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arxiv: 2605.30252 · v1 · pith:QW6EF4B3new · submitted 2026-05-28 · 🪐 quant-ph

Quantum optimization beyond QUBO for industrial logistics and scheduling

Juan F. R. Hernandez , Pavle Nikacevic , Enrique Solano , Chinonso Onah , Agneev Guin , Arne-Christian Voigt , Archismita Dalal This is my paper

Pith reviewed 2026-06-29 06:56 UTC · model grok-4.3

classification 🪐 quant-ph
keywords HUBOQUBOquantum optimizationvehicle routingschedulinglogisticscircuit depthfault-tolerant quantum computing
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0 comments X

The pith

HUBO formulations for logistics and scheduling use fewer qubits than QUBO encodings but add higher-order terms that increase circuit depth.

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

The paper investigates higher-order unconstrained binary optimization as a way to model industrial logistics and scheduling problems for quantum computers. It argues that these formulations represent complex constraints, such as correlated assembly rules, more compactly than quadratic forms, cutting the number of binary variables and thus the qubits required. At the same time the higher-order terms deepen the circuits needed for quantum optimization routines. Validation on classical solvers and small simulated instances supports the formulations, while a scaling analysis for vehicle routing shows the qubit advantage alongside depth constraints. The work concludes that hybrid quantum-classical methods or early fault-tolerant hardware offer the most realistic setting for these encodings.

Core claim

Formulating representative logistics and manufacturing problems as HUBO instances captures process intricacies difficult to express faithfully with QUBO while reducing the number of binary variables required in the quantum mapping, thereby lowering qubit demand, although the added higher-order interaction terms increase circuit depth and limit feasibility on current hardware.

What carries the argument

Higher-order unconstrained binary optimization (HUBO) formulations of logistics and scheduling problems, which encode interactions beyond quadratic order.

If this is right

  • HUBO encodings require fewer qubits than QUBO for the same logistics and scheduling instances.
  • Higher-order terms increase the depth of circuits used in bias-field digitized counterdiabatic quantum optimization.
  • Classical solvers confirm that the HUBO formulations correctly capture the original problem constraints.
  • Resource analysis for the capacitated vehicle routing problem shows improved qubit scaling relative to QUBO.
  • Practical deployment is constrained to hybrid quantum-classical workflows and early fault-tolerant regimes.

Where Pith is reading between the lines

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

  • The qubit-depth trade-off may shift in favor of HUBO once qubit counts become the dominant hardware limit rather than gate error rates.
  • Similar compact encodings could apply to other combinatorial problems where variable reduction outweighs interaction order.
  • Direct hardware benchmarks on mid-scale routing instances would test whether the depth penalty is offset by the qubit savings in practice.

Load-bearing premise

The added circuit depth from higher-order terms stays manageable inside hybrid quantum-classical workflows or early fault-tolerant hardware without violating gate fidelity and coherence limits.

What would settle it

A concrete measurement, on a fixed routing instance, showing that the total gate count required to implement the HUBO mapping exceeds available coherence time while the corresponding QUBO mapping remains within that time.

Figures

Figures reproduced from arXiv: 2605.30252 by Agneev Guin, Archismita Dalal, Arne-Christian Voigt, Chinonso Onah, Enrique Solano, Juan F. R. Hernandez, Pavle Nikacevic.

Figure 1
Figure 1. Figure 1: FIG. 1. High-level QUEST workflow from the set of breakers and surfers, through HUBO encoding and iterative BF-DCQO evolution, to [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Two-qubit gate operations needed to encode QUEST circuits [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Approximation ratios for simulated annealing quadratized [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Transpiled two-qubit gate counts for a subset of CVRP in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Estimated BF-DCQO quantum runtime versus qubit number [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Circuit fidelity of HUBO-based DCQO circuits versus qubit [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of constraint violations across all scheduling instances for di [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

