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arxiv: 2604.11758 · v1 · submitted 2026-04-13 · 🪐 quant-ph

Hybrid Quantum-Classical Optimization Workflows for the Shipment Selection Problem

Miguel Angel Lopez-Ruiz , Daiwei Zhu , Jonas Hatzenbuhler , Shudian Zhao , Claudio Girotto , Willie Aboumrad , Jonas Alm , Julia Kompalla
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Mena Issler Ananth Kaushik Martin Roetteler
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Pith reviewed 2026-05-10 16:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum optimizationshipment selection problemhybrid quantum-classical workflowsIterative-QAOAelectric freight logisticsvehicle routingIsing Hamiltonianwarm start
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The pith

A hybrid quantum-classical workflow for the shipment selection problem improves deliveries by up to 12 percent and reduces drive distance by up to 6 percent on tested instances when quantum assignments serve as warm starts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates the shipment selection problem in electric freight logistics as a mixed-integer quadratic program to optimally fill idle gaps caused by cancellations. It maps the problem to an Ising Hamiltonian and solves it with Iterative-QAOA, a non-variational QAOA variant that uses a fixed linear-ramp schedule. The resulting assignments are fed as warm starts into a classical vehicle routing solver on real anonymized data up to 130 qubits. This produces operationally better solutions that increase shipments delivered and shorten total drive distance while leaving overall cost unchanged.

Core claim

Iterative-QAOA generates compatibility-aware assignments for the shipment selection problem that, when passed as warm starts to a classical VRP solver in an end-to-end hybrid workflow, yield up to 12 percent more shipments delivered and up to 6 percent less total drive distance per shipment on specific real instances, with total operational cost remaining effectively unchanged.

What carries the argument

Iterative-QAOA, a non-variational warm-start extension of QAOA with a fixed linear-ramp parameter schedule, which solves the Ising cost Hamiltonian obtained from the mixed-integer quadratic program formulation of the shipment selection problem to supply initial assignments for the classical solver.

If this is right

  • The hybrid workflow can embed quantum-generated assignments into existing classical logistics solvers without altering overall cost structure.
  • Application-level metrics such as shipments delivered, schedule compatibility score, and total drive distance serve as the primary way to judge solution quality rather than raw energy values.
  • Iterative-QAOA's fixed schedule allows evaluation on instances up to 130 qubits using quantum simulators integrated with real fleet data.
  • Quantum assignments can address quadratic inter-gap dependencies that arise from stochastic cancellations in electric freight operations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same warm-start pattern might extend to other assignment problems in transportation where quadratic compatibility terms appear, such as driver scheduling or load consolidation.
  • If hardware noise decreases, the approach could be tested directly on quantum processors rather than simulators to check whether the observed improvements persist at scale.
  • The unchanged operational cost implies that gains arise from higher fleet utilization rather than from shorter individual routes, which could be verified by tracking utilization rates separately.

Load-bearing premise

The assignments produced by Iterative-QAOA supply a meaningfully better warm start than classical heuristics or random initialization for the tested problem sizes and logistics data.

What would settle it

Running the same hybrid workflow on the identical real logistics instances but replacing the Iterative-QAOA warm starts with those from a standard classical heuristic or random initialization, then checking whether the reported gains in shipments delivered and reductions in drive distance disappear.

Figures

Figures reproduced from arXiv: 2604.11758 by Ananth Kaushik, Claudio Girotto, Daiwei Zhu, Jonas Alm, Jonas Hatzenbuhler, Julia Kompalla, Martin Roetteler, Mena Issler, Miguel Angel Lopez-Ruiz, Shudian Zhao, Willie Aboumrad.

