Capacitated vehicle routing with time windows (CVRPTW) is a natural target for coherent Ising machines (CIMs), but a direct multi-vehicle arc encoding scales as O(mN2) and exceeds the variable budget of current CIM-compatible systems. We argue the bottleneck is encoding density, not expressiveness, and present LSQ, a hardware-aware sparse Quadratic Unconstrained Binary Optimization (QUBO) framework that decouples CVRPTW into a compact customer-to-route assignment QUBO and a classical intra-route ordering step under a soft no-wait service convention. LKH candidate edges compress the per-route edge space from O(N2) to O(KN), and a per-route dynamic-penalty subroutine encodes time-window sensitivities as binary variables in a round-wise outer loop. On a six-vehicle, 51-node reference instance curated from long-term operational data, LSQ shrinks the maximum single-submission QUBO from 15,300 arc variables to 342 logicalQUBOvariables (∼45× compression), cuts travel time by 22.9% (74 vs. 96), and cuts route duration by 11.2% (174 vs. 196) against an OR-Tools soft-window baseline at the same fleet size. OR-Tools retains an advantage on raw time-window penalty (1600 vs. 3540) and runtime; under the scalar operational cost ∑kT(πk)+∑iℓi(τi), OR-Tools is therefore the better single-objective solver, and the comparison is a multi-objective trade-off rather than a scalar dominance claim. Ablations confirm that the LKH prior recovers Held–Karp on a 15-customer TSP at 53 vs. 120 variables and that the dynamic-penalty encoding reduces compressed time-window loss by 15.25% at constant travel. All hardware claims refer to QUBO sizing on a Kaiwu/CIM-compatible backend, not physical CIM execution.
Capacitated vehicle routing with time windows (CVRPTW) is a natural target for coherent Ising machines (CIMs), but a direct multi-vehicle arc encoding scales as O(mN2) and exceeds the variable budget of current CIM-compatible systems. We argue the bottleneck is encoding density, not expressiveness, and present LSQ, a hardware-aware sparse Quadratic Unconstrained Binary Optimization (QUBO) framework that decouples CVRPTW into a compact customer-to-route assignment QUBO and a classical intra-route ordering step under a soft no-wait service convention. LKH candidate edges compress the per-route edge space from O(N2) to O(KN), and a per-route dynamic-penalty subroutine encodes time-window sensitivities as binary variables in a round-wise outer loop. On a six-vehicle, 51-node reference instance curated from long-term operational data, LSQ shrinks the maximum single-submission QUBO from 15,300 arc variables to 342 logicalQUBOvariables (∼45× compression), cuts travel time by 22.9% (74 vs. 96), and cuts route duration by 11.2% (174 vs. 196) against an OR-Tools soft-window baseline at the same fleet size. OR-Tools retains an advantage on raw time-window penalty (1600 vs. 3540) and runtime; under the scalar operational cost ∑kT(πk)+∑iℓi(τi), OR-Tools is therefore the better single-objective solver, and the comparison is a multi-objective trade-off rather than a scalar dominance claim. Ablations confirm that the LKH prior recovers Held–Karp on a 15-customer TSP at 53 vs. 120 variables and that the dynamic-penalty encoding reduces compressed time-window loss by 15.25% at constant travel. All hardware claims refer to QUBO sizing on a Kaiwu/CIM-compatible backend, not physical CIM execution.
This paper studies the cluster-assignment layer of a cluster-first-route-second decomposition for the capacitated pickup-and-delivery problem with time windows (CPDPTW), instead of a direct route-ordering quadratic unconstrained binary optimization (QUBO). Sample-based subspace diagonalization (SQD) degenerates to best-of-shots for any diagonal optimization Hamiltonian, so a useful transfer to routing requires non-diagonal relaxation. Quantum random access optimization (QRAO) provides this non-diagonality by encoding binary variables through non-commuting quantum random access code (QRAC) observables. CPDPTW-derived conflict graphs expose a routing-specific obstruction because dense objectives and penalty-encoded constraints destroy QRAO compression. We keep the Hamiltonian sparse by encoding only the sparsified objective and enforcing one-hot, capacity, and cluster time-budget feasibility by repair after rounding. We make no quantum-advantage claim. Under strong repair the classical pass alone reaches near-optimal cost at the tested sizes. The result is a regime map linking sparsification, achieved compression, linear-combination-of-unitaries (LCU) 1-norm, heavy-hex two-qubit depth, sampling drift, and final routing quality on Qiskit Aer, calibrated FakeTorino, and IBM Heron ibm_quebec. Across a 108-row ideal-calibrated-hardware evaluation grid spanning four to twelve pickup-delivery request pairs, the hardware slice has mean gap 0.0016 and mean device-vs-ideal total variation distance (TVD) 0.279; a decoder ablation at sizes six through twelve isolates the non-diagonal signal.
