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arXiv:1912.10499 · Physical Review Applied

Applying the Quantum Approximate Optimization Algorithm to the Tail-Assignment Problem

Pontus Vikstål, Mattias Grönkvist, Marika Svensson, Martin Andersson, Göran Johansson, Giulia Ferrini

Published Sep 3, 2020• 96 citations
arXivHTMLPDF

Abstract

Airlines today are faced with a number of large-scale scheduling problems. One such problem is the tail-assignment problem, which is the task of assigning individual aircraft to a given set of flights, minimizing the overall cost. Each aircraft is identified by the registration number on its tail fin. In this paper, we simulate the quantum approximate optimization algorithm (QAOA) applied to instances of this problem derived from real-world data. The QAOA is a variational hybrid quantum-classical algorithm recently introduced and likely to run on near-term quantum devices. The instances are reduced to fit on quantum devices with 8, 15, and 25 qubits. The reduction procedure leaves only one feasible solution per instance, which allows us to map the tail-assignment problem onto the exact-cover problem. We find that repeated runs of the QAOA identify the feasible solution with close to unit probability for all instances. Furthermore, we observe patterns in the variational parameters such that an interpolation strategy can be employed, which significantly simplifies the classical optimization part of the QAOA. Finally, we empirically find a relation between the connectivity of the problem graph and the single-shot success probability of the algorithm.

