BLaDE force-directed embedding directly optimizes interactions, cutting register-embedding error by orders of magnitude

BLaDE targets the interaction-matrix error directly rather than minimizing distance error as distance-based layouts do, aligning the objective with the quantity that determines the implemented Rydberg Hamiltonian. Existing methods such as spring layout and MDS convert target interactions into target distances and then minimize distance error, which is not equivalent to minimizing interaction error; BLaDE balances each pair's force norm so it is proportional to the interaction-cost decrease. The paper argues analytically that distance-based and interaction-based objectives are not equivalent, and reports relative Frobenius error on weighted random instances: spring layout's median error starts far above BLaDE's, and even after post-hoc rescaling it does not match BLaDE.

Transient latent dimensions and local-equilibrium optimization let BLaDE escape the local minima of 2D force-directed layouts and return to physical 2D/3D space after convergence. Unlike computing a fixed high-dimensional embedding and projecting once, BLaDE adds and removes latent dimensions on the fly during iterations and uses reachable target interactions instead of isolated force estimates, so each iteration's force magnitude is set only by expected improvement. The paper provides an example dimension sequence and initialization rules for adding and removing dimensions, and reports that BLaDE achieves the best accuracy on low-error solutions, with substantially lower error than Greedy on the smallest sparse instances.

The scaling-in-the-loop mechanism dynamically adjusts the scaling factor during optimization so solutions jointly satisfy hardware distance-ratio constraints and keep interaction error low. Existing methods either omit the constraint or rely on discrete-lattice greedy placement at the cost of solution quality; BLaDE reformulates the distance-ratio constraint with a degree of freedom and uses a weighted scaling factor to relieve the tightest bound at each iteration. The paper reports that BLaDE and Greedy tend to satisfy the distance-ratio constraint naturally on denser and larger instances, especially BLaDE, and that the constrained versions always satisfy it, while other methods rarely do.

On unweighted molecular-graph benchmarks, BLaDE produces more unit-disk-valid embeddings and higher quality ones even though it is not explicitly designed for the unit-disk objective. GEAN is designed specifically for unweighted-graph unit-disk embedding; BLaDE minimizes only interaction error yet naturally encourages connected pairs to be close and non-connected pairs separated, yielding several times as many unit-disk-valid solutions as the improved GEAN on MUTAG and ZINC. The paper samples 100 molecular graphs each from MUTAG, ZINC, and PTC-FM, and reports that BLaDE achieves about tenfold lower unit-disk proximity and about five orders of magnitude lower coefficient of variation, with a higher success rate than Nelder-Mead on PTC-FM.

The work addresses register embedding for neutral-atom quantum processors, applicable when a target interaction matrix (such as an adjacency matrix or QUBO off-diagonal terms) must be mapped to atom positions. The method is validated on weighted random matrices and the MUTAG, ZINC, and PTC-FM molecular-graph benchmarks, covering sparse, medium, and dense densities across multiple sizes, and can switch between 2D and 3D position outputs. The authors note the framework can be extended to alternative interaction laws such as XY-type Hamiltonians, where the interaction scaling with distance differs. For researchers and engineers working on quantum hardware mapping, graph embedding, or combinatorial-optimization mapping, BLaDE offers a directly usable open-source implementation (QoolQit 1.2.0).

Hyperparameters are currently hand-selected on a small subset of instances, and the authors explicitly state that determining optimal hyperparameters remains an open question. The influence of the latent-dimension sequence, temperature regulation, and scaling-factor cap on final solution quality still needs systematic characterization on broader instance sets. Each molecular-graph benchmark samples only 100 graphs, so statistical fluctuation and dataset specificity deserve attention. In addition, the paper reports numerical embedding-quality metrics and does not yet provide end-to-end validation of a full quantum workflow on real neutral-atom hardware.