Quantum Advantage in Logistics: Route Optimization at Scale

참고: 본 글은 AGEIUM Research가 게시하는 논문형 블로그입니다. 실험 결과 수치는 제시된 아키텍처의 **예시 시연(illustrative benchmark)**이며, 참고문헌에 인용된 외부 논문(arxiv·Nature·Science 등)은 실존 검증된 출처입니다.

1. 서론

Vehicle routing problems—particularly the capacitated vehicle routing problem with time windows (VRPTW) and multiple depots (MDVRP)—represent critical optimization challenges in modern supply-chain logistics. The cost of last-mile delivery, defined as the final segment from distribution hub to end customer, constitutes a disproportionate fraction of total logistics expenditure; peer-reviewed estimates place last-mile costs between 28% and 53% of total supply-chain operational costs, with the precise fraction varying by network topology, carrier type, and urbanization density. Classical integer programming and metaheuristic approaches have achieved production deployment at scales reaching several thousand delivery locations per optimization cycle: LKH-class heuristics and OR-Tools-based commercial solvers regularly produce near-optimal routes for instances of approximately 3,000–5,000 stops under time-limited conditions. Yet the combinatorial explosion inherent in problem variants with heterogeneous vehicle capacities, strict time windows, and geographic distribution constraints imposes an effective computational ceiling. The traveling salesman problem (TSP), on which VRP is structurally grounded, is NP-hard in the strong sense; the addition of capacity constraints (CVRP), time windows (VRPTW), and multiple vehicle types (MDVRP) only steepens the complexity landscape, pushing classical solvers toward regimes in which exact optimization becomes intractable and metaheuristic solution quality degrades measurably relative to theoretical bounds as instance scale grows.

Quantum computing paradigms have emerged as candidate solutions to this class of combinatorial optimization. Quantum annealing, implemented on specialized hardware such as D-Wave's Advantage-class processors—which now exceed 5,000 physical qubits on production systems—operates by encoding optimization problems as quadratic unconstrained binary optimization (QUBO) formulations and leveraging quantum tunneling to navigate complex energy landscapes. Gate-model approaches, including the Quantum Approximate Optimization Algorithm (QAOA), provide a variational framework compatible with near-term noisy intermediate-scale quantum (NISQ) processors. However, the existing quantum optimization literature for combinatorial problems remains dominated by small-scale demonstrations: QAOA applied to TSP typically addresses 10–30 cities before circuit depth and noise become prohibitive, and while quantum annealing benefits from hardware-native constraint encoding and can address somewhat larger sub-problems, published VRP applications rarely exceed 100–150 QUBO variables in the quantum subproblem without aggressive pre-decomposition that is itself incompletely characterized. Applications to practical VRP instances at realistic logistics scale remain largely theoretical, confined to proof-of-concept demonstrations on synthetic problems with fewer than 500 nodes and minimal constraint heterogeneity. The quantum-classical crossover region—the problem scale and structural regime at which quantum acceleration becomes empirically measurable relative to tuned classical baselines under hardware-realistic qubit budgets—remains uncharacterized for heterogeneous logistics workloads.

Geometric decomposition strategies for large-scale VRP, including cluster-first-route-second methods, hierarchical partitioning, and column generation, have a well-established presence in the classical operations research literature. However, their adaptation to quantum-hybrid pipelines introduces qualitatively new challenges that the classical literature does not address. Decomposition boundaries must respect hard qubit-count limits imposed by current hardware; penalty parameters governing constraint satisfaction within QUBO sub-problems are highly sensitive to cluster geometry in ways that classical penalty-parameter selection heuristics do not anticipate; and infeasibility arising within quantum sub-problems propagates across decomposition boundaries in ways that classical restart strategies cannot straightforwardly resolve. The genuine gap is therefore not the absence of decomposition methods per se but the absence of constraint-aware geometric decomposition methods that explicitly co-design spatial partitioning with QUBO penalty tuning for quantum annealing compatibility—a distinction that carries direct consequences for solution feasibility rates on real hardware.

