2021/10/02 by Norio Konno, Iwao Sato, Konno, Norio +5
Computer Science · Physics and Astronomy · #Quantum Computing Algorithms and Architecture #Quantum-Dot Cellular Automata #Quantum and electron transport phenomena
paper · pdf · doi:10.48550/arxiv.2110.00716
We consider a spectral analysis on the quantum walks on graph G=(V,E) with the local coin operators \Cu\u∈ V and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues κ,κ' and p=dim(ker(κ-Cu)) for any u∈ V with 1≤ p≤ δ(G), where δ(G) is the minimum degrees of G. We show that this quantum walk can be decomposed into a cellular automaton on ℓ2(V;ℂp) whose time evolution is described by a self adjoint operator T and its remainder. We obtain how the eigenvalues and its eigenspace of T are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on ℤd with the moving shift in the Fourier space.