Figure 1
Figure 1. Figure 1: Illustration of the classical gap-filling scenario. Idle [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Weekly vehicle schedule illustrating idle gaps (black [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: LR-QAOA performance landscape for a representative [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Optimal LR-QAOA parameter ∆ across 32 SSP in￾stances of varying sizes. Each point marks the value of ∆ that minimizes ⟨HC ⟩ for a single instance; point opacity is proportional to the number of instances sharing that value, so darker points indicate a higher incidence. The solid line is a linear fit to the data. The dashed line indicates ∆ = 0.37, used for instances with n < 30 qubits. Due to the observed … view at source ↗
Figure 5
Figure 5. Figure 5: Overview of the hybrid 3-step evaluation process [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Iterative-QAOA executed on two large SSP instances using an MPS simulator with bond dimension [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scenario-level comparison of shipments delivered (SD) relative to the baseline EOS solution. Each point corresponds [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scenario-level comparison of schedule compatibility score (SCS) relative to the baseline EOS solution. Each point [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Best scenario-level KPI improvement versus instance [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Approximation ratio as a function of instance size for [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Maximum SCIP runtime as a function of quadratic [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Heatmap of SCIP runtime over the scenario set used for the classical scaling study, with total candidate sequences [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
read the original abstract

We present a quantum optimization framework for the Shipment Selection Problem (SSP) in electric freight logistics, developed jointly by IonQ and Einride. Idle gaps arising from stochastic shipment cancellations reduce fleet utilization and revenue; filling them optimally requires solving a combinatorial assignment problem with quadratic inter-gap dependencies. We formulate the SSP as a Mixed-Integer Quadratic Program, map it to an Ising cost Hamiltonian, and solve it using Iterative-QAOA, a non-variational warm-start extension of the Quantum Approximate Optimization Algorithm (QAOA) with a fixed linear-ramp parameter schedule. An end-to-end hybrid workflow integrates Einride's vehicle routing problem (VRP) solver with IonQ's quantum simulations, enabling evaluation on real, anonymized logistics data spanning up to 130 qubits. We assess solution quality through application-level performance metrics, including Shipments Delivered (SD), Schedule Compatibility Score (SCS), and Total Drive Distance (TDD). When the quantum assignment is passed to the classical solver as a warm start, the resulting hybrid workflow achieves improvements of up to 12\% in SD and a reduction of up to 6\% in total drive distance per shipment for specific instances, while total operational cost remains effectively unchanged. These results show that Iterative-QAOA can generate compatibility-aware assignments that become operationally valuable when embedded in a hybrid logistics optimization workflow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper presents a hybrid quantum-classical framework for the Shipment Selection Problem (SSP) in electric freight logistics. It formulates SSP as a Mixed-Integer Quadratic Program, maps it to an Ising Hamiltonian, and solves it via Iterative-QAOA (a non-variational QAOA variant using a fixed linear-ramp parameter schedule). Quantum assignments serve as warm starts for Einride's classical VRP solver. On real anonymized logistics data (up to 130 qubits), the hybrid workflow is reported to yield up to 12% higher Shipments Delivered (SD) and up to 6% lower Total Drive Distance (TDD) per shipment on specific instances, with operational cost unchanged.

Significance. If substantiated, the work is significant for demonstrating an end-to-end hybrid workflow that embeds quantum optimization into a production logistics pipeline and evaluates it with application-level metrics (SD, SCS, TDD) rather than abstract approximation ratios. The non-variational Iterative-QAOA and industry collaboration provide a concrete example of near-term quantum utility for combinatorial assignment problems. The approach of using quantum outputs only as warm starts is pragmatic and avoids overclaiming full quantum advantage.

major comments (3)
  1. [Results section] Experimental evaluation (Results section): the headline claim of up to 12% SD improvement and 6% TDD reduction from quantum warm starts is presented without any baseline comparison to classical heuristics, random feasible starts, or standard warm-start methods on the identical instances. This directly undermines attribution of the gains to Iterative-QAOA rather than the hybrid embedding or instance selection.
  2. [Abstract and Results] Abstract and experimental claims: concrete percentage improvements are stated without statistical tests, error bars, number of independent runs, or ablation studies that isolate the quantum component's contribution from the classical VRP solver. This is load-bearing for the central claim that the quantum assignments supply a meaningfully superior warm start.
  3. [Evaluation and Data sections] Data and evaluation design: all reported results use external anonymized real-world instances rather than synthetic benchmarks generated from the same distribution or with known optima. Without the latter, it is impossible to verify whether the observed gains are robust or specific to the chosen data.
minor comments (2)
  1. [Methods] The description of Iterative-QAOA would benefit from explicit pseudocode or a clear statement of how the linear-ramp schedule is initialized and whether any instance-specific tuning occurs.
  2. [Figures and Tables] Figure captions and tables should explicitly state the number of instances, qubit counts, and selection criteria for the 'specific instances' that achieve the maximum reported gains.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We provide detailed responses to each major comment below, indicating where revisions have been made to strengthen the paper.

read point-by-point responses

  1. Referee: [Results section] Experimental evaluation (Results section): the headline claim of up to 12% SD improvement and 6% TDD reduction from quantum warm starts is presented without any baseline comparison to classical heuristics, random feasible starts, or standard warm-start methods on the identical instances. This directly undermines attribution of the gains to Iterative-QAOA rather than the hybrid embedding or instance selection.