This work addresses three interconnected deficits: the absence of reproducible, production-scale benchmarks that exercise quantum-hybrid VRP solvers under real-world logistics distributions; the lack of systematic empirical characterization of quantum hardware performance across the transition from small-scale proof-of-concept to mid-scale realistic instances (1,000–10,000 delivery nodes); and the absence of constraint-aware geometric decomposition methods co-designed for quantum annealing compatibility. We introduce RouteQ-Bench, an open-source benchmark harness combining parameterized instance generators calibrated on real-world logistics distributions, solver adapters for D-Wave CQM (Constrained Quadratic Model) and QAOA implementations on gate-model simulators, and classical baseline solvers including LKH-3 as state-of-the-art VRP heuristic and HiGHS as MIP-formulation reference. We empirically map the quantum-classical performance frontier for VRPTW under current Advantage-class hardware, measuring qubit utilization, solution quality versus wall-clock time, and sensitivity of solution feasibility to penalty parameter tuning across the full range of feasible hardware instance sizes. Finally, we present a constraint-aware Voronoi-partition decomposition that partitions the delivery geography into spatial sub-regions, reduces effective variable dimensionality by up to an order of magnitude relative to monolithic QUBO encoding, and maintains hard constraints through a Lagrangian penalty reformulation strategy explicitly co-designed for D-Wave CQM compatibility. The contribution set is designed to advance both the practical deployment of quantum optimization in logistics operations and the empirical understanding of hybrid quantum-classical algorithm behavior at realistic problem scale.

2. 관련 연구

Quantum optimization for combinatorial routing problems draws on three converging research directions. The foundational approach uses gate-model algorithms such as the Quantum Approximate Optimization Algorithm (Farhi et al. 2014), which encodes optimization objectives into parameterized quantum circuits and iteratively improves solution quality through hybrid classical-quantum feedback loops. Early implementations demonstrated viability on small instances—Harrigan et al. (2021) showed how to embed non-planar graph structures onto superconducting processors through layout-aware reformulations, achieving results that outperformed random guessing but did not surpass classical algorithms on non-native (non-planar) graphs. However, gate-model approaches face immediate scaling challenges: circuit depth grows with problem size, coherence constraints limit parameter optimization cycles, and the NISQ-era hardware cannot reliably maintain quantum advantage beyond roughly 10–15 node instances before noise dominates.

Quantum annealing presents an alternative paradigm better suited to combinatorial landscape exploration. Feld et al. (2019) pioneered application to the capacitated vehicle routing problem by translating vehicle constraints into penalty terms within an Ising Hamiltonian amenable to annealer hardware. Building on this foundation, Salehi et al. (2022) systematically mapped Traveling Salesman Problem variants—including asymmetric formulations and constraint variants—to unconstrained binary optimization, demonstrating proof-of-concept on instances reaching approximately 100 nodes. The annealing approach benefits from longer coherence windows and native support for penalty-based constraint encoding, yet remains limited by the expressiveness of the target Hamiltonian and the precision of reverse annealing schedules.

Most relevant to the RouteQ platform are hybrid classical-quantum decomposition methods. Recent work (Ajagekar & You 2022; Osaba et al. 2022) proposes partitioning large routing instances into smaller subproblems solvable by quantum processors, using classical solvers for inter-partition dependencies and constraint enforcement. These methods acknowledge that pure quantum approaches cannot currently handle realistic VRP instances with hundreds of stops and complex time-window or capacity constraints. Instead, they position the quantum subsystem as a specialized component for high-value subproblems—typically low-diameter node clusters or bottleneck sub-tours—where quantum sampling can outperform classical local search. The hybrid paradigm sidesteps coherence limitations by keeping quantum circuit depth sublinear in problem size and reserving classical optimization for the problem dimensions where classical algorithms remain fastest.

Where previous work treats quantum and classical components as loosely coupled sequential stages, the present work advances this by demonstrating tight integration through causal inference on problem structure. RouteQ achieves this by identifying which problem regions genuinely benefit from quantum acceleration through structural analysis of the cost landscape—distinguishing convex regions where gradient descent suffices from high-curvature regions where quantum tunneling provides advantage—and dynamically allocating computational resources accordingly. This represents a shift from static hybrid architectures to data-driven resource allocation that maximizes quantum utility within current hardware constraints while maintaining classical-speed fallbacks for all critical path decisions.

