Vehicle-Routing Clustering by Sparse Quantum Relaxation
Abstract
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.
Related
- Quantum Vehicle Routing Problem: a smart initialization strategy for ill-conditioned optimization landscapes
- Correcting Connectivity in Arc-Based QUBO Models for Fixed-Fleet Vehicle Routing
- Hierarchical Unsupervised Clustering of Nanobeam Electron Diffraction Data Using Sparse Bragg-Vector Representations
- Tight Communication Bounds for Distributed Algorithms in the Quantum Routing Model
- Quantum State Routing and Perfect State Transfer on Signed Graphs under Environmental Noise
BibTeX
@misc{moosavi2026vehiclerouting,
title = {{Vehicle-Routing Clustering by Sparse Quantum Relaxation}},
author = {Farzan Moosavi and Bilal Farooq},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19575},
eprint = {2609.19575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19575},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Vehicle-Routing Clustering by Sparse Quantum Relaxation
AU - Farzan Moosavi
AU - Bilal Farooq
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19575
UR - https://arxiv.org/abs/2609.19575
AB - 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.
ER -