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REVIEW 4 major objections 5 minor 1 cited by

Solving Large-Scale Vehicle Routing Problems with Hybrid Quantum-Classical Decomposition

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A two-level quantum-classical decomposition solves a 13-node, 5-vehicle routing problem—normally 156 qubits—with subcircuits of at most 6 qubits and depth 25, at costs comparable to classical heuristics.

desk verdict Resource reductions check out, but Eq. 9 omits MTZ subtour constraints, so QAOA solves a cycle cover rather than the declared VRP; the abstract's feasibility claim fails. read the letter →

arxiv 2507.05373 v1 pith:RPHRZCPI submitted 2025-07-07 quant-ph

Andrew Maciejunes , John Stenger , Dan Gunlycke , Nikos Chrisochoides This is my paper · ORCID
classification quant-ph MSC 68Q1281P6890C27 PACS 03.67.-a03.67.Ac
keywords vehicleroutingproblemquantumapproximateoptimizationalgorithmcircuitcuttingknittinggraphpartitioningtravelingsalesmanQUBOencodingamplitude
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that large Vehicle Routing Problems, which need too many qubits for near-term quantum computers in their standard encoding, can be solved by cutting them into smaller pieces at two levels. First, a graph partitioner splits the VRP into independent single-vehicle Traveling Salesman Problems; second, each TSP's QAOA circuit is cut into subcircuits whose expectation values are stitched back together classically. For a 13-node, 5-vehicle instance, the original 156-qubit, depth-534, 3432-two-qubit-gate circuit is reduced to subcircuits of at most 6 qubits, depth 25, and 16 two-qubit gates. The authors report that 5 of 6 tested TSP partitions matched brute-force optima, and that total route costs beat Google OR-Tools on 8- and 9-node instances while losing on their 7-node case, a failure they attribute to partitioning. If the claim holds, current small QPUs can approach VRPs previously out of reach for quantum simulators, which the authors call early evidence of quantum utility.

What carries the argument

The central mechanism is the two-level decomposition itself. Problem-Level Decomposition removes the depot node, uses the METIS graph partitioner to divide the remaining nodes into K connected subgraphs (one per vehicle), and reattaches the depot to each subgraph, turning the VRP into K independent TSPs. Circuit-Level Decomposition then takes each TSP's QAOA circuit, maps it through Qiskit's circuit-cutting add-on based on the CutQC method, splits it into fragments, runs each fragment separately, and reconstructs the expectation value of the cost Hamiltonian by classical knitting. The paper also contrasts two encodings: the standard QUBO edge encoding, where each edge is one qubit, and a newer amplitude encoding, where edges are basis states and far fewer qubits are needed but circuit depth is much higher.

What would settle it

Take a small VRP with a known optimal solution, run the PLD pipeline with the same K, and ask whether the optimal route's node assignment appears in any METIS partition; if the best summed TSP cost over those partitions is measurably above the true optimum, the decomposition's central feasibility claim fails.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a hybrid two-level decomposition makes a 13-node VRP with five vehicles—encoded with one qubit per edge, 156 qubits total—executable as a set of much smaller quantum circuits without sacrificing solution quality versus classical heuristics. Combining METIS problem-level partitioning with circuit-level cutting of a QAOA ansatz, the largest required subcircuit has 6 qubits, 25 layers of depth, and 16 two-qubit gates; this is a 96% cut in qubits, 95% in depth, and 99.5% in two-qubit gates relative to the full circuit. Five of the six small TSP instances produced by partitioning matched the brute-force optimum, and for the 8- and 9-node VRPs the summed decomposed cost was lower than Google OR-Tools' solution, while the 7-node case favored OR-Tools. The authors present this as evidence that VRPs previously too complex for quantum simulators are now within reach, with the caveat that the amplitude encoding, despite needing fewer qubits, could not be knitted back together with current tools.

Load-bearing premise

The load-bearing premise is that splitting the full VRP graph into independent single-vehicle TSPs with METIS keeps at least one high-quality route intact; the authors explicitly say it is not obvious how to partition without discarding good solutions, point to the 7-node case as a partition-related failure, and defer route-level results for the 13-node instance to a later draft.

