LUXE (Laser Und XFEL Experiment) is a new experiment in planning in Hamburg, which will study Quantum Electrodynamics at the strong-field frontier. LUXE intends to measure the positron production rate in this unprecedented regime by using, among others, a silicon tracking detector. The large number of expected positrons traversing the sensitive detector layers results in an extremely challenging combinatorial problem, which can become computationally expensive for classical computers. We investigate the potential future use of gate-based quantum computers for pattern recognition in track reconstruction, based on a quadratic unconstrained binary optimisation and a quantum graph neural network. This report introduces these approaches and compares their performance with a classical track reconstruction algorithm.