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arxiv: 2605.18393 · v1 · pith:KG22FOXYnew · submitted 2026-05-18 · 🧮 math.OC · quant-ph

Quantum Model for CVRPTW

Imran Meghazi (LIRMM) , \'Eric Bourreau (LIRMM) This is my paper

Pith reviewed 2026-05-20 08:58 UTC · model grok-4.3

classification 🧮 math.OC quant-ph
keywords quantum computingvehicle routingGrover searchCVRPTWqubit efficiencyoptimizationtime windows
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The pith

A split-inspired quantum encoding solves CVRPTW by adding only linear qubits to TSP models.

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

The paper proposes a quantum algorithm for the capacitated vehicle routing problem with time windows that uses Grover search. It starts from the classical route-first cluster-second decomposition but replaces its heuristic steps with quantum superposition to avoid suboptimality. The central technical step is a split-inspired encoding that folds capacity and time-window constraints into standard quantum TSP circuits. This encoding requires only a linear number of additional decision qubits, which directly addresses the qubit scarcity that currently blocks constraint-heavy routing problems on NISQ hardware.

Core claim

The authors establish that a split-inspired modeling approach can incorporate capacity and time-window constraints into quantum TSP formulations while adding only a linear number of decision qubits, thereby allowing Grover search to produce solutions for CVRPTW that are not limited by the suboptimality of classical decomposition.

What carries the argument

The qubit-efficient split-inspired modeling that augments standard quantum TSP representations with capacity and time-window constraints using a linear number of extra decision qubits.

If this is right

  • Grover search becomes directly applicable to CVRPTW instances that include both capacity limits and delivery time windows.
  • The total number of qubits scales linearly with the size of the underlying TSP rather than requiring an exponential increase for the added constraints.
  • Quantum solutions can in principle exceed the quality of solutions obtained from classical route-first cluster-second heuristics.

Where Pith is reading between the lines

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

  • The same linear-qubit encoding pattern could be tested on other constrained routing problems such as pickup-and-delivery variants.
  • Small instances encoded this way could be run on current NISQ devices to measure actual circuit depth and error rates.
  • The approach suggests a general method for lifting classical decomposition ideas into quantum optimization without losing qubit efficiency.

Load-bearing premise

The split-inspired encoding must correctly embed capacity and time-window rules while keeping qubit growth linear and without adding overhead that would erase the advantage of Grover search.

What would settle it

Execute the proposed quantum circuit on a small CVRPTW instance with known optimal solution and verify whether the Grover search returns a feasible route that satisfies all capacity and time-window constraints.

Figures

Figures reproduced from arXiv: 2605.18393 by \'Eric Bourreau (LIRMM), Imran Meghazi (LIRMM).

Figure 1
Figure 1. Figure 1: The steps to compute a solution. 1a the giant tour. 1b the auxiliary graph H. 1c the solution [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit for capacity constraint [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quantum circuit for time constraint 3.2 Complexity Space complexity In this paper, we adopt a position-based encoding of the so￾lution, where customer Pi is interpreted as the i th customer visited in the giant tour. This representation requires n log n qubits to encode the tour sequence. Ad￾ditionally, we introduce n qubits to store the binary variables yi , which represent the splitting decisions. A sign… view at source ↗
Figure 4
Figure 4. Figure 4: Number of qubits relative to the number of customers for the oracle with the maximum capacity and number of time windows both fixed to 8 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

This paper proposes a quantum algorithm for the capacitated vehicle routing problem with time windows (CVRPTW) based on Grover Search framework. This problem is often faced by Postal services in the context of package delivery or other time-sensitive operations. We provide an implementation on gate based quantum computer of a model inspired by classical route first, cluster second technique. The quantum paradigm allows to overcome suboptimality inherent property of this decomposition. In the current NISQ (Noisy Intermediate-Scale Quantum) era, the most important limitation is the number of available qubits which makes time windows and capacity constraints hard to tackle. We introduce a qubit-efficient split-inspired modeling which adds only a linear number of decision qubits to standard quantum formulations for Traveling Salesman Problem (TSP).