Editorial extensions

If this is right

  • A 13-node, five-vehicle VRP that needs 156 qubits and a depth-534 circuit can be run as subcircuits of 6 qubits, depth 25, and 16 two-qubit gates, so any current small QPU or simulator can attempt instances previously blocked by width.
  • If the partition preserves good routes, the same recipe applies to any VRP variant expressible as a QUBO: decompose by vehicle, solve each TSP with QAOA, cut the circuits, and sum the TSP costs.
  • The reported 5-of-6 TSP partitions matching brute force suggests decomposition does not always destroy solution quality on small instances; on 8- and 9-node VRPs the hybrid solution beat OR-Tools, so the method can be competitive with classical heuristics.
  • Because the amplitude encoding cuts qubits further (10-node VRP partitions to 4-5 qubits before cutting), progress in circuit-knitting overhead and depth cutting would directly extend this approach to qubit-frugal encodings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's own 7-node counterexample and its stated uncertainty about METIS partitioning imply that the method's practical reach will be set by the partitioner; replacing minimum-edge-cut with a cost- and hardware-aware partitioning criterion is the natural next step.
  • Because the paper had to drop QAOA from ten layers to one to keep the knitting overhead manageable, the resource-reduction numbers and the utility claim come from shallow circuits; deeper ansätze would require the circuit-knitting overhead reductions the authors cite as ongoing work.
  • The manuscript explicitly defers route-level results for the 13-node partitions to a later draft, so the headline 'too complex for simulators' claim at that size rests on resource counts plus a statement that the optimal route appeared rarely among 100,000 shots; readers should look for the completed route table before treating the 13-node demonstration as fully shown.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a two-level decomposition for solving Vehicle Routing Problems (VRP) with QAOA. A problem-level decomposition (PLD) uses METIS to partition a VRP into several single-vehicle TSPs, and a circuit-level decomposition (CLD) uses circuit cutting/knitting to reduce each TSP circuit to smaller subcircuits. The authors report large reductions in qubit count, circuit depth, and two-qubit-gate count for instances up to a 13-node, 5-vehicle VRP (from 156 qubits, depth 534, and 3432 CX gates to at most 6 qubits, depth 25, and 16 CX gates). They also compare solution costs with Google OR-Tools for n=7, 8, and 9 instances and describe an amplitude encoding that could not be fully executed due to cutting overhead. The abstract claims that these results demonstrate feasibility of solving previously intractable VRPs and provide early evidence of quantum utility.

Significance. If the resource-reduction claims are correct, the paper provides a concrete, directly checkable example of circuit cutting: Tables III and IV support the 96% qubit, 95% depth, and 99.5% two-qubit-gate reductions by simple arithmetic. The use of brute-force verification for small TSP partitions and the comparison with an external classical tool are appropriate methodological choices. However, the central feasibility claim is not supported by the reported experimental results. The submitted manuscript explicitly defers the 13-node solution-cost results, the amplitude-encoding results are declared incomplete, and the QUBO Hamiltonian in Eq. (9) does not enforce the MTZ subtour-elimination constraints that are needed for the problem to be a TSP/VRP rather than a cycle-cover problem. Consequently, the paper currently demonstrates that large resource reductions are possible on paper, but not that the decomposed quantum routine solves the declared routing problem or that its solutions are competitive. The significance of the paper as a feasibility demonstration therefore remains unestablished until these load-bearing gaps are addressed.

