QuantGraph: A Receding-Horizon Quantum Graph Solver

TL;DR

QuantGraph reframes trajectory optimization as a quantum search over discrete trajectory spaces and integrates it within a receding-horizon model-predictive-control framework to tackle large, dynamic graphs. It uses a two-stage architecture with a local stage to generate a warm-start threshold and a global Grover-adaptive-search stage to refine the solution, achieving a quadratic speedup over classical search while controlling computational burden. Empirical results on static, double-integrator, and cart-pole benchmarks demonstrate substantial search-space pruning, robust convergence, and improved discretization precision at fixed query budgets. The work highlights how closing the loop with classical control theory can stabilize and guide quantumSearch-based optimization, enabling scalable, high-precision planning for complex robotic and dynamical systems.

Abstract

Dynamic programming is a cornerstone of graph-based optimization. While effective, it scales unfavorably with problem size. In this work, we present QuantGraph, a two-stage quantum-enhanced framework that casts local and global graph-optimization problems as quantum searches over discrete trajectory spaces. The solver is designed to operate efficiently by first finding a sequence of locally optimal transitions in the graph (local stage), without considering full trajectories. The accumulated cost of these transitions acts as a threshold that prunes the search space (up to 60% reduction for certain examples). The subsequent global stage, based on this threshold, refines the solution. Both stages utilize variants of the Grover-adaptive-search algorithm. To achieve scalability and robustness, we draw on principles from control theory and embed QuantGraph's global stage within a receding-horizon model-predictive-control scheme. This classical layer stabilizes and guides the quantum search, improving precision and reducing computational burden. In practice, the resulting closed-loop system exhibits robust behavior and lower overall complexity. Notably, for a fixed query budget, QuantGraph attains a 2x increase in control-discretization precision while still benefiting from Grover-search's inherent quadratic speedup compared to classical methods.

Figures (4)

• Figure 1: Subsequent model-predictive-control (MPC) iterations: (a) In early model-predictive-control steps, discretized states deviate from the continuous optimum, with compounding errors $\delta x_1 < \delta x_2$ along the horizon. To limit computation over the full horizon $T$, the model-predictive-control framework optimizes only the first $N_c \ll T$ control actions, while beyond $N_c$ the control-inputs are held constant (the Fixed Horizon block). Although this truncation increases discretization drift, receding-horizon re-optimization corrects the tail error at each iteration, applying only the first control input before shifting the horizon forward, while there are significant gains in terms of computational efficiency. (b) In subsequent iterations, the model-predictive-control framework recomputes controls from the updated state, compensating for prior deviations and refining the trajectory. Through this iterative correction, the closed-loop trajectory converges toward the continuous-time optimum while maintaining tractable computational cost.
• Figure 2: A simple navigation problem as a directed graph with weighted transitions. QuantGraph is called to find the path associated with the lowest cost possible from node a to node h. The local-search step acts as a sub-optimal threshold that warm starts the global search effectively minimizing the required Grover-adaptive-search iterations needed for convergence.
• Figure 3: Planned trajectories for the double integrator: The local stage warm starts the global stage of QuantGraph that operates in a receding horizon fashion. The objective is to drive the double integrator to the target position. To demonstrate the consistency of our framework, we plot the results for ten runs. The local stage converges to a sub-optimal trajectory as expected, warm-starting the global stage that converges to a smooth optimal trajectory with low variance across runs.
• Figure 4: (a) Planned trajectories for the cart pole system. The primary objective is to swing up the underactuated pole mass to the upright position while minimizing displacement of the cart from its initial location (secondary objective). The plot shows the mean and standard deviation over ten independent runs. The negligible variance in the evolution of the pole angle $\left(\theta\right)$ highlights the consistency and reliability of our planning approach. (b) Schematic of the physical cart pole system, annotated with the variables used in Eq. \ref{['eq:cart_pole']}.