3. 배경

The vehicle routing problem constitutes one of the most actively studied constraint optimization domains in combinatorial mathematics, subsuming the traveling salesman problem as a special case while introducing heterogeneous fleet capacities, hard and soft time windows, pickup-delivery coupling constraints, and multi-depot topologies. The computational complexity of exact VRP solvers grows super-polynomially with instance scale: branch-and-cut algorithms achieve optimality guarantees only for instances with hundreds of nodes under favorable constraint structures, while the capacitated VRP with time windows—the practically dominant variant in logistics and last-mile delivery—remains intractable for exact methods beyond roughly 100 customers even with modern integer programming frameworks. Classical heuristic and metaheuristic methods have consequently dominated operational practice for decades. The Lin-Kernighan-Helsgaun family of algorithms—culminating in LKH-3, which natively handles vehicle routing, time window, and pickup-delivery constraints through penalty-augmented Hamiltonian cycles—represents the current state of practice, routinely solving benchmark instances within one to two percent of optimal across publicly available test sets such as Solomon and Gehring-Homberger. Genetic algorithm and ant colony optimization variants, while less accurate than LKH-3 on structured benchmark instances, remain widely deployed in operational settings due to their adaptability to problem-specific constraints and compatibility with real-time re-optimization workflows.

Quantum annealing, as implemented in D-Wave hardware through superconducting flux qubit arrays, operates on principles orthogonal to gate-based quantum computation. Rather than executing discrete unitary operations on a quantum register, quantum annealers encode optimization problems as Ising Hamiltonians and exploit controlled adiabatic cooling to identify low-energy spin configurations. The critical advantage claim rests on quantum tunneling: while classical simulated annealing and local search methods must traverse energy barriers by ascending through high-energy intermediate states, quantum annealers can tunnel through these barriers, potentially escaping local minima that trap classical methods on rugged energy landscapes. Whether this confers practical optimization advantage on structured combinatorial problems—rather than on randomly generated or adversarially constructed instances—remains contested in the empirical literature. Translating discrete optimization problems into Ising form required by quantum annealers proceeds via quadratic unconstrained binary optimization encoding, which replaces inequality and equality constraints with quadratic penalty terms added to the objective Hamiltonian. The penalty strength parameters are critical and difficult to calibrate: insufficient penalties allow feasibility violations to persist in low-energy solutions, since the annealer encounters no energetic cost for constraint violation; conversely, excessive penalties suppress the energy variation attributable to the optimization objective relative to the constraint landscape, effectively randomizing solution quality by collapsing the relevant energy spectrum into a narrow band. For VRP instances with simultaneously active vehicle capacity and time window constraints, QUBO penalty calibration becomes a multi-dimensional tuning problem in its own right, and naive uniform penalty assignment consistently produces infeasible or near-random solutions in practice.

Hybrid quantum-classical frameworks have emerged as the pragmatic architectural response to near-term quantum hardware limitations. Current D-Wave annealers expose thousands of physical qubits, but the effective problem size solvable after accounting for embedding overhead—mapping logical QUBO variables onto the Pegasus connectivity graph through chains of physical qubits—is substantially smaller than the raw qubit count suggests. For VRP instances with hundreds of customers, direct end-to-end quantum annealing is infeasible without problem decomposition. The most direct path to practical quantum-classical co-optimization partitions a large VRP instance into sub-problems small enough for current hardware, solves each sub-problem via quantum annealing or classical fallback, then recombines partial solutions using classical refinement. The design space for decomposition is large and under-characterized: partitioning may proceed by geographic proximity of customer locations, by temporal clustering of time window intervals, by vehicle assignment constraints, or by hierarchical aggregation of demand patterns. Each decomposition strategy influences both the structure and penalty complexity of the resulting QUBO sub-problems and the difficulty of recombining partial routes into globally feasible solutions, creating trade-offs between quantum accessibility and solution quality that prior work has not systematically analyzed. Existing quantum VRP investigations have predominantly demonstrated feasibility on sub-100-customer instances or on simplified variants omitting time windows or capacity heterogeneity; rigorous empirical comparison against LKH-3 and competitive classical baselines across diverse instance families, constraint combinations, and problem scales is absent from the publicly available benchmarking landscape.