major comments (4)
  1. [Section III-A and Conclusion] The Hamiltonian Hc in Eq. (9) is claimed to derive from the MTZ formulation in Section II-A, but Eqs. (4)-(7) enforce only in/out degree constraints and depot degree K. They do not include the MTZ subtour-elimination inequalities (e.g., u_i - u_j + n x_ij <= n-1 for i,j >= 1). As a result, the ground state of Hc can be a collection of disjoint cycles rather than a single Hamiltonian cycle. Since QAOA minimizes Hc, the reported 'optimal' routes need not be valid TSP/VRP solutions, and the solution-cost comparisons in Fig. 4 do not establish that the quantum routine solves the declared problem. This issue is load-bearing and must be fixed by adding proper subtour-elimination penalties, or by changing the encoding, and then rerunning the experiments.
  2. [Section II-B and IV] The paper's headline claim concerns a 13-node, 5-vehicle VRP, but the only 13-node results presented are the resource counts in Tables III and IV. The text states, 'We plan to include results for the 13 node partitions in our final draft,' which means the central feasibility claim—that this VRP can be solved with the proposed hybrid method—is not supported by any submitted result. Similarly, Section III-B reports that for the amplitude encoding the expectation value could not be reconstructed, so the abstract's statement that the method is demonstrated on amplitude encoding is not supported. The authors should either provide these missing results or substantially narrow the claims.
  3. [Section II-E and III-A] The validity of PLD rests on the assumption that partitioning the VRP into independent TSPs preserves enough of the optimal solution space, but this is not established. The authors acknowledge in Section IV that 'It is not obvious how to most efficiently partition the graph in a way that does not eliminate possible high quality solutions,' and Section III-A attributes the n=7 failure to graph partitioning. Without an analysis of the cost gap introduced by METIS partitions, or at least a systematic empirical study over multiple instances, the claim that the decomposition yields 'satisfactory results' is unsupported. The n=8 and n=9 cases where the quantum/PLD solution beats OR-Tools may reflect the partition chosen rather than the quantum optimization, and this ambiguity needs to be resolved.
  4. [Section II-E and III-A] The circuit-cutting pipeline reconstructs only the expectation value of the cost operator, not a solution eigenstate, as stated in Section II-E. The paper then presents solution costs and route frequencies in Section III-A. An expectation value over a reconstructed density matrix does not necessarily correspond to any feasible VRP route, especially when the cost operator is the incomplete QUBO from Eq. (9). The reported observation that valid paths appeared only 'a handful of times' in 100,000 shots further suggests the output distribution is not concentrated on valid routes. The authors should clarify exactly how the costs in Fig. 4 were computed and provide evidence that the reconstructed quantities correspond to feasible route costs.
minor comments (5)
  1. [Table V caption] The product notation 'pY p=0' is confusing and the index/appearance of p is not standard; the expression should be rewritten as a product over the QAOA layers, e.g., U(γ,β) = Π_{i=1}^{p} e^{-iγ_i H_c} e^{-iβ_i H_d} followed by the initial Hadamard layer.
  2. [Conclusion] The caption says '10.2 is the original seven node, two vehicle VRP' but the instance is a ten-node VRP; the text should say ten-node.
  3. [Section II-A] The conclusion states that the 13-node instance is reduced to circuits with 'a depth of only 55 gates,' but Tables III and IV report maximum depths of 18/45 before cutting and 10/25 after cutting for the 13-node partitions; the number 55 appears to be a typo and should be reconciled.
  4. [Section III-A] Equation (2) sums over i,j ∈ {1,...,n} but the depot is node 0 and later equations use 0; the indexing should be made consistent throughout the formulation.
  5. [Section III-A] The experimental setup for OR-Tools is not described: no time limit, solver parameters, or termination criteria are given, so the comparison in Fig. 4 is not reproducible. Please provide this information.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; central claims rest on external baselines and direct resource accounting.

full rationale

The paper's derivation chain is self-contained and non-circular. The resource reductions in Tables III-VI are direct arithmetic comparisons of qubit counts, circuit depths, and two-qubit gate counts before and after problem-level and circuit-level decomposition; they are not outputs of fitted parameters or predictions derived from the model itself. Solution-quality results are benchmarked against an external classical solver, Google OR-Tools, and individual TSP instances are checked by brute-force permutation enumeration, so the success criterion does not reduce to the paper's own construction. The QAOA objective in Eq. 9 is formed from the VRP cost and constraints in Eqs. 2-7, and the ansatz in Eq. 12 is the standard QAOA form, meaning the optimization target is not defined in terms of the reported outcome. The only citation with author overlap is [35] for the amplitude encoding; that encoding is presented as a secondary proof-of-concept, the paper explicitly states it could not reconstruct the expectation value for that encoding, and it is not used as evidence for the central 13-node QUBO claim. The acknowledged limitations, such as METIS potentially eliminating high-quality partitions and the deferred 13-node cost comparison, are correctness and validation risks rather than circularity: they do not make any claimed output equal to an input by construction. The missing MTZ subtour-elimination constraints in Eq. 9 are a substantive correctness concern about whether the quantum objective matches the declared TSP/VRP, but this is not a self-referential reduction. Overall, no load-bearing step reduces to a self-citation, a fitted parameter renamed as a prediction, or an ansatz smuggled in via citation.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper's central feasibility claim rests on three unverified domain assumptions: the partition preserves optimal VRP structure, circuit cutting reconstructs a faithful expectation value, and noiseless simulation represents hardware behavior. The free parameters are the penalty lambda and QAOA variational angles and layer count, none of which are reported with sensitivity analysis. No new physical entities are introduced.

