We introduce eigen-independency of certain unitary matrices of degree 2. We show that this property characterizes stationary measures of quantum walks on cycles with odd number of vertices. We also characterize the stationary measures of the two-state Hadamard walk (both moving-shift and flip-flop shit) on cycles with $p$ vertices, where $p$ is an odd prime number.We introduce eigen-independency of certain unitary matrices of degree 2. We show that this property characterizes stationary measures of quantum walks on cycles with odd number of vertices. We also characterize the stationary measures of the two-state Hadamard walk (both moving-shift and flip-flop shit) on cycles with $p$ vertices, where $p$ is an odd prime number.

Quantum Information and ComputationNov 1, 20240 citations

NON-UNIFORM MIXING OF QUANTUM WALK ON CYCLES

William Adamczak, Kevin Andrew, Leon Bergen +4 more

A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cnis not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2u, for u ≥ 3; or (b) n = 2uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle Cnis O(1/n)-uniform.A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cnis not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2u, for u ≥ 3; or (b) n = 2uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle Cnis O(1/n)-uniform.

International Journal of Quantum InformationDec 1, 200727 citations

p

vertices, where

p

is an odd prime number.