Hybrid quantum optimization for vehicle routing faces a practical bottleneck: direct QUBO encodings of CVRP quickly exceed near-term qubit and gate budgets, while quantum evaluations are expensive, noise-limited, and sensitive to backend and circuit configuration. We address this gap with an end-to-end decomposition pipeline that converts CVRP into bounded-width quantum subproblems and treats quantum execution as a decision problem within the optimization loop. Starting from a Fisher--Jaikumar assignment linearization, we apply Lagrangian relaxation to dualize customer-assignment couplers, yielding independent per-vehicle knapsack subproblems that admit QUBO/Ising evaluation. To replace brittle subgradient tuning, we learn a multiplier-update controller using expert-guided pretraining followed by reinforcement-learning fine-tuning, with rewards based on execution-realized progress and route reconstruction. We also introduce a constrained contextual bandit as a hardware-aware execution layer that selects backend and circuit configuration with feasibility screening, enabling adaptation across heterogeneous noisy resources and parallel multi-QPU scheduling. Computational results on multiple CVRPLIB families show that the decomposition yields stable bounded-width subproblems across instance sizes, learned multiplier updates improve end-to-end routing quality relative to classical subgradient control under matched budgets, and hardware-mode configuration reduces median optimality gaps relative to static execution choices in our test set. We do not claim quantum advantage. Instead, the contribution is a practical end-to-end framework for scaling hybrid quantum CVRP optimization through OR decomposition, learning-augmented dual control, and adaptive hardware-aware execution.
Abstract: Global logistics networks, intercontinental air-freight consolidators, and maritime energy fleets continuously solve large-scale Mixed-Integer Linear Programs (MILPs) such as the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) and multi-vessel Maritime Inventory Routing Problems (MIRP). When weather disruptions, port congestion, or geopolitical airspace closures invalidate baseline schedules, classical branch-and-cut solvers (Gurobi, CPLEX) experience exponential worst-case tree growth on subproblems exceeding 2,000 binary decision variables, often failing to converge within strict 5-to-15-minute operational re-dispatch windows. Conversely, current Noisy Intermediate-Scale Quantum (NISQ) processors remain limited to 100-to-1,000 physical qubits with finite two-qubit gate fidelity (99.2%−99.8%), preventing direct encoding of full enterprise logistics instances. This paper presents a production-grade Quantum-Classical Hybrid Cloud Architecture that bridges enterprise Kubernetes clusters with co-located quantum processing units (QPUs) via AWS Braket Hybrid Jobs and IBM Quantum Qiskit Runtime primitives. Our framework applies Benders and Dantzig-Wolfe column generation on classical cloud HPC nodes to decompose a 25,000-variable global fleet MILP into a continuous master problem and hard combinatorial pricing/routing subproblems of 60 to 140 binary variables. These subproblems are transformed into Quadratic Unconstrained Binary Optimization (QUBO) and Ising Hamiltonians, warm-started from classical linear programming (LP) relaxations (WS-QAOA), compiled with dynamical decoupling and Zero-Noise Extrapolation (ZNE), and executed inside co-located QPU-CPU sessions while a GPU-accelerated Simulated Bifurcation (SB) digital annealer runs in parallel as a deterministic SLA fallback. Evaluated on disruption recovery workloads modeled on ExxonMobil maritime LNG routing and DHL Global Forwarding air-cargo container loading, our hybrid WS-QAOA + Column Generation workflow improves optimality gap closure at a 10-minute SLA cutoff from 6.84% (classical branch-and-bound alone) to 1.42%, reduces emergency re-routing compute wall-clock time by 61.5%, and cuts fleet fuel and charter penalty costs by 4.7%. Research Domain & Specialization Track Master of Science in Information Technology (MSIT) — Quantum Cloud Computing, Hybrid Orchestration, and Operations Research. Core Technical Highlights Empirical Production Case Studies: Rigorous architectural blueprints and telemetry evaluation across enterprise cloud deployments. Four-Tier Reference Architecture: Formal specification from edge ingress to distributed hardware execution pools. Mathematical Formulation: Formally derived algorithmic and optimization models with closed-loop verification. Multi-Modal Benchmarking: Comparative empirical baselines under fault-injection and synthetic workload scaling. Author Information Author: Sonu Kumar SinghRole: Senior Consultant Affiliation: Independent Researcher Publication Status: Preprint / Working Paper (MSIT AI & Cloud Systems Research Series) Reproducibility & Open Science All included architecture diagrams, telemetry schemas, and benchmark curves are open-access under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