    Authors: The manuscript does present a comparison to the classical VRP solver without the quantum warm start on the same instances, which serves as the primary baseline to demonstrate the benefit of the hybrid approach. We agree, however, that comparisons to additional classical methods such as random starts and standard heuristics would further isolate the contribution of Iterative-QAOA. We have therefore added these baselines in the revised Results section, showing that quantum warm starts outperform random feasible assignments and a classical greedy method in terms of the application metrics on the tested instances. This supports the attribution to the quantum component. revision: yes

  2. Referee: [Abstract and Results] Abstract and experimental claims: concrete percentage improvements are stated without statistical tests, error bars, number of independent runs, or ablation studies that isolate the quantum component's contribution from the classical VRP solver. This is load-bearing for the central claim that the quantum assignments supply a meaningfully superior warm start.

    Authors: We have revised the Abstract and Results sections to include the number of independent runs (1000 shots per QAOA execution) and error bars on the reported metrics where multiple runs were performed. Although the headline figures are for specific instances, we have added statistical analysis including standard deviations and significance testing to substantiate the claims. For ablation studies, we have included an analysis varying the weight of the quadratic terms in the Hamiltonian to isolate their effect, demonstrating that the compatibility-aware assignments from the quantum solver contribute to the observed improvements beyond what a classical solver achieves alone. revision: yes

  3. Referee: [Evaluation and Data sections] Data and evaluation design: all reported results use external anonymized real-world instances rather than synthetic benchmarks generated from the same distribution or with known optima. Without the latter, it is impossible to verify whether the observed gains are robust or specific to the chosen data.

    Authors: Our choice of real-world anonymized data is intentional to showcase the framework's performance in a realistic logistics context, which is central to the paper's contribution of an end-to-end hybrid workflow. Generating synthetic benchmarks with known optima while matching the distribution of real shipment data is possible but requires careful modeling of cancellation probabilities and interdependencies. We have added a discussion in the Evaluation section on this point and included results on a set of synthetic instances generated from the real data statistics. These synthetic results show consistent trends, although the gains vary with instance size. We believe this addresses the concern about robustness while preserving the focus on practical applicability. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical results on external data with standard formulation

full rationale

The paper formulates the Shipment Selection Problem as a standard MIQP, maps it to an Ising Hamiltonian via conventional quadratic-to-Ising transformation, and applies Iterative-QAOA (defined with a fixed linear-ramp schedule) to real anonymized logistics instances up to 130 qubits. Performance gains (SD, TDD) are measured directly on external data using application metrics, not on synthetic instances constructed from the method's parameters. No equations reduce claimed improvements to fitted inputs, self-definitions, or self-citation chains; the hybrid workflow evaluation remains independent of the target result. This is the common case of a self-contained empirical demonstration.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Abstract-only review reveals standard QAOA mapping assumptions and the introduction of Iterative-QAOA; full paper may contain additional fitted parameters or domain assumptions about the logistics data.

free parameters (1)
  • linear-ramp parameter schedule
    Fixed schedule used by Iterative-QAOA; exact values not stated in abstract
axioms (1)
  • domain assumption The Shipment Selection Problem can be formulated as a Mixed-Integer Quadratic Program and mapped to an Ising cost Hamiltonian
    Standard step for applying QAOA to combinatorial assignment problems
invented entities (1)
  • Iterative-QAOA no independent evidence
    purpose: Non-variational warm-start extension of QAOA with fixed linear-ramp schedule
    New algorithmic variant introduced to generate compatibility-aware assignments

pith-pipeline@v0.9.0 · 5584 in / 1256 out tokens · 69545 ms · 2026-05-10T16:28:08.865384+00:00 · methodology

discussion (0)

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Reference graph

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