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Scientific Reports

Improving QAOA feasibility and hardware performance for vehicle routing via linear chain and ramp schedules

Talha Azfar, Ruimin Ke

Published Sep 10, 2026• 0 citations
Publisher

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is difficult to deploy on near-term hardware for constrained routing problems because feasibility constraints produce dense interaction graphs, deep transpiled circuits, and costly parameter search. We study this challenge for Vehicle Routing Problem (VRP) instances using a hardware-aware QAOA workflow that combines linear-ramp parameter initialization, Conditional Value-at-Risk (CVaR) optimization, and a Linear-Chain QAOA (LC-QAOA) surrogate that retains local ZZ interactions to reduce two-qubit depth. A matched full-QAOA versus LC-QAOA feasibility comparison is performed for 4-node instances, while the 5- and 6-node experiments characterize circuit executability and low-probability optimum recovery at the tested depths. In 4-node hardware runs, LC-QAOA increases mean feasible-sample probability from 0.182% to 0.917% and mean optimum-sample probability from 0.045% to 0.239%, giving approximately \(5\times\) improvement over full QAOA. At 6 nodes, a full-QAOA circuit at \(p=1\) already has large transpiled two-qubit depth, while LC-QAOA remains executable and samples CPLEX-certified optima in selected hardware runs at depths p =10–13, albeit with low optimum probability. Overall, the results identify LC-QAOA as a hardware-efficient surrogate heuristic that trades Hamiltonian completeness for lower depth and improved feasible-solution sampling on noisy quantum hardware.