The increasing complexity of industrial scheduling and transport routing problems motivates the study of alternative optimization formulations and computational paradigms. In this work, we study how higher-order unconstrained binary optimization (HUBO) formulations of such problems map onto quantum optimization workflows in both noisy and fault-tolerant regimes. We consider three representative logistics and manufacturing use cases and formulate each as a HUBO problem. This captures process intricacies, such as highly correlated assembly-line scheduling rules, which are difficult to express faithfully with the standard quadratic (QUBO) form, while at the same time reducing the number of binary variables required in the quantum mapping, thus lowering qubit demand. We compare the HUBO formulations with corresponding QUBO encodings, highlighting a key trade-off: while HUBO reduces qubit requirements through compact binary encoding, it introduces higher-order interaction terms that increase circuit depth, limiting feasibility on current quantum hardware. The proposed formulations are validated using classical solvers across several problem instances and benchmark small routing problem instances using bias-field digitized counterdiabatic quantum optimization in classical simulation. We complement these results with a resource and scalability analysis, focusing on the capacitated vehicle routing problem as a representative large-scale industrial use case. Our analysis indicates that while HUBO formulations offer advantages in qubit scaling compared to QUBO encodings, their practical implementation is constrained by gate fidelity, coherence, and circuit depth, making hybrid quantum-classical workflows and early fault-tolerant quantum hardware the most plausible settings for their practical use.

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 / 1 minor

Summary. The paper claims that HUBO formulations for three logistics and scheduling problems in industry can capture complex constraints better than QUBO while reducing the number of binary variables needed for quantum mapping, thereby lowering qubit requirements, although at the expense of higher circuit depth due to higher-order terms. The formulations are validated classically, small instances are benchmarked using bias-field digitized counterdiabatic quantum optimization in simulation, and a resource analysis is provided for the capacitated vehicle routing problem, suggesting applicability in hybrid quantum-classical or early fault-tolerant regimes.

Significance. If the quantitative aspects of the qubit reduction and depth increase hold as claimed, this work offers a concrete demonstration of the qubit-depth trade-off in mapping real-world industrial optimization problems to quantum computers. The classical validation and simulation benchmarks add credibility, and the focus on practical use cases like CVRP highlights the potential for HUBO in scenarios where variable reduction is critical.

major comments (2)
  1. Abstract: The central claim of reduced qubit demand via compact binary encoding in HUBO is presented without any specific numerical comparisons to QUBO for the three use cases or the CVRP analysis, which is load-bearing for assessing the practical advantage.
  2. Benchmarking and simulation: The results from classical simulation of the quantum algorithm for small routing problem instances are mentioned but without reported metrics such as success probability, energy landscapes, or comparison to other methods, undermining the ability to verify the feasibility claims.
minor comments (1)
  1. Abstract: The three representative logistics and manufacturing use cases are not named or described in the abstract, making it hard to contextualize the formulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comments point by point below, agreeing where revisions are warranted to improve clarity and verifiability.

read point-by-point responses

  1. Referee: Abstract: The central claim of reduced qubit demand via compact binary encoding in HUBO is presented without any specific numerical comparisons to QUBO for the three use cases or the CVRP analysis, which is load-bearing for assessing the practical advantage.

    Authors: We agree that the abstract would be strengthened by including specific numerical comparisons. The body of the manuscript provides detailed qubit-count comparisons between the HUBO and QUBO formulations for the three logistics and scheduling use cases, as well as the resource analysis for CVRP. We will revise the abstract to incorporate representative numerical examples of qubit reductions drawn from these sections. revision: yes

  2. Referee: Benchmarking and simulation: The results from classical simulation of the quantum algorithm for small routing problem instances are mentioned but without reported metrics such as success probability, energy landscapes, or comparison to other methods, undermining the ability to verify the feasibility claims.

    Authors: The referee correctly identifies that the simulation section mentions the benchmarking of small instances with bias-field digitized counterdiabatic quantum optimization but does not report quantitative metrics such as success probabilities, energy landscapes, or comparisons to alternative methods. We will add these specific results and comparisons to the revised manuscript to enable verification of the feasibility claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript defines HUBO and QUBO encodings as explicit mappings from problem constraints (assembly rules, routing capacities, etc.) to polynomial objective functions. These mappings are stated directly in the text, validated by classical solvers on concrete instances, and compared via explicit metrics (binary variable count, interaction order, estimated circuit depth). No parameter is fitted to a data subset and then relabeled as a prediction; no uniqueness theorem or ansatz is imported via self-citation as the sole justification for the central claim; the qubit-vs-depth trade-off is derived from the stated encodings themselves rather than from any self-referential reduction. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard quantum computing assumptions about hardware regimes and classical solver capabilities for validation; no new free parameters, axioms beyond domain standards, or invented entities are introduced.

axioms (1)
  • domain assumption Standard assumptions of quantum computing hardware performance in noisy and fault-tolerant regimes, including gate fidelity and coherence limits.
    Invoked when discussing feasibility constraints on circuit depth without further derivation.

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