The Mixing Matrix of a Continuous-Time Quantum Walk

Usage

# S3 method for ctqwalk
mixing_matrix(object, t, ...)

Arguments

object: an instance of class ctqwalk.
t: it will be returned the mixing matrix at time t.
...: further arguments passed to or from other methods.

Value

mixing_matrix() returns the mixing matrix of the CTQW evaluated at time t.

Details

Let $U(t)$ be the time evolution operator of the quantum walk at time $t$, then the mixing matrix is given by

$$M(t) = U(t) \circ \overline{U(t)}$$

$M(t)$ is a doubly stochastic real symmetric matrix, which encodes the probability density of the quantum system at time $t$.

More precisely, the $(M(t))_{ab}$ entry gives us the probability of measuring the standard basis state $|b \rangle$ at time $t$, given that the quantum walk started at $|a \rangle$.

Examples

walk <- ctqwalk(matrix(c(0,1,0,1,0,1,0,1,0), nrow=3))

# Returns the mixing matrix at time t = 2*pi, M(2pi)
mixing_matrix(walk, t = 2*pi)
#>             [,1]      [,2]        [,3]
#> [1,] 0.005025663 0.1317325 0.863241848
#> [2,] 0.131732489 0.7365350 0.131732489
#> [3,] 0.863241848 0.1317325 0.005025663