unitary_matrix.ctqwalk {qwalkr}

The Unitary Time Evolution Operator of a Continuous-Time Quantum Walk

Description

The Unitary Time Evolution Operator of a Continuous-Time Quantum Walk

Usage

## S3 method for class 'ctqwalk'
unitary_matrix(object, t, ...)

Arguments

object

an instance of class ctqwalk.

t

it will be returned the evolution operator at time t.

...

further arguments passed to or from other methods.

Details

If |\psi(t) \rangle is the quantum state of the system at time t, and H the Hamiltonian operator, then the evolution is governed by the Schrodinger equation

\frac{\partial}{\partial t}|\psi(t) \rangle = iH|\psi(t) \rangle

and if H is time-independent its solution is given by

|\psi(t) \rangle = U(t)|\psi(0) \rangle = e^{iHt}|\psi(0) \rangle

The evolution operator is the result of the complex matrix exponential and it can be calculated as

U(t) = e^{iHt} = \sum_r e^{i t \lambda_r}E_r

in which H = \sum_r \lambda_r E_r.

Value

unitary_matrix() returns the unitary time evolution operator of the CTQW evaluated at time t.

See Also

ctqwalk(), unitary_matrix(), act_eigfun()

Examples

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

# Returns the operator at time t = 2*pi, U(2pi)
unitary_matrix(walk, t = 2*pi)