As consequences of disruptions in railway traffic affect passenger experience and satisfaction, appropriate rerouting and/or rescheduling is necessary. These problems are known to be NP-hard, given the numerous restrictions of traffic nature. With the recent advances in quantum technologies, quantum annealing has become an alternative method to solve such optimization problems. To use quantum annealing, the problem needs to be encoded in quadratic unconstrained binary optimization (QUBO) or higher-order binary optimization (HOBO) formulation that can be recast as a QUBO. This paper introduces QUBO and HOBO representations for rescheduling problems of railway traffic management; the latter is a new approach up to our knowledge. This new approach takes into account not only the single-track lines but also the double- and multi-track lines, as well as stations composed of tracks and switches. We consider the conditions of minimal headway between trains, minimal stay on stations, track occupation, and rolling stock circulation. Furthermore, a hybrid quantum-classical procedure is presented that includes rerouting. We demonstrate the proof of concept implementation on the D-Wave Quantum Processing Unit and D-Wave hybrid solver.

We are in the noisy intermediate-scale quantum (NISQ) devices’ era, in which quantum hardware has become available for application in real-world problems. However, demonstrations of the usefulness of such NISQ devices are still rare. In this work, we consider a practical railway dispatching problem: delay and conflict management on single-track railway lines. We examine the train dispatching consequences of the arrival of an already delayed train to a given network segment. This problem is computationally hard and needs to be solved almost in real time. We introduce a quadratic unconstrained binary optimization (QUBO) model of this problem, which is compatible with the emerging quantum annealing technology. The model’s instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including the conventional solution of a linear integer version of the model as well as the solution of the QUBO model using a tensor network-based algorithm. Our preliminary results illustrate the degree of difficulty of real-life railway instances for the current quantum annealing technology. Moreover, our analysis shows that the new generation of quantum annealers (the advantage system) does not perform well on those instances, either.

In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the $d{th}$ order cumulant can be presented in the form of a $d$-dimensional tensor, the algorithm is presented using tensor operations. The algorithm provided in the paper takes advantage of supersymmetry of cumulant and moment tensors. We show that the proposed algorithm considerably reduces the computational complexity and the computational memory requirement of cumulant calculation as compared with existing algorithms. For the sizes of interest, the reduction is of the order of $d!$ compared to the naive algorithm.