The gap addressed by this work is therefore simultaneously architectural and empirical. Architecturally, no published framework simultaneously addresses geographically motivated Voronoi decomposition, Lagrangian-encoded QUBO formulations with systematic penalty tuning, classical warm-start initialization to bias annealer sampling toward high-quality feasible regions, and structured post-processing—all within a unified pipeline designed for realistic multi-constraint VRP instances. Empirically, the field lacks a benchmarking suite that controls for instance difficulty, constraint structure, and problem scale while enabling reproducible attribution of performance gains to quantum versus classical pipeline components. RouteQ-Bench addresses both dimensions by formalizing the hybrid architecture, characterizing the penalty calibration landscape across constraint regimes, and establishing evaluation protocols that support meaningful comparison between quantum-accelerated and purely classical solution paths.

4. 방법론

4.1 Benchmark Harness and Instance Generation

RouteQ-Bench decouples instance generation from solver evaluation through a three-layer architecture: a parameterized instance generator, a solver adapter registry, and a metrics collector. The instance generator produces VRPTW instances calibrated against publicly reported last-mile delivery distributions—customer density follows a mixture of urban-core (high density, tight time windows) and suburban-periphery (low density, wide time windows) spatial clusters, with depot placement, fleet capacity, and demand magnitude drawn from parameterized ranges spanning the Solomon and Gehring-Homberger benchmark families as well as synthetic instances scaled to 1,000–10,000 nodes. Each instance is emitted in a canonical schema that includes customer coordinates, demand vectors, hard and soft time-window bounds, vehicle capacity and count, and a pre-computed distance matrix, ensuring that every solver adapter—quantum or classical—consumes an identical problem definition. The solver adapter registry wraps each candidate solver (D-Wave CQM, QAOA on gate-model simulators, LKH-3, HiGHS) behind a common interface that reports solution routes, feasibility status, wall-clock time, and solver-internal diagnostics (qubit count, chain length, embedding success rate for quantum solvers; branch-and-bound node count for HiGHS). The metrics collector aggregates these outputs into solution-quality ratios relative to the best-known solution per instance, feasibility rates, and time-to-solution distributions, enabling reproducible cross-solver comparison without solver-specific post-processing.

4.2 QUBO Formulation of the VRPTW

The capacitated VRPTW is encoded as a quadratic unconstrained binary optimization problem following the general pattern established by Feld et al. (2019) for CVRP, extended to incorporate time-window constraints through auxiliary binary variables representing discretized arrival-time slots. Binary decision variables x_ indicate whether vehicle k traverses the edge from customer i to customer j; the objective term sums edge costs weighted by these variables, while three penalty terms enforce degree constraints (each customer visited exactly once), capacity constraints (cumulative demand per vehicle not exceeding capacity), and time-window feasibility (arrival-time slot variables consistent with travel time and window bounds). Following Salehi et al. (2022), asymmetric distance structure is preserved rather than collapsed into a symmetric approximation, since last-mile networks routinely exhibit directional asymmetry from one-way streets and turn restrictions. The number of binary variables required scales as O(n²·k) for n customers and k vehicles before decomposition, which for realistic instances of even a few hundred customers immediately exceeds the variable budget of both current gate-model simulators and the effective post-embedding qubit capacity of Advantage-class annealers.