free parameters (3)
  • penalty strength lambda in QUBO Hamiltonian (Eq. 9)
    Set by the researcher to enforce constraints; no sensitivity analysis or chosen value is reported. The QAOA solution quality depends directly on this parameter.
  • QAOA variational angles gamma and beta = not reported
    Optimized by COBYLA for each instance; no values, convergence criteria, or number of function evaluations are given. These determine whether the measured expectation value corresponds to the optimal route.
  • QAOA layer count p = 10 or 1
    The layer count is reduced from 10 to 1 to make circuit cutting feasible, with acknowledged degradation in solution quality. The choice is instance-dependent and not systematically justified.
assumptions (3)
  • domain assumption The optimal VRP solution can be approximated by independently solving the TSP partitions produced by METIS, and the partition does not eliminate high-quality routes.
    Central to PLD; Section II-B and the Discussion admit this is unresolved and caused the n=7 failure.
  • domain assumption Circuit cutting and knitting reconstruct the expectation value of the cost Hamiltonian with sufficient fidelity for QAOA optimization.
    Invoked in Section II-E; no reconstruction error is quantified. Qiskit Addon Cutting recovers expectation values, not eigenstates, so the link to valid routes is assumed.
  • domain assumption The noiseless Aer simulator results transfer to the behavior of a real QPU relevant for quantum utility.
    All executions are on a noiseless simulator (Section II-D), but the conclusion speaks to near-term QPU feasibility. Noise effects are not modeled.

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Cite this review

Pith. "Pith review of Solving Large-Scale Vehicle Routing Problems with Hybrid Quantum-Classical Decomposition." pith.science (2026). https://pith.science/paper/RPHRZCPI

@misc{pith2026250705373,
  author       = {Pith},
  title        = {Pith review of: Solving Large-Scale Vehicle Routing Problems with Hybrid Quantum-Classical Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPHRZCPI}},
  note         = {Machine review of arXiv:2507.05373}
}
read the original abstract

We present a two-level decomposition strategy for solving the Vehicle Routing Problem (VRP) using the Quantum Approximate Optimization Algorithm. A Problem-Level Decomposition partitions a 13-node (156-qubit) VRP into smaller Traveling Salesman Problem (TSP) instances. Each TSP is then further cut via Circuit-Level Decomposition, enabling execution on near-term quantum devices. Our approach achieves up to 95\% reductions in the circuit depth, 96\% reduction in the number of qubits and a 99.5\% reduction in the number of 2-qubit gates. We demonstrate this hybrid algorithm on the standard edge encoding of the VRP as well as a novel amplitude encoding. These results demonstrate the feasibility of solving VRPs previously too complex for quantum simulators and provide early evidence of potential quantum utility.

Figures

Figures reproduced from arXiv: 2507.05373 by the authors.

Figure 1
Figure 1. High-level overview of two-level decomposition [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) shows a VRP instance of n = 4 and K = 2. Vehicle 1 (purple) follows the path 0 → 1 → 0. Vehicle 2 (red) follows the path 0 → 2 → 3 → 0. Node 0, highlighted green, represents the depot. It should be noted that this path is equivalent to 0 → 3 → 2 → 0 due to the symmetry of the problem. (b) illustrates a VRP instance of n = 4 and K = 2. The solution violates both constraints. Vehicle 1 (purple) does not start and … view at source ↗
Figure 3
Figure 3. Graph decomposition for a 7-node VRP instance with 2 vehicles. (a) The full graph with 7 cities, where the green node [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: For the larger 8, 9, and 13 node graphs, the classical [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: This graph only shows the problem level decomposition [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Forward citations

Cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.