The Generalized Average Mixing Matrix of a Continuous-Time Quantum Walk

Usage

# S3 method for ctqwalk
gavg_matrix(object, R, ...)

Arguments

object: a representation of the quantum walk.
R: samples from the random variable $R$ (For performance, it is recommended at most 10000 samples).
...: further arguments passed to or from other methods.

Value

gavg_matrix() returns the generalized average mixing matrix as a square matrix of the same order as the walk.

Details

Let $M(t)$ be the mixing matrix of the quantum walk and $R$ a random variable with associated probability density function $f_R(t)$. Then the generalized average mixing matrix under $R$ is defined as

$$\widehat{M}_R := \mathbb{E}[M(R)] = \int_{-\infty}^{\infty} M(t)f_R(t)\textrm{d}t$$

Examples

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

# Return the average mixing matrix under a Standard Gaussian distribution
gavg_matrix(walk, rnorm(1000))
#>           [,1]      [,2]      [,3]
#> [1,] 0.5600789 0.2461794 0.1937416
#> [2,] 0.2461794 0.5076411 0.2461794
#> [3,] 0.1937416 0.2461794 0.5600789