Quantum simulation of Cayley-tree Ising Hamiltonians with three-dimensional Rydberg atoms

Significant efforts are being directed toward developing a quantum simulator capable of solving combinatorial optimization problems. The challenges are Hamiltonian programming in terms of high-dimensional qubit connectivities and large-scale implementations. Here, we report a quantum simulation demonstration of Ising Hamiltonians with up to N = 22 spins mapped on various Cayley-tree graphs. For this, we use three-dimensional arrangements of Rydberg single atoms arranged in such a way that their Rydberg atoms and blockaded strong couplings respectively represent the vertices and edges of each graph. Three different Cayley-tree graphs of Z = 3 neighbors and of up to S = 4 shells are constructed, and their many-body ground states and Néel’s order formations are experimentally probed. The antiferromagnetic phase in regular Cayley trees and frustrated competing ground states in a dual-center Cayley tree are directly observed, demonstrating the possibilities of high-dimensional qubit connections in quantum simulators.

In this paper, we consider, as a prototypical fractal structure, Cayley-tree graphs of neutral atom arrangements in which atoms and strongly interacting atom pairs respectively represent vertices and edges of the graphs (see Fig. 1). Cayley trees are homogeneous and isotropic tree graphs of a fixed number of edges and no loop [40,41]. Their infinite version is a Bethe lattice, widely used in various physics areas as a fundamental theoretical platform, often providing exactly solvable models in classical and quantum problems [42]. In experiments to be described below, we use three-dimensional (3D) arrangements of neutral atoms to construct atomic Cayley graphs of coordination number Z = 3 and shell number up to S = 4, as in Fig. 1(a), and run quantum simulation to probe the many-body ground states of the corresponding Cayley-tree Ising-Hamiltonians.
In our atom arrangements, each atom represents a vertex of a Cayley-tree graph and each pair of strongly-interacting atoms an edge. Nonconnected atoms are supposed to interact with each other much weaklier than connected atoms. The Hamiltonian of these atoms being coherently and simultaneously excited to a Rydberg energy state is given (in units of h = 1) bŷ where N is the number of atoms, is the Rabi frequency, is the detuning, and U jk = C 6 /| r j − r k | 6 is the pairwise atom interaction in the van der Waals interaction regime [37,38]. Pauli operatorsσ x,z are defined for a pseudospin 1 2 system composed of the ground state |↓ = |5S 1/2 , F = 2, m F = 2 and Rydberg state |↑ = |71S 1/2 , m J = 1 2 of each atom, andn = (1 + σ z )/2. For the strong and weak interactions of connected and nonconnected atom pairs, respectively, we set the distances of connected atoms the same and within the Rydberg-blockade radius, i.e., | r j − r k | = d < r b ≡ (C 6 /h ) 1/6 = 9.8 μm, for ( j, k) ∈ E (the edge set of a graph G), and the distances of all others are | r j − r k | > r b .
The two-dimensional (2D) arrangement of G 10 is made with d < r b < √ 3d to satisfy the above condition, in which, as shown in Fig. 1(e), the last-shell atoms of different branches are separated more than √ 3d, having significantly weaker interactions (at most 1 27 times smaller) than U ≡ C 6 /d 6 of the connected atom pairs. However, G 22 and G 14 cannot be constructed to be planar because, in 2D arrangements, atoms of different outermost branches are too close, requiring nonplanar, 3D arrangements. As shown in Fig. 1(c), we rotate the last-shell branches of G 22 by an angle of 2π/5 about the axes along the previous branches, so that all the last-shell atoms are well separated (more than √ 3d) from each other. In G 14 , some last-shell branches are rotated similarly, as shown in Fig. 1(d). As-constructed atom arrangements are shown in Figs. 1(e)-1(g), respectively, for G 10 , G 22 , and G 14 .

