R ) (cid:105) (cid:104) g r λ | + Ω ( µ ) λ ( R )2 | µ ( R ) (cid:105) (cid:104) r λ g | (cid:35) + h . c ., where the first line contains terms coupling the pair ground state | g g (cid:105) to the single excited pair states, and thesecond line couples the singly excited states to the molecular states. The effective coupling strength to the molecularstates, Ω ( µ ) λ ( R ) = Ω λ c ( µ ) rλ ,rλ ( R ) , (B8)with λ = λ , λ
2, is reduced with a factor c ( µ ) rλ ,rλ ≤
1, and is additionally dependent on the distance.The light shifts of the pair ground state presented in the main text, Fig. 3(d), are calculated by numericallydiagonalizing Hamiltonian (B7). Here, however, it is instructive to analyze the light shifts perturbatively. In particular,we consider the case where there is one dominant molecular state, denoted µ = ¯ µ , with significant effective lasercoupling strength and lying close to 0 in energy. This is precisely the situation described in the main text in thevicinity of the point of closest approach of the molecular wells to the pair ground state, as depicted in Fig 3(a)-(c) inthe main text. Figure 6 illustrates the relevant pair states, their energies and the couplings between them.To second order in the laser coupling, the light shift of the pair ground state is constant and independent of distance, E (2) = 12 Ω λ (cid:15) λ + 12 Ω λ (cid:15) λ , (B9)where we have defined (cid:15) λ = Ω λ / λ . Thus far, the laser light shift is merely a single particle effect. Interactions enterin fourth order perturbation theory, when we consider processes involving couplings to the molecular state | ¯ µ ( R ) (cid:105) .The resulting contribution to the light shift is (ignoring terms contributing to the single particle light shift) E (4)int = (cid:15) µ δ (¯ µ ) ( R ) , (B10)where we have defined (cid:15) ¯ µ = Ω λ Ω (¯ µ ) λ δ (¯ µ ) ( R ) (cid:18) λ + 1∆ λ (cid:19) = Ω λ Ω (¯ µ ) λ δ (¯ µ ) ( R ) (cid:18) λ + 1∆ λ (cid:19) . (B11)Clearly, choosing the laser detunings such that δ (¯ µ ) ( R ) becomes small, boosts the interaction strength. The aboveperturbative expression is valid as long as δ (¯ µ ) ( R ) (cid:29) (cid:15) λ Ω (¯ µ ) λ . A particular situation where the perturbative treatmentbreaks down occurs when a molecular state crosses the zero energy level. At such a point, pairs of Rydberg atomsare resonantly excited by the laser, instead of the intended weak admixture. The system parameters in the main textare chosen such that this situation is avoided, by ensuring that no significant resonances occur at lattice distances.We included checking the ’cross’ potentials, i.e. the molecular potentials for the case λ λ | (cid:103) g g (cid:105) , becomes | (cid:103) g g (cid:105) = | g g (cid:105) + (cid:15) λ | r λ g (cid:105) + (cid:15) λ | g r λ (cid:105) − (cid:15) ¯ µ | ¯ µ ( R ) (cid:105) + O (cid:32) Ω λ ,λ ∆ λ ,λ (cid:33) , (B12)where we have ignored the normalization, and used δ (¯ µ ) ( R ) (cid:28) ∆ λ ,λ to truncate the expansion after the third term.Assuming for simplicity a single decoherence rate Γ for all Rydberg states, we see that the second and third termin Eq. (B12) each introduce an effective decoherence rate (cid:15) λ Γ to the dressed pair state, whereas the third termintroduces a decoherence rate 2 (cid:15) µ Γ, where the factor 2 in front stems from the fact that the state | ¯ µ ( R ) (cid:105) has twoparticles in the excited state. The total decoherence per particle is evidentlyΓ eff = 12 ( (cid:15) λ + (cid:15) λ + 2 (cid:15) µ )Γ . (B13)The figure of merit for realizing fully coherent operation of the quantum annealer, i.e. the ratio of interactionstrength versus decoherence rate (per particle), is now readily computed from the results obtained above. Using Eqs.(B10) and (B13), we have that E (4)int Γ eff = 2 (cid:15) µ δ (¯ µ ) ( R )( (cid:15) λ + (cid:15) λ + 2 (cid:15) µ )Γ . (B14)For the system parameters employed in the main text we thus obtain: • λ λ | r (cid:105) = | P / , m J = − / (cid:105) , for which the single particle lifetime is τ = 1 / Γ = 54 µ s[38], and at the minimum of the selected potential well δ (¯ µ ) ( R ) = 2 . c (¯ µ ) (cid:39) .
32, leading to a finalfigure of merit E (4)int Γ eff (cid:39) . × , • λ λ | r (cid:105) = | P / , m J = − / (cid:105) , with single particle lifetime τ = 1 / Γ = 75 µ s [38], and δ (¯ µ ) ( R ) = 5 . c (¯ µ ) (cid:39) .
28, leading to a final figure of merit E (4)int Γ eff (cid:39) . × , • λ , λ C : simultaneous dressing of Rb to | r (cid:105) = | P / , m J = − / (cid:105) and Cs to | r C (cid:105) = | P / , m J = − / (cid:105) , with averaged single particle lifetime τ C = 1 / Γ C = 50 µ s [38], and δ (¯ µ ) ( R ) = 10 . c (¯ µ ) (cid:39) .
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