DOI
10.1038/s41598-026-70696-y

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The Quantum Approximate Optimization Algorithm (QAOA) is difficult to deploy on near-term hardware for constrained routing problems because feasibility constraints produce dense interaction graphs, deep transpiled circuits, and costly parameter search. We study this challenge for Vehicle Routing Problem (VRP) instances using a hardware-aware QAOA workflow that combines linear-ramp parameter initialization, Conditional Value-at-Risk (CVaR) optimization, and a Linear-Chain QAOA (LC-QAOA) surrogate that retains local ZZ interactions to reduce two-qubit depth. A matched full-QAOA versus LC-QAOA feasibility comparison is performed for 4-node instances, while the 5- and 6-node experiments characterize circuit executability and low-probability optimum recovery at the tested depths. In 4-node hardware runs, LC-QAOA increases mean feasible-sample probability from 0.182% to 0.917% and mean optimum-sample probability from 0.045% to 0.239%, giving approximately (5\times) improvement over full QAOA. At 6 nodes, a full-QAOA circuit at (p=1) already has large transpiled two-qubit depth, while LC-QAOA remains executable and samples CPLEX-certified optima in selected hardware runs at depths p =10–13, albeit with low optimum probability. Overall, the results identify LC-QAOA as a hardware-efficient surrogate heuristic that trades Hamiltonian completeness for lower depth and improved feasible-solution sampling on noisy quantum hardware.
Abstract The Quantum Approximate Optimization Algorithm (QAOA) is known to struggle on near-term hardware due to constraint handling, parameter search overhead, and tight depth limits. We address this with a hardware-aware co-design: a linear-ramp parameter schedule paired with a Linear-Chain (LC) ansatz that restricts ZZZZZZ interactions to nearest neighbors, lowering swaps and depth. We benchmark Linear-Ramp LC-QAOA against standard QAOA, Two-Step QAOA, and Grover-Mixer QAOA on small VRP instances across ideal simulation, Aer simulation, and IBM Eagle/Heron runs, measuring feasibility, solution quality, and hardware efficiency. Linear-ramp improves convergence and feasibility; LC reduces two-qubit gates and boosts noise robustness. Grover-Mixer and Two-Step can aid constraints in theory but are hampered in practice by deeper circuits. On hardware, overall feasibility remains <1<1<1\%, yet LC-QAOA with XpXmXpXmXpXm dynamical decoupling more than doubles feasibility and recovers the optimum in several trials. We also show efficient scaling, including a larger case with ramp-initialized LC-QAOA plus CVaR. The results underscore hardware-aligned ansätze and structured schedules as a promising path.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading framework for quantum combinatorial optimization. The Vehicle Routing Problem (VRP), a core problem in logistics and transportation, is a natural application target, but it poses a major feasibility challenge for standard QAOA because feasible solutions occupy only a tiny fraction of the search space, and the conventional Pauli-XXX mixer can disrupt partial solution structures that satisfy key local constraints. To address this issue, we propose a constraint-aware QAOA framework with two complementary components. First, we design a lightweight initialization strategy that encodes a selected subset of simple yet informative local one-hot constraints into the initial state, thereby reducing the initial superposition space and increasing the probability mass on states with important local structure. Second, we introduce a hybrid XY-XXX mixer that preserves the constraint structure imposed at initialization while retaining exploratory flexibility over the remaining unconstrained degrees of freedom during QAOA evolution. We evaluate the proposed framework against standard QAOA under three progressively more realistic regimes: ideal statevector simulation, finite-shot sampling, and noisy finite-shot sampling. Across all regimes, the proposed method consistently achieves lower average energy and higher feasible-solution ratios than standard QAOA, indicating more effective guidance toward structurally valid, lower-cost VRP solutions. However, the performance gap narrows in the noisy regime. Because this setting adopts a hardware-inspired error model based on near-best-reported laboratory-level qubit gate and readout fidelities, the observed attenuation suggests that the practical advantage of the more structured mixer is likely to grow as quantum hardware improves and error rates decline.
The Vehicle Routing Problem (VRP) is a fundamental combinatorial optimization challenge with broad applications in logistics and transportation. In this work, we present a quantum-assisted framework that integrates the Quantum Approximate Optimization Algorithm (QAOA) with a link-based formulation of VRP. Our approach encodes flow conservation and subtour elimination directly into the cost Hamiltonian, preserving graph structure while minimizing resource requirements for practical hardware implementation. We design and implement the full pipeline on a gate-based quantum computer, including problem formulation, encoding, circuit synthesis, and execution on IBM Quantum System One. Experimental results on small VRP instances highlight the effects of penalty scaling, coefficient normalization, and circuit depth on solution feasibility under hardware noise. While scalability remains constrained by circuit complexity and decoherence, the study demonstrates a practical pathway for implementing VRP on quantum hardware and identifies methodological directions for advancing near-term quantum optimization.
Transportation systems such as urban logistics, vehicle routing, and infrastructure planning require solving large-scale combinatorial optimization problems under complex constraints. Problems such as the vehicle routing problem (VRP), traveling salesman problem (TSP), and facility location problem (FLP) involve large discrete search spaces and the need to generate multiple feasible solutions in real time. In this work, we develop a hardware-grounded hybrid quantum optimization framework that uses Approximate Quantum Compilation (AQC) to compress early segments of digitized adiabatic evolution into shallow circuits. The compressed prefix is combined with variational layers, enabling a systematic study of how initialization, circuit depth, and expressivity interact on near-term quantum hardware. All experiments are performed on an IBM gate-based quantum computer, and circuits are evaluated as stochastic generators of candidate transportation plans. Results show that moderate prefix compression reduces two-qubit gate depth while maintaining or improving feasible solution discovery, particularly for routing problems. These benefits depend on compatibility between the compressed prefix and the variational ansatz: while standard QAOA effectively leverages AQC initialization, linear-chain QAOA shows limited improvement. Overall, this work demonstrates that hybrid AQC-QAOA methods provide a practical pathway for hardware-efficient quantum optimization, positioning quantum algorithms as candidate generators within transportation decision-making workflows.
We propose a hierarchical quantum approximate optimization framework for solving large-scale Vehicle Routing Problems (VRP) using Quantum Approximate Optimization Algorithm (QAOA). The method decomposes a VRP instance into balanced clusters of customer nodes. We formulate intra-cluster routing as Open loop Traveling Salesman Problems (OTSPs), and inter-cluster routing as a reduced VRP over the cluster representatives and depot. We then map the sub-problems to Ising Hamiltonians and solve with both standard and multi-angle QAOA variants at fixed depth p=3, and merge them to produce a routing path for the original VRP. Additionally, to improve solution feasibility and success probability, we introduce a polynomial-time post-processing protocol that samples candidate bit-strings from the QAOA output using a probability threshold and performs exhaustive local 1 and 2 bit-flip searches around these candidates. Benchmarking on 100 randomly generated 13-node, two-vehicle VRP instances, we show that the post-processed standard-QAOA implementation achieves high success rates and approximation ratios within 1.2-1.5 compared to classical optimizer (Gurobi) solutions, while requiring only 12 logical qubits per subproblem instead of 156 qubits for a direct edge-based encoding. These results provide a proof-of-concept demonstration that hierarchical decomposition, shallow QAOA, and local bit-flip repair can offer a scalable and resource-efficient pathway toward larger VRP instances on near-term quantum devices.
Intelligent transportation systems (ITS) are a critical component of Industry 4.0 and 5.0, particularly having applications in logistic management. One of their crucial utilization is in supply-chain management and scheduling for optimally routing transportation of goods by vehicles at a given set of locations. This paper discusses the broader problem of vehicle traffic management, more popularly known as the Vehicle Routing Problem (VRP), and investigates the possible use of near-term quantum devices for solving it. For this purpose, we give the Ising formulation for VRP and some of its constrained variants. Then, we present a detailed procedure to solve VRP by minimizing its corresponding Ising Hamiltonian using a hybrid quantum-classical heuristic called Quantum Approximate Optimization Algorithm (QAOA), implemented on the IBM Qiskit platform. We compare the performance of QAOA with classical solvers such as CPLEX on problem instances of up to 15 qubits. We find that performance of QAOA has a multifaceted dependence on the classical optimization routine used, the depth of the ansatz parameterized byppp, initialization of variational parameters, and problem instance itself.