Iterative Optimization with Partial Convergence Guarantees on Neutral Atom Quantum Computers

Abstract

Neutral atom quantum computers (NAQCs) have emerged as a promising platform for solving the maximum weighted independent set (MWIS) problem. However, analog quantum approaches face two key limitations: constraints of the atomic layout on realizable graph geometries and the absence of performance guarantees. We introduce Lp-Quts, a hybrid quantum-classical framework that integrates an NAQC sampler into a classical cutting-plane algorithm. At each iteration, a relaxed linear program (RLP) bounds the MWIS and induces a reduced graph from which independent sets are sampled using an analog quantum sampler. A novel sample-informed separation problem guides odd-cycle cut selection and accelerates convergence. For t-perfect graphs, Lp-Quts inherits polynomial-time convergence guarantees from the classical theory of cutting planes. We evaluate our approach on instances with up to 300 vertices -- a scale that exceeds the capabilities of current NAQC hardware. In this regime, Lp-Quts reaches solutions within 5--10\% of optimality, outperforming direct analog quantum protocols and greedy baselines under equal sampling budgets. As expected, simulated annealing remains the strongest sample-based solver at this scale. These results demonstrate how quantum samplers can be effectively embedded within classical optimization frameworks to deliver near-optimal solutions with reduced quantum resources while preserving formal guarantees.

Figures (6)

• Figure 1: Overview of Lp-Quts. The algorithm iterates steps 2--6 to tighten the upper (blue - RLP solutions) and lower (yellow - best independent set sampled) bounds on the MWIS, as shown in step 7). After the RLP is solved in step 2), we use the dual solution $\Vec{\pi}$ to generate the reduced graph $G_{\mathrm{RLP}}$ where edges correspond to all the tight inequalities in the optimal solution to the RLP. Then, in step 4), this graph is embedded on a NAQC and a laser pulse is used to sample independent sets. After postprocessing the samples $\mathcal{S}$ on the original graph $G$ in step 5), the bounds are improved step 7), and a separation problem is solved to identify a new cutting plane, i.e. an odd-cycle, for the RLP (step 6). The bottom panel of 6) illustrates our sample-informed separation problem. $\mathcal{S}_t$ represents the space of all possible solutions with $\vec{x}_{\mathrm{RLP}}$ as the current best one. Green, pink, and brown cuts separate this solution from the integer hull ($\mathcal{S}_I$; red), with sampled solutions marked by red "X"s. The RLP cost $\varepsilon_{\mathrm{RLP}}$ distinguishes the worst (green) cut but cannot separate the pink and brown cuts. Incorporating the sampler-based cost $\varepsilon_s(\mathcal{S})$ favors the brown cut because it is in closer proximity to the sampled solutions. The brown cut is facet-defining and resolves the MWIS.
• Figure 2: a) Ratio of the number of edges in the final $G_{\mathrm{RLP}}$ to that of the original graph $G$ for varying graph sizes $N$ and edge densities $p$. Each data point is the average of 10 random instances. b) Number of iterations to convergence for series--parallel graphs of different sizes (an example of such graph is shown in the inset). Series--parallel graphs are examples of t-perfect graphs and we observe polynomial scaling with system size, as expected, while seeing that our modified separation cost (blue) converges faster than the standard approach (orange), with the performance gap widening for larger graphs. c) Final optimality gap between the RLP solution and the true optimum. For t-perfect graphs, convergence to the optimal solution for all system sizes is observed as expected. For general graphs, the gap grows exponentially with system size and is larger for denser graphs. This reflects that odd-cycle inequalities alone are not sufficient in this general case, and additional cutting planes inequalities are required.
• Figure 3: Normalized optimality gap $1 - R$, where $R = C_{\mathrm{best}}/C_{\mathrm{opt}}$ is the ratio between the cost of the best sampled solution and the optimal cost $C_{\mathrm{opt}}$, for Lp-Quts and three comparative samplers for the MIS (top) and MWIS (bottom) problems on 10 synthetic Erdős--Rényi instances. The optimal cost $C_{\mathrm{opt}}$ is obtained by solving the integer linear program of Eq. \ref{['eq-psp']} using the Gurobi solver gurobi. Each column corresponds to a different edge probability $p$, and each subplot shows results for varying graph sizes $N$ (x-axis). All methods are compared using the same total sampling budget, matched to that of Lp-Quts, given by $N_{\text{it}} \cdot N_{\text{shots}}$, where $N_{\text{it}}$ is the number of iterations and $N_{\text{shots}}$ is the number of shots per iteration (specified in the legend). Lp-Quts uses at most $\min(N, 40)$ qubits and performs as well as or better than QSOL, which has the same pulse design as Lp-Quts with an improved embedding heuristics applied to the full graph (thus using $N$ qubits). Lp-Quts tackles larger instances beyond the limits of open-source classical emulators ($\sim 50$ qubits) and current neutral-atom quantum computers (100--200 qubits wurtz2023aquilaqueras256qubitneutralatom) achieving significantly superior performance than Greedy for both MIS and MWIS problems and near-optimality for the largest MWIS instances tested, though SA outperforms it at this scale.
• Figure 4: Probability $P_{\text{MIS}}$ (top) and $P_{\text{MWIS}}$ of obtaining the optimal solution among the sampled solutions (bottom) versus the maximum subgraph size for 10 random MIS and MWIS instances of size $N=50$ with varying edge density (columns). We run Lp-Quts (blue) and compare it with Lp-Greedy (purple) and Lp-SA (red) to assess whether comparable solution quality can be achieved purely classically. All methods use $N_{\text{shots}} = 100$ shots per iteration. As an additional benchmark, we include QSOL (green) on the full graph ($N_{\text{max-qubits}} = 50$), adjusted to use the same total number of samples as Lp-Quts. We show the individual instances as scatter points and the mean with a black diamond. Consistently, Lp-Quts outperforms its classical counterparts.
• Figure 5: Sample-to-target (STT($\epsilon$)) energy as a function of graph size $N$ and edge density $p$ (columns) for MWIS instances. We compare Lp-Quts and the three classical solvers over the same total sample budget for each instance. The first and second rows correspond to $\epsilon = 1\%$ and $\epsilon = 5\%$, respectively. Instances in which no sample reached the target energy are omitted. To ensure fair comparisons, the averages for QSOL, Lp-Quts, and SA are computed over instances where all methods return well-defined STT values.