+ docs for time_evol.R, avg_evol.R, matrices.R

R/avg_evolution.R

@@ -1,4 +1,4 @@
-#' Get the Average Mixing Matrix of a Quantum Walk
+#' The Average Mixing Matrix of a Quantum Walk
 #'
 #' @param object a representation of the quantum walk.
 #' @param ... further arguments passed to or from other methods.
@@ -10,25 +10,22 @@
 avg_matrix <- function(object, ...) UseMethod("avg_matrix")
 
 
-#' Get the Average Mixing Matrix of a Continuous-Time Quantum Walk
+#' The Average Mixing Matrix of a Continuous-Time Quantum Walk
 #'
-#' @param object a representation of the quantum walk.
-#' @param ... further arguments passed to or from other methods.
-#'
-#' @returns The average mixing matrix of the continuous-time quantum walk.
-#'
-#'    Let \eqn{M(t)} be the mixing matrix of the quantum walk, then the
-#'    average mixing matrix is defined as
+#' @details Let \eqn{M(t)} be the mixing matrix of the quantum walk, then the average mixing matrix is defined as
 #'
 #'    \deqn{\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. It is possible to prove that
+#'    and encodes the long-term average behavior of the walk. Given the Hamiltonian
+#'    \eqn{H = \sum_r \lambda_r E_r}, it is possible to prove that
 #'
 #'    \deqn{\widehat{M} = \sum_r E_r \circ E_r}
 #'
-#'    in which \eqn{E_r} is the orthogonal projector onto the \eqn{\lambda_r}-eigenspace.
+#' @param object a representation of the quantum walk.
+#' @param ... further arguments passed to or from other methods.
 #'
-#'    `avg_matrix` returns the matrix from the above calculation.
+#' @returns `avg_matrix()` returns the average mixing matrix
+#'   as a square matrix of the same order as the walk.
 #'
 #' @export
 #'
@@ -48,7 +45,7 @@ avg_matrix.ctqwalk <- function(object, ...){
   return (M)
 }
 
-#' Get the Generalized Average Mixing Matrix of a Quantum Walk
+#' The Generalized Average Mixing Matrix of a Quantum Walk
 #'
 #' @param object a representation of the quantum walk.
 #' @param ... further arguments passed to or from other methods.
@@ -60,21 +57,20 @@ avg_matrix.ctqwalk <- function(object, ...){
 gavg_matrix <- function(object, ...) UseMethod("gavg_matrix")
 
 
-#' Get the Generalized Average Mixing Matrix of a Continuous-Time Quantum Walk
+#' The Generalized Average Mixing Matrix of a Continuous-Time Quantum Walk
+#'
+#' @details Let \eqn{M(t)} be the mixing matrix of the quantum walk and \eqn{R} a random variable
+#'   with associated probability density function \eqn{f_R(t)}. Then the generalized average mixing
+#'   matrix under \eqn{R} is defined as
+#'
+#'    \deqn{\widehat{M}_R := \mathbb{E}[M(R)] = \int_{-\infty}^{\infty} M(t)f_R(t)\textrm{d}t}
 #'
 #' @param object a representation of the quantum walk.
 #' @param R samples from the random variable \eqn{R} (For performance, it is recommended at most 10000 samples).
 #' @param ... further arguments passed to or from other methods.
 #'
-#' @returns The generalized average mixing matrix of the continuous-time quantum walk.
-#'
-#'    Let \eqn{M(t)} be the mixing matrix of the quantum walk and \eqn{f_R(t)} the probability
-#'    density function associated with \eqn{R}, then the generalized average mixing matrix under \eqn{R}
-#'    is defined as
-#'
-#'    \deqn{\widehat{M}_R := \mathbb{E}[M(R)] = \int_{-\infty}^{\infty} M(t)f_R(t)\textrm{d}t}
-#'
-#'    `gavg_matrix` returns an approximation of the above matrix by the Monte Carlo approach.
+#' @returns `gavg_matrix()` returns the generalized average mixing matrix
+#'   as a square matrix of the same order as the walk.
 #'
 #' @export
 #'

R/matrices.R

@@ -11,6 +11,8 @@
 #'
 #' @export
 #'
+#' @seealso [qwalkr::tr()], [qwalkr::trdot()], [qwalkr::cartesian()]
+#'
 #' @examples
 #' # Return the all-ones matrix of order 5.
 #' J(5)
@@ -29,6 +31,7 @@ J <- function(n){
 #'   then \eqn{tr(A) = \sum_{i=1}^{n}a_{ii}}.
 #'
 #' @export
+#' @seealso [qwalkr::J()], [qwalkr::trdot()], [qwalkr::cartesian()]
 #'
 #' @examples
 #' A <- rbind(1:5, 2:6, 3:7)
@@ -51,12 +54,13 @@ tr <- function(A){
 #'
 #'   \deqn{\langle A, B \rangle := tr(A^*B)}
 #' @export
+#' @seealso [qwalkr::J()], [qwalkr::tr()], [qwalkr::cartesian()]
 #'
 #' @examples
 #' A <- rbind(1:5, 2:6, 3:7)
 #' B <- rbind(7:11, 8:12, 9:13)
 #'
-#' # Calculate the trace inner product of A and B
+#' # Compute the trace inner product of A and B
 #' trdot(A, B)
 trdot <- function(A, B){
   return (tr(Conj(t(A)) %*% B))
@@ -80,6 +84,7 @@ trdot <- function(A, B){
 #'   \deqn{A(G \times H) = A(G) \otimes I_{m\ x\ m} + I_{n\ x\ n} \otimes A(H)}
 #'
 #' @export
+#' @seealso [qwalkr::J()], [qwalkr::tr()], [qwalkr::trdot()]
 #'
 #' @examples
 #' P3 <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
@@ -103,4 +108,3 @@ cartesian <- function(G, H=NULL){
 }
 
