IQM Quantum Computers and Deutsche Bahn have demonstrated a hybrid quantum-classical algorithm for rolling stock scheduling, using a superconducting quantum processor to process real operational data from major German cities
IQM Quantum Computers and Deutsche Bahn have reported the execution of a hybrid quantum-classical optimization algorithm on real-world railway scheduling data, using IQM's Emerald superconducting quantum processor. The experiment focused on the rolling stock planning problem: assigning physical train units to scheduled trips while minimizing operational costs and meeting maintenance requirements. This task is representative of a class of industrial scheduling problems that are computationally challenging for classical algorithms, especially when constraints such as maintenance intervals and distance caps are included.
Device and Dataset
The study used operational data from Deutsche Bahn's IT subsidiary, DB Systel, covering 190 scheduled train trips across Cologne, Munich, Berlin, Frankfurt, and Hamburg over a two-day period. To map the scheduling constraints into a form suitable for quantum processing, the problem was encoded as a Maximum-Weight Independent Set (MWIS) on a conflict graph. In this representation, each node corresponds to a feasible closed train cycle-incorporating mandatory two-hour maintenance stops in Hamburg and a 4,000 km distance limit-while edges connect cycles that cannot be assigned simultaneously because they overlap on the same trip.
Hybrid Algorithm and Experimental Method
Due to the large search space-approximately 98,500 feasible cycles in the full MWIS graph-the researchers implemented a quantum divide-and-conquer approach. The classical component iteratively selected subgraphs of manageable size, typically around 20 nodes, prioritizing those with high passenger-trip density. The quantum subroutine then applied the Quantum Approximate Optimization Algorithm (QAOA) at depth p=1 to these subgraphs, generating partial solutions. A classical post-processing step pruned conflicting assignments to ensure valid independent sets before updating the global problem state.
Results and Scaling Behavior
The hybrid framework was able to generate feasible train schedules for the real dataset using current superconducting quantum hardware, without requiring fault-tolerant quantum processors. Benchmarking with classical exact solvers showed a statistically significant trend: as the size of the quantum-processed subgraphs increased, the algorithm produced schedules with fewer empty kilometers-distance traveled without passengers. The reported p-value for this scaling relationship was 1.04 × 10⁻⁹, indicating a robust effect within the tested regime. The study also suggests that as quantum processors improve in qubit count, connectivity, and gate fidelity, the same algorithmic structure could yield better scheduling outcomes without major redesign.
Limitations and Future Directions
The demonstration was limited to deterministic, offline planning and did not address real-time disruption management, which would require dynamic reassignment of rolling stock during service interruptions. The MWIS mapping and divide-and-conquer approach are not unique to railway scheduling and could, in principle, be adapted to other industrial optimization problems, but the current implementation remains constrained by the number of qubits and the noise levels of present-day superconducting devices. The research has not yet been independently replicated, and the results should be interpreted as a proof-of-principle rather than evidence of quantum advantage or commercial readiness.
This work follows other recent efforts to benchmark quantum hardware on classically challenging tasks, such as the demonstration of quantum processors performing verifiable computations beyond the reach of leading supercomputers, as discussed in a recent Science Report article. However, the present study remains within the regime where classical verification is possible, and the quantum component is integrated as part of a hybrid workflow rather than replacing classical optimization entirely.
Hybrid quantum-classical algorithms like QAOA are designed to exploit the strengths of both quantum and classical processors. In these approaches, a quantum device prepares and measures parameterized quantum states, while a classical optimizer updates the parameters based on measurement outcomes. The depth of the quantum circuit (here, p=1) is limited by hardware noise and coherence time, which constrains the complexity of problems that can be addressed. As quantum hardware improves, deeper circuits and larger subgraphs may become accessible, but the transition from laboratory demonstration to practical utility will depend on advances in error rates, qubit connectivity, and integration with classical infrastructure.