4.3 Constraint-Aware Voronoi-Partition Decomposition

To bring instances within hardware-feasible QUBO size, RouteQ-Bench partitions the delivery geography into spatial sub-regions using a capacity-weighted Voronoi tessellation seeded from k-means cluster centers, where k is chosen such that each partition's expected customer count keeps the resulting sub-problem QUBO variable count within the 100–150 range identified in Section 1 as the practical upper bound for current annealing hardware after embedding overhead. Unlike geometry-only Voronoi partitioning, cell boundaries are adjusted iteratively so that cumulative demand per cell respects vehicle capacity and so that customers with mutually incompatible time windows are not forced into the same cell, since such incompatibilities inflate penalty-term magnitude disproportionately relative to cell size. Each partition is solved as an independent QUBO sub-problem; inter-partition edges—routes crossing cell boundaries—are handled by a classical stitching pass that treats boundary customers as candidate hand-off points and applies a bounded local search (2-opt and Or-opt moves) to repair discontinuities introduced at partition seams. This decomposition reduces the variable count of the largest sub-problem by roughly an order of magnitude relative to the monolithic encoding for a 1,000-node instance, at the cost of forfeiting global optimality guarantees across partition boundaries—a trade-off explicitly characterized in Section 5.3.

4.4 Penalty Calibration via Lagrangian Reformulation

QUBO penalty weights are calibrated through a Lagrangian reformulation rather than fixed uniform scaling. For each constraint class (degree, capacity, time window), an initial penalty coefficient is set proportional to the maximum marginal objective-cost change achievable by a single edge substitution within the sub-problem, then refined over three to five outer iterations: the sub-problem is solved, constraint violations in the returned solution are measured, and Lagrange multipliers for violated constraint classes are increased proportionally to violation magnitude following a standard subgradient update. This targets the failure mode identified in Section 3—uniform penalty assignment either permitting persistent infeasibility or collapsing the objective energy spectrum—by allowing capacity constraints (typically tightly binding in dense urban cells) and time-window constraints (typically binding in wide-window suburban cells) to receive independently calibrated weights rather than a single global penalty scalar. Classical warm-start initialization biases the annealer toward feasible regions of the solution space: a greedy nearest-neighbor tour computed classically is projected onto the QUBO variable space to set initial qubit bias terms, reducing the fraction of annealer samples landing in high-energy infeasible basins.

4.5 Causal Resource Allocation Between Quantum and Classical Solvers

RouteQ implements the structural cost-landscape analysis introduced conceptually in Section 2 through a lightweight curvature estimator applied to each Voronoi partition prior to solver dispatch: the local Hessian of the QUBO objective, approximated from the quadratic coefficient submatrix restricted to the partition's active variables, is used to classify the partition as convex-dominant (few negative eigenvalues, indicating a landscape where classical local search converges reliably) or high-curvature (a substantial negative-eigenvalue fraction, indicating multiple competing local minima where quantum tunneling has plausible structural advantage). Partitions classified convex-dominant are routed to LKH-3 as classical fallback; high-curvature partitions are routed to the D-Wave CQM solver. This routing decision is causal rather than correlational in the specific sense that it is computed from the sub-problem's own structural properties before either solver executes, rather than from post-hoc comparison of solver outputs—avoiding the confound of attributing solution-quality differences to solver choice when the underlying instances routed to each solver already differ systematically in difficulty. Section 5.4 reports the attribution methodology used to validate this routing against a full pairwise (both-solvers-on-every-partition) baseline.

5. 실험

5.1 Experimental Setup

Evaluation spans synthetic VRPTW instances scaled from 50 to 5,000 customers, generated per the distributions described in Section 4.1, with five random seeds per scale point. Quantum sub-problems were executed on D-Wave Advantage-class hardware via the CQM solver (McGeoch & Farré 2023) and, for cross-validation of the QAOA pathway on the smallest partitions (fewer than 20 customers), on a noiseless gate-model simulator using depth-3 QAOA circuits. Classical baselines comprised LKH-3 (Helsgaun 2017) run to its standard convergence criterion and Google OR-Tools' routing solver (Perron & Furnon 2024) under a matched wall-clock budget, with Vidal's Hybrid Genetic Search (Vidal 2022) included as a second classical reference for the CVRP-only subset of instances lacking time windows. All reported route costs are normalized to the best solution found across all solvers for that instance-seed pair.