DOI
10.1103/physrevapplied.14.034009

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Airlines today are faced with a number of large-scale scheduling problems. One such problem is the tail-assignment problem, which is the task of assigning individual aircraft to a given set of flights, minimizing the overall cost. Each aircraft is identified by the registration number on its tail fin. In this paper, we simulate the quantum approximate optimization algorithm (QAOA) applied to instances of this problem derived from real-world data. The QAOA is a variational hybrid quantum-classical algorithm recently introduced and likely to run on near-term quantum devices. The instances are reduced to fit on quantum devices with 8, 15, and 25 qubits. The reduction procedure leaves only one feasible solution per instance, which allows us to map the tail-assignment problem onto the exact-cover problem. We find that repeated runs of the QAOA identify the feasible solution with close to unit probability for all instances. Furthermore, we observe patterns in the variational parameters such that an interpolation strategy can be employed, which significantly simplifies the classical optimization part of the QAOA. Finally, we empirically find a relation between the connectivity of the problem graph and the single-shot success probability of the algorithm.
Airline companies struggle every day trying to schedule crew, flights and aircraft.Tail Assignment is the problem of allocating individual aircraft to a set of flights, while ensuring multiple constraints and aiming to minimise an objective function, such as operational costs.Given the enormous amount of possibilities and constraints involved, this problem has been a study case for the last decade.Many solutions have emerged using classical computing, but with limitations regarding performance.Quantum Annealing (QA) is a heuristic technique for finding global minimum energy levels over an energy landscape using quantum mechanics.Due to its characteristics, it has been proved to have a clear advantage in solving some complex optimisation problems, being a promising technique to apply in multiple fields.In this study, Tail Assignment Problem was set as a Quadratic Unconstrained Binary Optimisation (QUBO) model, using two different techniques, and was solved using a classical and two hybrid solvers.Tests were run based on extractions from real-world data, analysing the performance of the implementation in terms of time, scalability and quality (i.e., the lowest operational costs) of the obtained solutions.We concluded that using a library for modelling the problem as well as considering individual flights rather than pre-aggregating them in strings can be a bottleneck when it comes to its scalability.Furthermore, we found out that there was a higher probability of obtaining better solutions for this problem using one of the hybrid solvers when compared with the classical heuristic algorithms such as Simulated Annealing (SA).These findings can serve as foundations for further studies.
We present a method that integrates any quantum algorithm capable of finding solutions to integer linear programs into the branch-and-price algorithm, which is regularly used to solve large-scale integer linear programs with a specific structure. The role of the quantum algorithm is to find integer solutions to subproblems appearing in branch-and-price. Obtaining optimal or near-optimal integer solutions to these subproblems can increase the quality of solutions and reduce the depth and branching factor of the branch-and-price algorithm and hence reduce the overall running time. We investigate the viability of the approach by considering the tail assignment problem and the quantum approximate optimization algorithm (QAOA). Here, the master problem is the optimization problem set partitioning or its decision version exact cover and can be expressed as finding the ground state of an Ising spin glass Hamiltonian. For exact cover, our numerical results indicate that the required algorithm depth decreases with the number of feasible solutions for a given success probability of finding a feasible solution. For set partitioning, on the other hand, we find that for a given success probability of finding the optimal solution, the required algorithm depth can increase with the number of feasible solutions if the Hamiltonian is balanced poorly, which in the worst case is exponential in the problem size. We therefore address the significance of properly balancing the objective and constraint parts of the Hamiltonian. We empirically find that the approach is viable with QAOA if polynomial algorithm depth can be realized on quantum devices.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm to solve binary-variable optimization problems. Due to its expected robustness to systematic errors and the short circuit depth, it is one of the promising candidates likely to run on near-term quantum devices. We project the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives, both for exact or approximate solution. When comparing approximate solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we introduce performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. By selecting a suitable optimizer for the variational parameters and reducing the number of function evaluations, QAOA performance improves by up to 2 orders of magnitude compared to previous estimates. Especially for 3-regular random graphs, this setting decreases the performance gap with classical alternatives.
Abstract The quantum approximate optimization algorithm (QAOA) is a hybrid quantum–classical algorithm to solve binary-variable optimization problems. Due to the short circuit depth and its expected robustness to systematic errors it is a promising candidate likely to run on near-term quantum devices. We simulate the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives. When comparing solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we utilize performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. In particular for three-regular random graphs, QAOA performance shows improvement by up to two orders of magnitude compared to previous estimates, strongly reducing the performance gap with classical alternatives. This was possible by reducing the number of function evaluations per iteration and optimizing the variational parameters on small graph instances and transferring to large via training. Because QAOA’s performance guarantees are only known for limited applications and contexts, we utilize a framework for the search for quantum advantage which incorporates a large number of problem instances and all three classical solver modalities: exact, approximate, and heuristic.
Abstract Airline itinerary and scheduling represent some of the most difficult computational problems that exist in transportation. The problems require choosing the best flight routes while determining schedules and allocating resources through various constraints which include regulatory requirements and maintenance schedules and dynamic demand patterns. The solution of small and medium-scale instances has been achieved through mixed-integer programming and heuristic methods including simulated annealing and genetic algorithms and branch-and-price but these methods become impractical for larger real-world instances due to exponential scaling issues. Quantum computing has introduced fresh solutions which enable the resolution of these problems. Quantum mechanical phenomena particularly superposition and entanglement enable optimization algorithms to examine an exponentially expanded solution space simultaneously. Our research applies a basic airline scheduling problem to the Traveling Salesman Problem (TSP) before converting it into a Quadratic Unconstrained Binary Optimization (QUBO) problem. The Quantum Approximate Optimization Algorithm (QAOA) serves as a hybrid quantum-classical method to solve this formulation. The experimental evaluation of a 4-city scenario shows that quantum-based optimization generates performance levels comparable to traditional methods and indicates that improved quantum hardware development will provide computational benefits for complex airline scheduling problems. This research holds dual importance because it can minimize operational expenses and improve scheduling performance and it establishes fundamental principles for complex combinatorial problem optimization across the transportation industry. The research establishes fundamental principles for quantum-enabled applications that will revolutionize airline operations and other transportation networks by solving critical scalability and solution quality problems.
The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems.Despite its promise for near-term quantum applications, not much is currently understood about the QAOA's performance beyond its lowestdepth variant.An essential but missing ingredient for understanding and deploying the QAOA is a constructive approach to carry out the outer-loop classical optimization.We provide an in-depth study of the performance of the QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit nonadiabatic operations.Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasioptimal p-level QAOA parameters in O½polyðpÞ time, whereas the standard strategy of random initialization requires 2 OðpÞ optimization runs to achieve similar performance.We then benchmark the QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps.The comparison reveals that the QAOA can learn via optimization to utilize nonadiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps.Finally, we provide a realistic resource analysis on the experimental implementation of the QAOA.When quantum fluctuations in measurements are accounted for, we illustrate that optimization is important only for problem sizes beyond numerical simulations but accessible on near-term devices.We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.