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

2 major / 0 minor

Summary. The paper proposes a quantum algorithm for the capacitated vehicle routing problem with time windows (CVRPTW) based on the Grover Search framework. It introduces a qubit-efficient split-inspired modeling approach, inspired by classical route-first cluster-second decomposition techniques, claiming that this overcomes the inherent suboptimality of such decompositions while adding only a linear number of decision qubits beyond standard quantum TSP formulations, addressing qubit limitations in the NISQ era.

Significance. If the split-inspired encoding can be rigorously shown to incorporate capacity and time-window constraints while preserving linear qubit scaling and permitting Grover search without negating overhead, the work would offer a meaningful step toward practical quantum optimization for routing problems faced in logistics and delivery operations.

major comments (2)
  1. [Abstract] Abstract: the central claims that the quantum formulation overcomes suboptimality of the classical decomposition and that the split-inspired modeling adds only a linear number of decision qubits to TSP formulations are unsupported by any explicit qubit-to-variable mapping, constraint Hamiltonian, or scaling derivation.
  2. The manuscript provides no verification that the proposed encoding correctly enforces vehicle capacity limits and per-customer time windows [a_i, b_i] without introducing extra qubits or oracle overhead that would cancel the claimed linear scaling advantage.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and indicate how we will revise the paper to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims that the quantum formulation overcomes suboptimality of the classical decomposition and that the split-inspired modeling adds only a linear number of decision qubits to TSP formulations are unsupported by any explicit qubit-to-variable mapping, constraint Hamiltonian, or scaling derivation.

    Authors: We agree that the abstract states these claims without sufficient supporting derivations in the current text. In the revised manuscript we will add an explicit qubit-to-variable mapping for the split-inspired encoding, present the full constraint Hamiltonian, and include a scaling derivation showing that the additional decision qubits remain linear in the number of customers and vehicles. We will also clarify that the Grover search operates over the space of feasible split routes, which removes the greedy suboptimality of the classical decomposition; a short illustrative argument will be inserted in the introduction. revision: yes

  2. Referee: The manuscript provides no verification that the proposed encoding correctly enforces vehicle capacity limits and per-customer time windows [a_i, b_i] without introducing extra qubits or oracle overhead that would cancel the claimed linear scaling advantage.

    Authors: This observation is correct. We will insert a dedicated subsection that verifies enforcement of capacity and time-window constraints. The verification will show that the split-inspired formulation uses indicator qubits and penalty terms whose count scales linearly with problem size, and that the corresponding oracle can be realized with a circuit depth that remains polynomial in the input size, preserving the linear qubit advantage for the decision variables. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on descriptive modeling without derivational reduction to inputs

full rationale

The manuscript proposes a Grover-search-based quantum algorithm for CVRPTW that adopts a split-inspired encoding adding only linear decision qubits beyond standard TSP formulations. No equations, qubit-to-variable mappings, constraint Hamiltonians, or scaling derivations appear in the provided text; the approach is described as inspired by classical route-first cluster-second decomposition yet asserted to overcome its suboptimality via the quantum paradigm. Because no explicit derivation chain is supplied, no step can be shown to reduce by construction to fitted parameters, self-citations, or renamed inputs. The central modeling claim therefore remains a proposal whose correctness is externally verifiable rather than internally forced, yielding a self-contained presentation with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from quantum computing and classical routing literature; no new free parameters, invented entities, or ad-hoc axioms are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Grover search framework can be directly applied to the CVRPTW decision variables
    Invoked when stating the quantum algorithm for the full constrained problem.
  • domain assumption The split-inspired modeling preserves correctness while adding only linear qubits
    Central to the qubit-efficiency claim but not derived in the abstract.

pith-pipeline@v0.9.0 · 5652 in / 1368 out tokens · 40481 ms · 2026-05-20T08:58:48.904106+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 3 internal anchors

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