Requirements for Early Quantum Utility and Quantum Utility in the Capacitated Vehicle Routing Problem

Chinonso Onah; Kristel Michielsen

arxiv: 2509.11469 · v2 · pith:CBPBRMM7new · submitted 2025-09-14 · 🪐 quant-ph · cs.CE· physics.app-ph· physics.comp-ph· physics.data-an

Requirements for Early Quantum Utility and Quantum Utility in the Capacitated Vehicle Routing Problem

Chinonso Onah , Kristel Michielsen This is my paper

Pith reviewed 2026-05-21 22:18 UTC · model grok-4.3

classification 🪐 quant-ph cs.CEphysics.app-phphysics.comp-phphysics.data-an

keywords Capacitated Vehicle Routing ProblemNISQ hardwarequantum advantageQUBO encodingHOBO encodingresource estimationdecision diagramvehicle routing

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The pith

A resource framework shows early quantum utility for CVRP is unlikely on NISQ hardware even with efficient encodings.

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

The paper develops an encoding-agnostic framework that calculates exact qubit and gate needs for CVRP instances and compares them against real device performance data. It concludes that quantum solutions remain out of reach for noisy intermediate-scale hardware, even when using the most compact direct encodings. The method produces a single decision diagram with three go/no-go thresholds that classify any given CVRP problem and hardware pair. For the Golden-5 benchmark the framework reports a sharp contrast between encodings, with one using far fewer resources than the other. If the estimates hold, meaningful quantum advantage in vehicle routing would require new ways to break problems into smaller pieces rather than solving them directly.

Core claim

What carries the argument

The decision diagram defined by the quantum feasibility point together with the qubit-feasibility and gate-feasibility lines, which accepts any CVRP encoding and any hardware error profile to produce a go or no-go outcome.

If this is right

HOBO encodings need only 7685 qubits for Golden-5 while comparable QUBO encodings exceed 200000 qubits.
Quantum advantage for CVRP will most likely require new problem decomposition methods rather than direct mappings.
The same three figures of merit can rank any CVRP instance against any given quantum device profile.
NISQ hardware falls short of the resource thresholds even in the most favorable encoding scenarios.
The framework supplies a concrete way to identify which CVRP instances are closest to becoming quantum-feasible.

Where Pith is reading between the lines

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

The same counting approach could be reused for related routing problems such as the traveling salesman problem by swapping in their specific encodings.
Hybrid solvers that split large instances into quantum-solvable subproblems become a natural next target once the framework flags direct encodings as insufficient.
Plugging improved error rates or higher qubit counts from future devices into the diagram would give a quantitative timeline for when advantage might appear.

Load-bearing premise

The framework assumes closed-form qubit and gate counts plus published device benchmarks are enough to judge feasibility without extra costs from error mitigation, compilation, or classical pre- and post-processing.

What would settle it

A working demonstration that solves the Golden-5 CVRP instance on hardware with roughly 8000 qubits in less time or with better quality than the best classical heuristic would falsify the claim that early utility is unlikely.

Figures

Figures reproduced from arXiv: 2509.11469 by Chinonso Onah, Kristel Michielsen.

**Figure 1.** Figure 1: Feasibility maps for HOBO-encoded CVRP instances. Each green dot represents a Capacitated Vehicle Routing Problem (CVRP) instance placed by its required logical-qubit count N (horizontal axis) and two-qubit-gate depth D (vertical axis) when encoded with the higher-order binary optimisation (HOBO) Hamiltonian. The vertical dashed line marks the projected qubit budget Nmax for next-generation gate-based dev… view at source ↗

**Figure 2.** Figure 2: Quantum-resource requirements and feasibility outlook for solving proto [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

We introduce a transparent, encoding-agnostic framework for determining when the Capacitated Vehicle Routing Problem (CVRP) can achieve early quantum advantage. Our analysis shows this is unlikely on noisy intermediate scale quantum (NISQ) hardware even in best case scenarios that use the most qubit-efficient direct encodings. Closed-form resource counts, combined with recent device benchmarks, yield three decisive go/no-go figures of merit: the quantum feasibility point and the qubit- and gate-feasibility lines, which place any CVRP instance on a single decision diagram. Contrasting a direct QUBO mapping with a space-efficient higher-order (HOBO) encoding reveals a large gap. Applied to early-advantage benchmarks such as Golden-5, our diagram shows that HOBO circuits require only 7,685 qubits, whereas comparable QUBO encodings still exceed 200,000 qubits. In addition to identifying candidate instances for early quantum advantage in CVRP, the framework provides a unifying go/no-go metric that ingests any CVRP encoding together with any hardware profile and highlights when quantum devices could challenge classical heuristics. Quantum advantage in CVRP would likely require innovative problem decomposition techniques.