 
-

R/time_evolution.R

@@ -1,5 +1,5 @@
 
-#' Get the Unitary Time Evolution Operator of a Quantum Walk
+#' The Unitary Time Evolution Operator of a Quantum Walk
 #'
 #' @param object a representation of the quantum walk.
 #' @param ... further arguments passed to or from other methods.
@@ -12,16 +12,9 @@
 unitary_matrix <- function(object, ...) UseMethod("unitary_matrix")
 
 
-#' Get the Unitary Time Evolution Operator of a Continuous-Time Quantum Walk
+#' The Unitary Time Evolution Operator of a Continuous-Time Quantum Walk
 #'
-#' @param object an instance of class `ctqwalk`.
-#' @param t it will be returned the evolution operator at time `t`.
-#' @param ... further arguments passed to or from other methods.
-#'
-#' @returns The unitary time evolution operator of the continuous-time quantum walk
-#'   at time `t`.
-#'
-#'   If \eqn{|\psi(t) \rangle} is the quantum state of the system at time \eqn{t}, and
+#' @details If \eqn{|\psi(t) \rangle} is the quantum state of the system at time \eqn{t}, and
 #'   \eqn{H} the Hamiltonian operator, then the evolution is governed by
 #'   the Schrodinger equation
 #'
@@ -38,8 +31,12 @@ unitary_matrix <- function(object, ...) UseMethod("unitary_matrix")
 #'
 #'    in which \eqn{H = \sum_r \lambda_r E_r}.
 #'
-#'    `unitary_matrix` returns the result of the above calculation for a
-#'    provided `t`.
+#' @param object an instance of class `ctqwalk`.
+#' @param t it will be returned the evolution operator at time `t`.
+#' @param ... further arguments passed to or from other methods.
+#'
+#' @returns `unitary_matrix()` returns the unitary time evolution operator of the
+#'   CTQW evaluated at time `t`.
 #'
 #' @export
 #' @seealso [qwalkr::ctqwalk()], [qwalkr::unitary_matrix()],
@@ -56,7 +53,7 @@ unitary_matrix.ctqwalk <- function(object, t, ...){
   return (Ut)
 }
 
-#' Get the Mixing Matrix of a Quantum Walk
+#' The Mixing Matrix of a Quantum Walk
 #'
 #' @param object a representation of the quantum walk.
 #' @param ... further arguments passed to or from other methods.
@@ -69,16 +66,10 @@ unitary_matrix.ctqwalk <- function(object, t, ...){
 mixing_matrix <- function(object, ...) UseMethod("mixing_matrix")
 
 
-#' Get the Mixing Matrix of a Continuous-Time Quantum Walk
+#' The Mixing Matrix of a Continuous-Time Quantum Walk
 #'
-#' @param object an instance of class `ctqwalk`.
-#' @param t it will be returned the mixing matrix at time `t`.
-#' @param ... further arguments passed to or from other methods.
-#'
-#' @returns The mixing matrix of the continuous-time quantum walk.
-#'
-#'    Let \eqn{U(t)} be the time evolution operator of the
-#'    quantum walk at time \eqn{t}, then the mixing matrix is given by
+#' @details Let \eqn{U(t)} be the time evolution operator of the quantum walk at
+#'    time \eqn{t}, then the mixing matrix is given by
 #'
 #'    \deqn{M(t) = U(t) \circ \overline{U(t)}}
 #'
@@ -89,8 +80,12 @@ mixing_matrix <- function(object, ...) UseMethod("mixing_matrix")
 #'    of measuring the standard basis state \eqn{|b \rangle} at time \eqn{t}, given that
 #'    the quantum walk started at \eqn{|a \rangle}.
 #'
-#'    `mixing_matrix` returns the result of the above calculations for a
-#'    provided `t`.
+#' @param object an instance of class `ctqwalk`.
+#' @param t it will be returned the mixing matrix at time `t`.
+#' @param ... further arguments passed to or from other methods.
+#'
+#' @returns `mixing_matrix()` returns the  mixing matrix of the CTQW
+#'   evaluated at time `t`.
 #'
 #' @export
 #' @seealso [qwalkr::ctqwalk()], [qwalkr::mixing_matrix()]