The Average Mixing Matrix of a Continuous-Time Quantum Walk

The Average Mixing Matrix of a Continuous-Time Quantum Walk

Usage

# S3 method for ctqwalk
avg_matrix(object, ...)

Arguments

object

a representation of the quantum walk.

...

further arguments passed to or from other methods.

Value

avg_matrix() returns the 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, then the average mixing matrix is defined as

$$\widehat{M} := \lim_{T \to \infty} \frac{1}{T}\int_{0}^T M(t)\textrm{d}t$$

and encodes the long-term average behavior of the walk. Given the Hamiltonian \(H = \sum_r \lambda_r E_r\), it is possible to prove that

$$\widehat{M} = \sum_r E_r \circ E_r$$

See also

ctqwalk(), avg_matrix()

Examples

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

# Return the average mixing matrix
avg_matrix(walk)
#>       [,1] [,2]  [,3]
#> [1,] 0.375 0.25 0.375
#> [2,] 0.250 0.50 0.250
#> [3,] 0.375 0.25 0.375