Quantum algorithms

Quantum algorithms are step-by-step procedures for solving problems on quantum hardware. This page introduces the algorithms most relevant to the ATM optimisation problems studied in the JANUS project, with emphasis on near-term approaches suitable for current and near-future quantum devices.

Quantum annealing

Quantum annealing is a metaheuristic designed to find low-energy states of a physical system described by an Ising Hamiltonian. The system starts in a known ground state of a simple Hamiltonian and is gradually evolved toward the Hamiltonian whose ground state encodes the solution to the problem. The adiabatic theorem guarantees that if the evolution is slow enough, the system remains in its instantaneous ground state throughout — yielding the optimal solution.

In practice, finite temperature, noise, and limited anneal times mean that annealers find near-optimal rather than provably optimal solutions. Despite these limitations, quantum annealers have been applied to problems including airspace sectorisation, gate assignment, and drone collision avoidance — making them a natural fit for the JANUS exploration.

QAOA — Quantum Approximate Optimisation Algorithm

QAOA is a gate-model algorithm for combinatorial optimisation. It alternates between applying a problem Hamiltonian (encoding the objective) and a mixing Hamiltonian (driving exploration), with tunable parameters that are classically optimised. The depth of this alternating structure — the number of layers p — controls the approximation quality.

For p = 1, QAOA applies a single round of mixing; as p grows, the solution quality approaches the optimal. This tunability makes QAOA well-suited to NISQ devices, where circuit depth is limited by gate fidelities and decoherence times. QAOA can be applied to any QUBO or Ising problem, including those arising from ATM/UTM use cases.

VQE — Variational Quantum Eigensolver

VQE is a hybrid quantum-classical algorithm designed to find the ground state of a Hamiltonian. A parameterised quantum circuit (ansatz) prepares a trial state; the quantum computer measures its energy; a classical optimiser adjusts the parameters to minimise that energy.

VQE was originally developed for quantum chemistry but is applicable to any problem that can be expressed as a Hamiltonian minimisation, including optimisation problems formulated as Ising models. Its shallow circuit requirements make it one of the most promising algorithms for near-term quantum advantage.

Hybrid quantum-classical workflow

All three algorithms above share a common pattern: quantum hardware handles the exploration of large solution spaces, while classical processors manage parameter optimisation, constraint encoding, and result interpretation. This hybrid quantum-classical approach is the dominant paradigm for NISQ-era computing.

In the JANUS project, this workflow is applied to selected ATM use cases: classical baselines are established (WP5), operational constraints are translated into quantum- native formulations (WP6), and quantum solutions are benchmarked against their classical counterparts. The results inform the research community about which problems are genuinely amenable to quantum speedup and under what conditions.