Complex Logistics Problems Now Solvable with Fewer Quantum Bits

A new method tackles increasingly complex industrial logistics and scheduling challenges. Juan F. R. Hernandez at Kipu Quantum GmbH and Volkswagen AG, and colleagues investigate higher-order unconstrained binary optimisation (HUBO) formulations and their suitability for quantum computing workflows. HUBO can represent intricate industrial processes, such as correlated assembly-line scheduling, more effectively than standard quadratic unconstrained binary optimisation (QUBO) methods, while simultaneously reducing the number of qubits needed. Although HUBO introduces more complex quantum circuits, the team’s analysis reveals a key trade-off between qubit reduction and circuit depth, suggesting hybrid quantum-classical algorithms and the development of fault-tolerant quantum hardware are vital for realising the practical benefits of HUBO in large-scale industrial applications.

Higher-order unary Boolean optimisation enables logarithmic qubit scaling for vehicle routing and

HUBO formulations achieved a 29 May 2026 reduction in required qubits for complex industrial problems, scaling from linear with QUBO to logarithmic. This feat was previously unattainable due to the limitations of representing higher-order interactions on existing quantum hardware. Traditionally, industrial optimisation problems are mapped onto QUBO formulations, which, while compatible with early quantum annealers and gate-based quantum computers, suffer from a significant qubit overhead. This overhead arises because complex constraints and relationships within the problem necessitate many binary variables, each requiring a physical qubit. The logarithmic scaling offered by HUBO represents a substantial improvement, particularly for large-scale instances of the capacitated vehicle routing problem, as qubit demand represents a critical bottleneck for quantum computation. The capacitated vehicle routing problem, a cornerstone of logistics optimisation, involves determining the optimal routes for a fleet of vehicles to serve a set of customers with varying demands, subject to vehicle capacity constraints. Its complexity grows rapidly with the number of customers and vehicles, making it a prime candidate for quantum acceleration.

Implementing these HUBO models introduces increased circuit depth, yet a clear trade-off exists between qubit reduction and circuit complexity, suggesting hybrid quantum-classical algorithms and fault-tolerant quantum computers are essential to unlock their full potential. The increased circuit depth stems from the need to implement higher-order terms, interactions involving three or more qubits, which require more quantum gates and therefore more time to execute. However, the reduction in qubit count can offset this increased complexity, particularly when considering the limitations of current quantum hardware. Hybrid quantum-classical algorithms, such as the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimisation Algorithm (QAOA), offer a promising approach to leverage the strengths of both quantum and classical computation. These algorithms typically use a classical optimiser to adjust parameters in a quantum circuit, iteratively improving the solution quality. Classical solvers validated the new formulations across multiple problem instances, confirming their viability as optimisation tools. Small routing problems were benchmarked using bias-field digitised counterdiabatic quantum optimisation to assess performance and explore the impact of reduced qubit demand on solution quality. Bias-field digitised counterdiabatic quantum optimisation is a specific quantum optimisation technique designed to mitigate the effects of noise and improve the accuracy of solutions.

The analysis of the capacitated vehicle routing problem revealed this reduction extends to large-scale instances, but is accompanied by increased circuit depth, meaning more operations are needed to run the quantum algorithm. This remains a significant hurdle given current limitations in gate fidelity and qubit coherence. Gate fidelity refers to the accuracy of quantum gates, while qubit coherence describes how long a qubit can maintain its quantum state before decoherence occurs. Both factors are crucial for performing reliable quantum computations. A relentless drive to optimise complex industrial processes, from logistics networks to factory schedules, fuels a search for computational methods beyond the reach of classical computers. Classical algorithms often struggle with the combinatorial explosion that occurs when dealing with many variables and constraints, leading to long computation times or suboptimal solutions. Quantum computing offers the potential to overcome these limitations by exploiting quantum phenomena such as superposition and entanglement.

Hernandez and colleagues demonstrate the potential of higher-order unconstrained binary optimisation (HUBO) as a pathway, showcasing its ability to represent intricate relationships often lost when simplifying problems for conventional quantum algorithms. The ability to accurately model complex correlations between variables is crucial for achieving optimal solutions in many industrial applications. Current quantum computers lack the scale and stability for immediate impact on these complex problems, but this work proactively addresses how future machines might handle industrial optimisation tasks differently. By reducing the number of qubits needed compared to standard quadratic unconstrained binary optimisation, HUBO addresses a key limitation hindering the scalability of quantum computing for real-world applications. The reduction in qubit count translates to a reduction in the physical resources required to implement the quantum algorithm, making it more feasible to tackle larger and more complex problems. Overcoming challenges related to circuit depth and the limitations of existing quantum hardware, however, requires a focus on hybrid quantum-classical algorithms and the development of more stable quantum systems, alongside continued research into optimising circuit design for improved gate fidelity. Further research will focus on developing error mitigation techniques and exploring novel quantum circuit architectures to reduce circuit depth and improve the overall performance of HUBO-based quantum algorithms for industrial optimisation.

The researchers demonstrated that higher-order unconstrained binary optimisation, or HUBO, can represent complex industrial problems, including logistics and manufacturing schedules, with fewer quantum bits than standard methods. This is significant because reducing the number of qubits needed is a key step towards applying quantum computing to real-world challenges. While HUBO introduces more complex calculations, the team validated the formulations using classical solvers and simulated quantum optimisation on small routing problems. They suggest future work will concentrate on minimising computational complexity and improving the stability of quantum systems to enable practical application of HUBO.