Solving optimization problems with local light shift encoding on Rydberg quantum annealers

TL;DR

This work presents a non-unit-disk, locally detuned Rydberg encoding to solve Max-Cut and MIS via a quantum annealer, achieving linear scaling in qubit resources and enabling 2D realizations with up to 15 nodes. By mapping graph weights to distance-dependent Rydberg interactions and using site-specific detunings, the authors reduce the problem to finding the many-body ground state of a corresponding Ising Hamiltonian, then steer the system along an optimized, time-dependent protocol shaped by both gradient and non-gradient control methods. The approach yields near-optimal solutions (R close to 1) for prototype graphs and outperforms fast simulated annealing in comparable regimes, with runtimes in the microsecond range and demonstrated resilience to modest laser-noise. The scheme is broadly applicable to QUBO-type problems and can be extended to other hardware with local qubit control and global driving, though 2D geometry imposes connectivity constraints that may be alleviated by 3D layouts and further noise-robust optimization.

Abstract

We provide a non-unit disk framework to solve combinatorial optimization problems such as Maximum Cut (Max-Cut) and Maximum Independent Set (MIS) on a Rydberg quantum annealer. Our setup consists of a many-body interacting Rydberg system where locally controllable light shifts are applied to individual qubits in order to map the graph problem onto the Ising spin model. Exploiting the flexibility that optical tweezers offer in terms of spatial arrangement, our numerical simulations implement the local-detuning protocol while globally driving the Rydberg annealer to the desired many-body ground state, which is also the solution to the optimization problem. Using optimal control methods, these solutions are obtained for prototype graphs with varying sizes at time scales well within the system lifetime and with approximation ratios close to one. The non-blockade approach facilitates the encoding of graph problems with specific topologies that can be realized in two-dimensional Rydberg configurations and is applicable to both unweighted as well as weighted graphs. A comparative analysis with fast simulated annealing is provided which highlights the advantages of our scheme in terms of system size, hardness of the graph, and the number of iterations required to converge to the solution.

Figures (6)

• Figure 1: Setup of weighted Max-Cut and MIS problems. The figure in panel (a) is the problem graph for Max-Cut with weights $w_{ij}$ on the edges (such that $\omega_{34} > \omega_{14} > \omega_{24} >\omega_{12}$), and the figure in panel (d) is the problem graph for MIS with weights $w_i$ on the vertices (such that $\omega_{3} > \omega_{1} > \omega_{4} > \omega_{2}$). Panels (b) and (e) show the solution to the corresponding problems. The dashed curve in (b) indicates the cut, dividing the graph into two sets, red and blue vertices. In (e), red vertices constitute the MIS. Panels (c) and (f) correspond to an atomic configuration where the shaded blue region around each atom indicates the local detuning and the global Rabi frequency used in the setup. Each atom is subjected to specific values of detuning depending on the weights which are encoded in the distance-dependent interactions between the atoms as indicated with the solid-black arrow. Distance between atoms in the Rydberg configuration in the case of the Max-Cut problem follows $r_{34} < r_{14} < r_{24} < r_{12}$ and in the case of the MIS problem follows $r_{34} < r_{14} < r_{12} < r_{24}$.
• Figure 2: Left panels (a - e) are for the Max-Cut problem and right panels (f - j) are for the MIS problem. (a) and (f) show weighted prototype graphs of size $5$ with corresponding solution graphs. Panels (b) and (g) show optimal protocols for the Rabi frequency (solid dark red) and the local detunings (dotted black symbols) with time. The local detuning of each atom is controlled by varying a single time-dependent parameter $\Delta_G(t)$ which is explained in the main text. Maximum and minimum speed in detuning change for the Max-Cut problem is $47.4 MHz / \mu s$ and $15.4 MHz / \mu s$ respectively, and for the MIS problem, it is $24.5 MHz / \mu s$ and $4.7 MHz / \mu s$ respectively. The expectation value $E$ of the problem Hamiltonian with respect to the instantaneous state (solid blue) and the fidelity $F$ (dashed green) of the instantaneous state with respect to the ground state are shown in panels (c) and (h), where $R$ indicates the approximation ratio. (d) and (i) show the ordered energies of the instantaneous eigenstates, with a color bar indicating the population of the basis states during the protocol. The population of the basis states of the Hamiltonian at final time $t = T$ is shown at three different times during the protocol in (e) and (j). As shown in the middle panels of (e) and (j), $12$ states and $9$ states out of $32$ are populated in the middle of the protocol. The output at the end of the protocol captures all the degenerate states corresponding to all the degenerate Max-Cut/MIS solutions.
• Figure 3: Solutions to the Max-Cut (a-d) and MIS (e-h) problems for a graph of size $15$ and degree $5$ using the optimal quantum annealing. Weighted prototype problem graphs with solution graphs for the Max-Cut problem are shown in (a) and are shown for the MIS problem in (e). In (a), vertex 11 has a degree of 5 while in (e), vertex 4 has a degree of 5. (b) and (f) show the optimal protocol for the Rabi frequency depicted by the solid dark red curve and $\Delta_G(t)$ (defined similar to Fig. \ref{['N5']}) depicted by the dashed black curve, with time. The maximum and minimum speeds for detuning change, for Max-Cut protocols, are $28.8 MHz / \mu s$ and $5.1 MHz / \mu s$ respectively, and, for MIS protocols are $1.8 MHz / \mu s$ and $0.8 MHz / \mu s$ respectively. The expectation value $E$ (solid blue) of the problem Hamiltonian with respect to the instantaneous state and fidelity $F$ (dashed green) of the instantaneous state with respect to the ground state is shown in (c) and (g), where $R$ indicates the approximation ratio. The population of states at three different times ($t=3.50 \mu s$, $t=7.10 \mu s$, and $t=10.65 \mu s$) are shown in (d,h).
• Figure 4: Comparison of optimized simulated annealing (SA) and Localised Optimal-control for Quantum Annealing in a Loop (LOQAL) for Max-Cut (a,b) and MIS (c,d) problem. Approximation ratio error $1-R$ with respect to system size $N$ is shown in (a,c), and for the hardness parameter $HP$ in (b,d).
• Figure 5: The figure shows the results using the optimal quantum annealing with noisy protocols to solve the Max-Cut problem for a graph of size $10$. The weighted prototype graph for the Max-Cut problem is shown in (a) along with the solution graph. The optimal protocol for the Rabi frequency (solid dark red) and the factor controlling the detunings $\Delta_G$ (dashed black) are shown in (b) where noise is added to the laser parameters of the pre-optimized protocol and in (d) where noise is added during the optimization of the parameters. The shaded regions represent the fluctuations in the laser parameters for each run that were chosen from a random distribution. (c) and (e) show the corresponding expectation value (solid blue) of the problem Hamiltonian with respect to the instantaneous state ($E$) and the fidelity $F$ (dashed green) of the instantaneous state with respect to the ground state for both cases, where $R$ indicates the approximation ratio.