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.

Desk Editor's Note private letter to a colleague

The paper supplies a clear decision diagram for CVRP quantum feasibility that contrasts QUBO and HOBO encodings on Golden-5, but the resource estimates omit compilation, error mitigation, and classical overheads so the thresholds are optimistic lower bounds.

read the letter

The main contribution is an encoding-agnostic framework that turns closed-form qubit and gate counts plus device benchmarks into three go/no-go lines on one diagram. For the Golden-5 instance it shows a HOBO encoding needing roughly 7,685 qubits while a comparable QUBO mapping exceeds 200,000. That concrete gap and the plug-in structure for any encoding or hardware profile are the parts that feel new and potentially useful for quick scoping of early-advantage candidates. The authors also state plainly that quantum advantage in CVRP will probably need decomposition techniques beyond direct encoding. That honesty about the limits is welcome. The diagram itself is simple enough that a practitioner could adapt it without much trouble. What the work does well is make the comparison between encodings transparent and tie it directly to published hardware numbers rather than leaving everything abstract. On the soft spots, the closed-form counts treat the derived resources as directly usable against coherence and error-rate data. In practice, mapping to limited-connectivity hardware adds SWAP overhead and extra gates, error mitigation multiplies the effective depth or shot count, and classical pre- and post-processing for instance decomposition and solution extraction adds wall-clock cost. None of these appear folded into the feasibility lines, so the reported thresholds are best-case lower bounds. The abstract gives the final numbers without derivation steps or sensitivity checks, which makes it hard to judge how much the lines would move under modest changes in device parameters. The central claim that CVRP is unlikely on NISQ even in best-case direct encodings therefore rests on assumptions that need more explicit justification. This paper is aimed at researchers working on quantum optimization for routing problems and at logistics groups that want a first-cut feasibility check. A reader already thinking about QUBO versus higher-order encodings for vehicle routing would get concrete numbers to compare against their own hardware targets. I would send it to peer review because the unifying diagram idea is worth discussion and the benchmark application is specific enough to be testable, even though the overhead analysis and derivation details would need strengthening in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a transparent, encoding-agnostic framework for assessing early quantum utility in the Capacitated Vehicle Routing Problem (CVRP). Using closed-form resource counts for direct QUBO and higher-order (HOBO) encodings combined with device benchmarks, it generates three go/no-go metrics—the quantum feasibility point, qubit-feasibility line, and gate-feasibility line—placed on a decision diagram. The analysis concludes that achieving early quantum advantage for CVRP on NISQ hardware is unlikely even under best-case direct encodings, with the Golden-5 instance requiring 7,685 qubits in HOBO versus over 200,000 in QUBO. The framework is intended to ingest any encoding and hardware profile to highlight potential for quantum devices to challenge classical heuristics.

Significance. If the closed-form counts hold after accounting for practical implementation details, the framework could serve as a useful benchmark for evaluating quantum approaches to CVRP and other routing problems. The large gap identified between QUBO and HOBO encodings underscores the importance of encoding choice, and the decision diagram provides a clear visual tool for determining feasibility. This could guide experimental efforts toward promising instances and encodings.

major comments (1)

The feasibility analysis is based on closed-form qubit and gate counts that do not include additional resources required for transpilation to hardware with limited connectivity, error mitigation (such as zero-noise extrapolation or probabilistic error cancellation), or classical pre- and post-processing. These omissions mean the reported thresholds represent an optimistic lower bound, potentially shifting the quantum feasibility point and the position of instances like Golden-5 on the decision diagram.

minor comments (2)

It would be beneficial to provide the explicit derivation steps for the closed-form resource counts in a dedicated section or appendix to allow verification.
The manuscript should clarify how the device benchmarks are selected and whether sensitivity analysis was performed on the error rates and coherence times used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We have carefully reviewed the major comment and provide a point-by-point response below. Where the comment identifies a genuine limitation in the current presentation, we have revised the manuscript to incorporate additional discussion and caveats.

read point-by-point responses

Referee: The feasibility analysis is based on closed-form qubit and gate counts that do not include additional resources required for transpilation to hardware with limited connectivity, error mitigation (such as zero-noise extrapolation or probabilistic error cancellation), or classical pre- and post-processing. These omissions mean the reported thresholds represent an optimistic lower bound, potentially shifting the quantum feasibility point and the position of instances like Golden-5 on the decision diagram.