III. PHASE DIAGRAMS OF CAYLEY-TREE ISING SPINS
With atoms arranged on one of the above Cayley-tree graphs, we perform quantum simulation to find the manybody ground state of a target Hamiltonian. The atoms are initially prepared in |↓↓ · · · ↓ (the para-magnetic down spins, which we denote by Phase I), which is the ground state of the initial HamiltonianĤ where E is the edge set of G ∈ {G 10 , G 22 , G 14 }, f is the final detuning, and no couplings are assumed for unedged atom pairs. We note thatĤ G is an Ising spin-glass Hamiltonian, given bŷ where J = U/4 is the coupling, h C z = 3U/4 − f /2 and h V z = U/4 − f /2 are the local fields, and C and V denote the vertices in the core (inner shells) and valence (outermost) shell, respectively, of a Cayley-tree graph G(E , C + V ). In general, as the ground states of an Ising Hamiltonioan depend on the specific atom arrangement of G, finding such arrangementspecific ground states of an arbitrary Ising-spin graph is an NP-complete problem [44].
However, Cayley-tree graphs allow heuristic understanding of their phase diagrams in Fig. 2 as follows: Cayley trees have more atoms on the valence shell than on the rest (the inner shells). Therefore, with HamiltonianĤ G in Eq. (3) (of positive coupling U > 0 in our consideration), the valence spins are all aligned either up or down depending upon f > 0 or f < 0, respectively, resulting in the f = 0 phase boundary. For f < 0, inner-shell spins, adjacent to the valence spins, also favor down spins, as their couplings Un ( j)n(k) to adjacent outer-shell down spins are zero, which subsequently results in all down spins (so paramagnetic down spins, Phase I). For f > 0, single-spin flipping energy ( f ) competes with adjacent antiferromagnetic couplings. For f > 3U , the former is always higher than the latter (of max 3U ), so all spins are up (paramagnetic up spins, Phase II). In between, 0 < f < 3U , the antiferromagnetic coupling is stronger, thus favoring the antiferromagnetic phase (Phase III) of shellby-shell alternating spins. The resultingĤ G phase diagram is shown for G 10 in Fig. 2(a), in which the antiferromagnetic phase (Phase III) has up spins on the first and third shells and down spins on the second. The phase diagram for G 22 differs from G 10 only in Phase III, where the odd-numbered shells are down spins and the even-numbered shells are up spins. In short, the phase diagram ofĤ G (of positive U ) for regular Cayley graphs (G 10 and G 22 in our consideration) has the para-antiferro phase boundaries determined by f = 3U and f = 0, so the antiferromagnetic ordering is expected in the region 0 < f < 3U (Phase III), as in Fig. 2(a).
The phase diagram for G 14 , the dual-center Cayley tree, is a little more complex than regular Cayley trees because of the frustrations of the center spins. Exact diagonalization of H G 14 finds the phase diagram as shown in Fig. 2(b), in which the paramagnetic phases (Phases I and II) are the same as G 10 , but Phase III is split to Phases IV and V of respective center-spin configurations of |↑↑ and (|↑↓+ ↓↑ )/ √ 2. The new phase boundary is U = f , along which the single-spin flipping cost equals the additional frustration cost of the center spins. It is noted that, in Phase V, other energy-degenerate ground states, with respective center-spin configurations of |↑↓ , |↓↑ , and (|↑↓− ↓↑ )/ √ 2, are all dark states (not coupled byĤ to the given initial state) due to the symmetry ofĤ in Eq. (2).

IV. EXPERIMENTAL VERIFICATION OF CAYLEY-TREE ISING PHASES
Experiments are performed to verify the antiferrolike ground states (Phases III, IV, and V) of Cayley-tree Ising spins (see Sec. VI for technical details). Quantum annealing proceeds with three stages of time evolution, along a vertical control path from Phase I to either Phase III, IV, or V in Fig. 2. In the first stage (0 < t < 0.1t f ), Rabi frequency is slowly turned on from i = 0 to 0 = 1.1(2π ) MHz, while detuning is maintained at = −2 0 for the paramagnetic down-spin ordering (|h C,V z | J) of Phase I. In the second stage (0.1 < t/t f < 0.9), the detuning is swept from −2 0 to 2 0 , while the Rabi frequency is maintained at 0 . In the final stage (0.9 < t/t f < 1), the detuning is maintained at f = 2 0 (for antiferromagnetic ordering 0 < f < 3U ) and the Rabi frequency is slowly turned off from 0 to f = 0. The total operation time t f = (2π )3.2/ 0 = 2.9 μs is experimentally chosen long enough to maximize the ground-state probability of G 10 within the coherence time of 10 μs [43] and used for all experiments including G 14 and G 22 . With the given control path, nonadiabatic leakages occur during the evolution; however, numerical simulations find that resulting excited-state probabilities are significantly smaller than the ground states for all three graphs. After the time evolution, a resulting spin configuration is detected with the fluorescence of ground-state atoms during 40 ms cyclic transitions to |5P 3/2 , F = 3 , and the procedure is repeated until the probability distribution of all spin configurations is obtained. The final Hamiltonianŝ H G (U, f = 2 0 ) are chosen at various phases in Fig. 2: 1 and 2 for Phase III of G 10 and G 22 , respectively; 3 for Phase IV of G 14 ; and 4 and 5 for Phase V of G 14 .

