High-dimensional graphs convolution for quantum walks photonic applications

Abstract: Quantum random walks represent a powerful tool for the implementation of various quantum algorithms. We consider a convolution problem for the graphs which provide quantum and classical random walks. We suggest a new method for lattices and hypercycle convolution that preserves quantum walk dynamics. Our method is based on the fact that some graphs represent a result of Kronecker's product of line graphs. We support our methods by means of various numerical experiments that check quantum and classical random walks on hypercycles and their convolutions. Our findings may be useful for saving a significant number of qubits required for algorithms that use quantum walk simulation on quantum devices.