Quantum Computing for Optimizing Aircraft Loading

TL;DR

This work tackles aircraft loading optimization, a constrained combinatorial problem with NP-hard characteristics, by developing MAL-VQA, a shallow, multi-angle quantum algorithm suited for near-term ion-trap QPUs. The method encodes the problem into $MN$ qubits and uses a CVaR-based cost with penalties to handle hard and soft constraints without slack qubits, while offloading constraint evaluation to a classical processor. Demonstrations on IonQ Aria and Forte hardware show optimal or near-optimal solutions for instances up to 28 qubits, with robust performance against initial guesses and varying problem constraints, highlighting potential scalability as quantum hardware improves. The approach offers a practical pathway for quantum-assisted optimization in aviation logistics and similar knapsack-like problems, combining hardware-aware circuit design with advanced cost-function strategies.

Abstract

The aircraft loading optimization problem is a computationally hard problem with the best known classical algorithm scaling exponentially with the number of objects. We propose a quantum approach based on a multi-angle variant of the QAOA algorithm (Multi-Angle Layered Variational Quantum Algorithm (MAL-VQA)) designed to utilize a smaller number of two qubit gates in the quantum circuit as compared to the standard QAOA algorithm so that the quantum optimization algorithm can be run on near-term ion-trap quantum processing units (QPU). We also describe a novel cost function implementation that can handle many different types of inequality constraints without the overhead of introducing slack variables in the quantum circuit so that larger problems with complex constraints may be represented on near-term QPUs which have low qubit counts. We demonstrate the performance of the algorithm on different instances of the aircraft loading problem by execution on IonQ QPUs Aria and Forte. Our experiments obtain the optimal solutions for all the problem instances studied ranging from 12 qubits to 28 qubits. This shows the potential scalability of the method to significantly larger problem sizes with the improvement of quantum hardware in the near future as well as the robustness of the quantum algorithm against varying initial guesses and varying constraints of different problem instances.

Figures (12)

• Figure 1: The Aircraft Loading Problem: A set of $n$ cargo containers of up to three different sizes is available for loading. Standard size containers (1) occupy a single position, half size containers (2) may share a single position, whereas double size containers (3) occupy two adjacent positions. Each container in this set has an individual mass $m$, which lies in between the empty mass and the maximum mass of each container type. Typically, the combined maximum masses of all containers exceed the aircraft's payload capacity.
• Figure 2: Bipartite graph with nodes representing containers (red vertices) and slots (green vertices). Highlighted edges indicate container-to-slot assignments.
• Figure 3: Architecture of the quantum circuit used in the study. a) The quantum circuit consists of alternating layers of parametrized single qubit rotation $R_Y$ gates and entanglement blocks U. The U blocks consist of parametrized $R_{ZZ}$ gates between different qubits. In this study, only one entanglement block U was used sandwiched between two layers of $R_Y$ gates, b) The arrangement of $R_{ZZ}$ gates within a single block U used in this study shown for a 5 qubit circuit.
• Figure 4: Only 1 edge can be selected (shown in green, left image). Two selected edges from the same container not allowed (shown in green with a red cross, right image)
• Figure 5: Cost function behavior for constraints. The error function multiplied by a penalty is used to implement the constraint. It is approximately linear for small violations of the constraint and gradually saturates at the value of the penalty. The soft constraints are modelled with a softer error function than the hard constraints.