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.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.

arXiv:1912.10499Physical Review AppliedSep 3, 202096 citations

Dynamical reduction models

Angelo Bassi, GianCarlo Ghirardi

arXiv:quant-ph/0302164Physics ReportsMay 19, 2003725 citations

iSTAR: an algebraic-collapse framework for variational reduction in quantum-inspired continuous Ising solvers

Bowen Liu, Dongmei Xiao

Continuous Ising solvers embed a discrete optimization problem into a continuous dynamical system and recover the spin configuration by sign readout, but dense interaction evaluation gives an $O(N^2)$ -per-step cost. We show that this cost is not intrinsic: during late-stage simulated bifurcation the trajectory collapses onto a lower-dimensional active subspace, and saturated coordinates can be eliminated exactly by a variational frozen-set identity whose couplings fold into an induced field on the unresolved subsystem. We prove large-parameter recovery for the external-field quartic model, the hard-box limit of ballistic confinement, and a robust-margin freezing criterion. The resulting algorithm, iSTAR (Ising Stable-set Tail-Aware Reduction), exploits this collapse by detecting stabilized coordinates and continuing only on the active tail. An online certified implementation on the G-set benchmark preserves the same-seed baseline in all runs and removes on average 64.4% of the dense interaction work.Continuous Ising solvers embed a discrete optimization problem into a continuous dynamical system and recover the spin configuration by sign readout, but dense interaction evaluation gives an $O(N^2)$-per-step cost. We show that this cost is not intrinsic: during late-stage simulated bifurcation the trajectory collapses onto a lower-dimensional active subspace, and saturated coordinates can be eliminated exactly by a variational frozen-set identity whose couplings fold into an induced field on the unresolved subsystem. We prove large-parameter recovery for the external-field quartic model, the hard-box limit of ballistic confinement, and a robust-margin freezing criterion. The resulting algorithm, iSTAR (Ising Stable-set Tail-Aware Reduction), exploits this collapse by detecting stabilized coordinates and continuing only on the active tail. An online certified implementation on the G-set benchmark preserves the same-seed baseline in all runs and removes on average 64.4% of the dense interaction work.

math.NACross-listedarXiv:2607.05448Jul 7, 20260 citations

Applying Quantum Annealing to the Tail Assignment Problem

Luís Noites Martins

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.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.

Jul 23, 20204 citations

Continuous Ising solvers embed a discrete optimization problem into a continuous dynamical system and recover the spin configuration by sign readout, but dense interaction evaluation gives an

O(N^2)

-per-step cost. We show that this cost is not intrinsic: during late-stage simulated bifurcation the trajectory collapses onto a lower-dimensional active subspace, and saturated coordinates can be eliminated exactly by a variational frozen-set identity whose couplings fold into an induced field on the unresolved subsystem. We prove large-parameter recovery for the external-field quartic model, the hard-box limit of ballistic confinement, and a robust-margin freezing criterion. The resulting algorithm, iSTAR (Ising Stable-set Tail-Aware Reduction), exploits this collapse by detecting stabilized coordinates and continuing only on the active tail. An online certified implementation on the G-set benchmark preserves the same-seed baseline in all runs and removes on average 64.4% of the dense interaction work.