Leveraging Analog Neutral Atom Quantum Computers for Diversified Pricing in Hybrid Column Generation Frameworks

TL;DR

This work studies how analog neutral-atom quantum computers can accelerate hybrid column generation for fleet assignment by designing specialized embeddings and pulse schedules to sample many high-quality, diverse MWIS solutions at the pricing sub-problem. The authors introduce SA-Embedder for improved problem embeddings into unit-disk graphs and two pulse strategies (QSAMP and QSOL) to modulate quantum fluctuations during sampling, complemented by a fast Make_Diff post-processing to ensure non-degenerate samples. Benchmark results on synthetic instances show that while classical solvers like ILP+DIV often perform best, the quantum approach with SA-embedder and Make_Diff can be competitive and, in some regimes, superior, especially for larger subproblems; Gurobi remains a strong competitor but can stall in local minima. The findings demonstrate the potential of hybrid quantum-classical CG workflows on NISQ hardware for industrial CO problems, providing concrete runtime estimates and outlining pathways for practical deployment and further improvements.

Abstract

In this work, we develop new pulse designs and embedding strategies to improve the analog quantum subroutines of hybrid column generation (CG) algorithms based on neutral-atoms quantum computers (NAQCs). These strategies are designed to improve the quality and diversity of the samples generated. We apply these to an important combinatorial optimization (CO) problem in logistics, namely the fleet assignment. Depending on the instance tested, our quantum protocol has a performance that is either comparable or worse than the best classical method tested, both in terms of the number of iterations and final objective value. We identify the cause of these suboptimal solutions as a result of our quantum protocol often generating high-quality but degenerate samples. We address this limitation by introducing a greedy post-processing technique, Make\_Diff, which applies bit-wise modifications to degenerate samples in order to return a non-degenerate set. With this modification, our quantum protocol becomes competitive with an exact solver for the subproblem, all the while being resilient to state preparation and measurements (SPAM) errors. We also compare our CG scheme with a Gurobi solver and find that it performs better on over 50\% of our synthetic instances and that, despite Gurobi having a more extensive runtime. These improvements and benchmarks herald the potential of deploying hybrid CG schemes on NISQ devices for industrially relevant CO problems.

Figures (10)

• Figure 1: Workflow of the tested hybrid classical-quantum column generation approach for the fleet assignment problem, which uses a neutral atom quantum computer (NAQC) as a sampler of many, good, and diverse potential variables to add to the classical workflow. In the top left, a fleet assignment problem is represented as a conflict graph; nodes represent available tours, colors compatible vehicle classes, and the cost table represents the tour costs (vehicle costs not shown). After building the RMP with an initial set of columns $\Lambda'$, one obtains its optimal dual solution, which serves to construct a series of maximum weighted independent set problems (called pricing subproblems, PSPs) that are each sent to the NAQC. There, we use our register and pulse design strategies to taylor the dynamics in order to measure a set of good and diverse independent sets for the PSPs. These are post-processed, and then the ones passing a threshold are accepted in the column set $\Lambda'$. The process terminates when no improving column is found. An example of the primal solution after a single CG iteration is shown under '(Re)Build RMP'.
• Figure 2: Schematic of SA-Embedder, our simulated-annealing-based protocol to embed a given graph $G$ into a user-specified layout graph $H$, resulting in an embedding graph $G_H$.
• Figure 3: Comparison of the SA-Embedder and "Spring Free" embedding methods, as described in the text. We compare the average final value of $C_2(G_H, G)$ (see Eq. \ref{['eq:cost']}) for 100 random Erdos-Renyi connected graphs $G$ with varying edge probability $p$. The three graphs correspond to varying sizes of the graph $G$, with respectively $10$, $15$ and $20$ nodes. The layout used for SA-Embedder is triangular. We see that, consistently, SA-Embedder recovers better embeddings which have less extra edges in the final graph $G_H$.
• Figure 4: Comparison of quality (a) and diversity (b) of the $M=1000$ measurement samples for the two pulse strategies (QSOL, see (c) top, and QSAMP, see (c) bottom). Quality is evaluated through the approximation ratio $\alpha(\mathcal{S})$, while diversity is evaluated through the bitwise difference metric of Eq. \ref{['eq:diversity']} Benchmarks are done on 100 random Erdos-Rényi graphs with $N=30$ nodes and varying edge probability $p$, with node weights $w_i$ drawn uniformly from $[1, 10]$. Any obtained bitstring is passed through the classical post-processing. We compare results with and without SPAM errors. While QSAMP delivers similar approximation ratio to QSOL across all edge probabilities, it delivers a more diverse set of samples for small $p$.
• Figure 5: Comparison of pulse and register design protocols for the quantum methods tackling the PSP in the CG workflow for the fleet assignment problem. Each column corresponds to an instance class $(|V|, |K_v|)$ of the fleet assignment problem, as described in Subsection \ref{['subsec:instance']}. (Top): approximation ratio $C_{\text{method}}/C_{\text{ILP+DIV}}$, where we see that, as the instance size $|K_v|$ increases, methods using SA-embedder lead to lower (better) approximation ratios. (Bottom): Average number of iterations $N_{\text{iters}}$ for the CG workflow. We see that QSAMP pulse designs lead to the smallest number of iterations (i.e. faster termination of the workflow).