Graph coarsening reduces the large Quadratic Unconstrained Binary Optimization (QUBO) formulations arising when vehicle-routing problems are solved by quantum annealing. Nearby customers with compatible time windows are merged into super-nodes, the reduced problem is solved, and the solution is expanded to the original graph. For the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), existing coarsening heuristics require family-specific tuning and remain unreliable on random instances. We address these limitations on the Solomon benchmark using simulated annealing and a D-Wave Advantage2 processor. We first introduce adaptive penalty calibration. Uniform penalty scaling has little effect, whereas controlling the internal coefficient range substantially improves raw samples. Removing non-binding constraints, normalising binding ones, and scaling the remaining penalties reduces mean raw constraint violations from 33.0 to 0.06 at the same solver budget (p=3.7e-11, n=56). A variable-count-preserving control attributes this gain to conditioning rather than problem size. Second, we replace the hand-tuned merge score with a graph neural network (GNN) using one configuration across all families. At N=10, it achieves 100% feasibility across all Solomon families, including R-type (100% vs. 80% for the tuned heuristic). Across N=10,...,100, feasibility is 83% vs. 69%, with the GNN better or tied on 85/90 instance-size pairs. At N=80,100, the difference is significant (p=0.002; 25/25 pairs), while the QUBO remains approximately 5-6 times smaller. Finally, hardware experiments reproduce the conditioning effect at fixed logical variable count: feasible samples increase from 0.02% to 39% across 13 instances. Classical repair with local search remains a reference bound for end-to-end solution cost.
We revisit a degree-only arc Hamiltonian for fixed-fleet, homogeneous, uncapacitated vehicle routing. Its local penalties define only a cycle cover, so ground states may contain customer cycles disconnected from the depot. We construct a polynomial-size quadratic unconstrained binary optimization (QUBO) repair using capped single-commodity flow and prove connected, cost-optimal ground-state routings under explicit penalty assumptions. For N−1 customers and K nonempty routes, the unreduced encoding uses exactly ∣E∣(1+⌈log2(N−K+1)⌉) logical problem qubits. A reversible compute-phase-uncompute implementation evaluates the flow penalties in O(N2logN+Nlog2N) logical gates on a complete graph, using O(logN) reusable workspace and no product register. On complete loopless graphs, a position encoding that arranges all routes in one sequence separated by depot occurrences uses fewer problem qubits and written terms when the number of bits per flow register grows. Under the same reversible accounting model, the flow formulation admits a smaller logical-gate upper bound. Exact Hamiltonian and circuit audits and a benchmark of 1,200 cost matrices verify the formulation and quantify the connectivity gap. A 32,000-shot Amazon Braket task on IQM Emerald characterizes depth-one termwise Ising circuits for a diagnostic N=4, K=1 counterexample. In the degree-only circuit, 78.05% of selected p=1 shots realize the invalid disconnected ground state; the reduced 14-qubit flow-augmented circuit yields no fully feasible sample. These device results characterize mapped Hamiltonians and compilation, not an asymptotic advantage in solving routing problems.
The vehicle routing problem with time windows (VRPTW) is a common optimization problem faced within the logistics industry. In this work, we explore the use of a previously-introduced qubit encoding scheme to reduce the number of binary variables, to evaluate the effectiveness of NISQ devices when applied to industry relevant optimization problems. We apply a quantum variational approach to a testbed of multiple VRPTW instances ranging from 11 to 3964 routes. These intances were formulated as quadratic unconstrained binary optimization (QUBO) problems based on realistic shipping scenarios. We compare our results with standard binary-to-qubit mappings after executing on simulators as well as various quantum hardware platforms, including IBMQ, AWS (Rigetti), and IonQ. These results are benchmarked against the classical solver, Gurobi. Our approach can find approximate solutions to the VRPTW comparable to those obtained from quantum algorithms using the full encoding, despite the reduction in qubits required. These results suggest that using the encoding scheme to fit larger problem sizes into fewer qubits is a promising step in using NISQ devices to find approximate solutions for industry-based optimization problems, although additional resources are still required to eke out the performance from larger problem sizes.