Authors: We agree that the closed-form counts constitute optimistic lower bounds. The framework was deliberately designed to isolate the resource requirements arising from the encoding itself, enabling direct, transparent comparisons between QUBO and HOBO mappings without conflating them with hardware-specific overheads. This baseline approach highlights the substantial gap between encodings (e.g., 7,685 vs. >200,000 qubits for Golden-5) and shows that even the most qubit-efficient direct encoding already exceeds near-term device capabilities. We acknowledge that transpilation, error mitigation, and classical processing will add further overheads that can only increase the required resources. In the revised manuscript we have added a dedicated paragraph in Section 4 (Discussion) that explicitly states these omissions, references typical overhead factors from the literature, and notes that such factors would move the quantum feasibility point and the position of Golden-5 farther into the “no-go” region of the decision diagram. We also clarify that the framework is extensible and can ingest refined resource models that include these overheads once they become available. revision: yes

Circularity Check

0 steps flagged

No significant circularity in resource-count derivation

full rationale

The paper derives closed-form qubit and gate counts directly from standard QUBO and HOBO mappings applied to CVRP instances, then combines these with independently published device benchmarks to locate instances on a decision diagram. No equation reduces the feasibility thresholds or go/no-go figures to a parameter fitted inside the paper, and no load-bearing step relies on a self-citation whose validity is assumed rather than externally verified. The framework remains self-contained against external data sources.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard assumptions from quantum resource estimation literature and published device benchmarks; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Published device benchmarks for error rates and coherence times are representative of the hardware that would run the mapped CVRP circuits.
The three feasibility lines are constructed by combining closed-form counts with these external benchmarks.

pith-pipeline@v0.9.0 · 5746 in / 1369 out tokens · 32030 ms · 2026-05-21T22:18:15.756462+00:00 · methodology

discussion (0)

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Optimal, Qubit-Efficient Quantum Vehicle Routing via Colored-Permutations
quant-ph 2026-04 unverdicted novelty 7.0

A qubit-efficient colored-permutation encoding for CVRP enables Constraint-Enhanced QAOA to recover verified optimal solutions on benchmarks without additional capacity qubits.
Qubit-Scalable CVRP via Lagrangian Knapsack Decomposition and Noise-Aware Quantum Execution
quant-ph 2026-04 unverdicted novelty 6.0

A hybrid quantum framework decomposes CVRP into bounded-width knapsack subproblems, trains a reinforcement learning controller for Lagrangian multipliers, and uses a contextual bandit to adapt quantum hardware executi...
Quantum optimisation in cities: Limitations and prospects of urban transport systems
math.OC 2026-04 accept novelty 2.0

Quantum methods lack proven benefits for full-scale urban transport optimization and are best suited to small combinatorial subproblems within hybrid classical-quantum frameworks.

Reference graph

Works this paper leans on

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work page doi:10.1007/s11128-020-02692-8 2020

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Benchmarking gate-based quantum computers,

K. Michielsen, M. Nocon, D. Willsch, F. Jin, T. Lippert, and H. De Raedt, “Benchmarking gate-based quantum computers,”Computer Physics Communica- tions 220, 44–55 (2017). DOI: 10.1016/j.cpc.2017.06.011

work page doi:10.1016/j.cpc.2017.06.011 2017

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Error Mitigation for Short-Depth Quantum Circuits,

K. Temme, S. Bravyi, and J. M. Gambetta, “Error Mitigation for Short-Depth Quantum Circuits,”Physical Review Letters, vol. 119, 180509, 2017

work page 2017

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QOPTLib: A Quantum Computing Oriented Benchmark for Combinatorial Optimization Problems,

E. Osaba and E. Villar-Rodriguez, “QOPTLib: A Quantum Computing Oriented Benchmark for Combinatorial Optimization Problems,” inBenchmarks and Hybrid Algorithms in Optimization and Applications, SpringerNature Singapore, 2023,pp. 49–63

work page 2023

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Hybrid Quan- tum Solvers in Production: how to succeed in the NISQ era?,

E. Osaba, E. Villar-Rodriguez, A. Gomez-Tejedor, and I. Oregi, “Hybrid Quan- tum Solvers in Production: how to succeed in the NISQ era?,”arXiv preprint arXiv:2401.10302, 2024

work page arXiv 2024

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Noise tailoring for scalable quantum computation via randomized compiling,

J. J. Wallman and J. Emerson, “Noise tailoring for scalable quantum computation via randomized compiling,”Phys. Rev. A , 94:052325, 2016. 10 C. Onah and K. Michielsen A Appendix A.1 Opportunities for Quantum Utility in CVRP The high-value nature of CVRP indicates an immense potential for quantum utility but what does it take to realize this? It is already...

work page 2016