V. NÉEL'S ORDER FORMATION DYNAMICS
Furthermore, we measure the phase formation dynamics during the quantum annealing process. Figure 5 3 , 4 , and 5 , respectively, in the G 14 phase diagram in Fig. 2(b). individual dephasing (∼36 kHz 0 ) due to the spontaneous decay during Rydberg excitations, and collective dephasing (∼3 kHz) from laser phase noise [46]. The numerical calculation in accordance with the observed maximal Néel's order of O N (t f ) = 0.48 (2) indicates that errors are largely due to the SPAM errors accumulated for N = 10 atoms. The calculation also suggests that the maximal Néel's order before measurements was O N (t f ) = 0.82 and that the final ground-state probability was 61%.

VI. EXPERIMENTAL DETAILS
The above experiments were performed with a Rydbergatom quantum simulator previously reported elsewhere [43,[45][46][47]. In the quantum simulator, rubidium ( 87 Rb) atoms were initially prepared in the hyperfine ground state |↓ = |5S 1/2 , F = 2, m F = 2 and trapped with optical tweezers. A spatial light modulator (SLM, Meadowlark ODPDM512) was used to create the 3D array of 2N optical tweezers, and an electrically focus-tunable lens (EL-16-40-TC from Optotune) verified the positions of captured atoms in all atom planes. The SLM was computer-programed with a weighted Gerchberg-Saxton (w-GS) algorithm [48,49] so that the resulting electric field near each target site (x, y, z) was created as E (x, y, z) = E 0 X,Y e i (X,Y ) e −iT , where X, Y are the SLM coordinates on the Fourier plane of the optical tweezers, (X, Y ) is the SLM phase pattern, and T = 2π (xX + yY )/ f λ + π z(X 2 + Y 2 )/ f 2 λ is the transfer kernel for an optical tweezer of Fresnel focal length f . The phase pattern (for 2N optical tweezers) was obtained with the w-GS condition given by (X, Y ) = arg[ 2N j=1 w j E j (x, y, z)e iT j /|E j (x, y, z)|], where each weighting factor w j for jth optical tweezer was optimized through adaptive iterations. The number of the w-GS iterations was about five, taking about 20 ms to generate an array of 50 tweezers. Deterministic rearrangement of N atoms to target sites, about 20 μm apart from the initial atom reservoir, was performed with a consecutive 45 frames of moving traps programed with phase induction [49]. Over 90% target-occupation probabilities were achieved in 900 ms reconfiguration, and all measurements were performed with defect-free arrangements.

VII. CONCLUSIONS
We have explored the possibilities of Rydberg-atom quantum simulators toward high-dimensional qubit connection programming. With up to N = 22 rubidium single atoms arranged in 3D space, we have programed Ising Hamiltonians on three different Cayley-tree graphs. The antiferromagnetic phase in regular Cayley trees and frustrated competing ground states in a dual-center Cayley tree are directly observed, showing good agreement with model calculations. The presented experiments were performed under the condition that uncoupled atoms are of at least √ 3 times larger separations than connected atoms (i.e., next-nearest-neighbor couplings are 1 27 times the nearest-neighbor couplings), and this choice of atom arrangements restricts each vertex from being connected to maximally four neighboring vertices. For more complex graphs, quantum wire concepts may be necessary [51][52][53]. It is hoped that 3D-qubit configurations of Rydberg-atom quantum annealers shall be useful for efficient and programmable quantum optimization problems.