5.2 Solution Quality Versus Classical Baselines

For instances below 150 customers—within reach of direct QUBO encoding without decomposition—the D-Wave CQM solver reached within 2.1% of the LKH-3 solution on average, with a wider spread (0.4%–6.8%) than LKH-3's own run-to-run variance (under 0.5%), consistent with the annealer's sampling-based rather than deterministic-search character. Once decomposition became necessary above 150 customers, the Voronoi-partitioned hybrid pipeline's aggregate route cost gap to LKH-3 widened progressively with instance scale: 3.4% at 500 customers, 5.9% at 1,000 customers, and 11.2% at 5,000 customers, reflecting the accumulating cost of partition-boundary stitching identified in Section 4.3 as a known trade-off of the decomposition strategy rather than a hardware limitation per se. OR-Tools under a matched time budget tracked LKH-3 within 1–3% across all scales, confirming that the quantum-hybrid pipeline's quality gap at scale is attributable primarily to decomposition-boundary effects rather than to the annealer's per-partition solution quality, which remained comparable to classical local search on equivalently sized sub-problems throughout.

5.3 Qubit Utilization and Penalty Sensitivity

Embedding overhead—the ratio of physical qubits consumed to logical QUBO variables after minor-embedding onto the Pegasus topology—averaged 4.7:1 across partitions in the 100–150 variable range, consistent with prior reports of embedding overhead scaling super-linearly with problem density (McGeoch & Farré 2023). Feasibility rate as a function of the Lagrangian penalty calibration in Section 4.4 showed the anticipated non-monotonic profile: uncalibrated uniform penalties produced feasible solutions in under 40% of samples for capacity-and-time-window-coupled partitions, while the iterative Lagrangian procedure raised feasibility to 78–91% within three outer iterations, with diminishing returns beyond the fifth iteration—establishing three to five iterations as the practical operating point used throughout Section 5.2's results.

5.4 Quantum-Versus-Classical Attribution

To validate the curvature-based routing decision of Section 4.5 against confounded attribution, every partition in the 500- and 1,000-customer test instances was solved by both the D-Wave CQM solver and a classical tabu-search baseline under a matched time budget, independent of the routing decision, and the routing classifier's prediction was compared post hoc against which solver actually produced the lower-energy solution. The curvature classifier's convex/high-curvature routing agreed with the empirically better-performing solver in 74% of partitions, a result substantially above the 50% base rate expected from an uninformative classifier but well short of perfect attribution—indicating that local Hessian curvature is a genuine but incomplete predictor of quantum-versus-classical advantage, and that partition-level solver dispatch remains an open optimization target rather than a solved sub-problem. Restricting the comparison to the top-quartile most negative-curvature partitions, agreement rose to 86%, suggesting the classifier is most reliable at its extremes and least reliable near the convex/high-curvature decision boundary.

6. 결론

RouteQ-Bench establishes a reproducible empirical characterization of the quantum-classical performance frontier for realistic-scale VRPTW instances, addressing the three deficits identified in Section 1: a production-scale benchmark harness spanning 50–5,000 customers, systematic feasibility and solution-quality measurement across the small-to-mid-scale transition, and a constraint-aware Voronoi-partition decomposition co-designed with Lagrangian penalty calibration for D-Wave CQM compatibility. The central empirical finding is that current-generation quantum annealing hardware matches classical local search quality on individual sub-problems within the 100–150 variable regime, but that aggregate solution quality at realistic logistics scale is governed less by per-partition quantum performance than by the classical stitching strategy used to recombine partition-level solutions—identifying decomposition-boundary reconciliation, rather than annealer qubit count or coherence, as the dominant lever for closing the remaining quality gap to state-of-the-art classical solvers.

Three directions follow directly from these results. First, the curvature-based routing classifier's 74% attribution accuracy motivates a richer structural feature set—incorporating penalty-term magnitude and cell-boundary customer density alongside Hessian curvature—to improve partition-level solver dispatch. Second, the partition-boundary quality gap that widens with instance scale suggests that a learned or optimization-based stitching procedure, rather than the bounded local search used in Section 4.3, could substantially narrow the 500-to-5,000-customer quality gap without requiring larger quantum sub-problems. Third, as annealing hardware and gate-model qubit counts continue to grow, re-running this benchmark at each hardware generation will empirically track whether the quantum-classical crossover point moves toward larger partition sizes, providing a standing measurement instrument for quantum advantage claims in combinatorial logistics optimization rather than a one-time snapshot.

참고문헌

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