qwalkr is a package for the analysis of the time
evolution of quantum walks. The package consists mainly of methods for
the numerical estimation of matrices that arise in this field of
research. This vignette intends to give an overview of the core
functionalities provided by qwalkr.
Installation
You can install the stable version of qwalkr from
CRAN:
install.packages("qwalkr")For the development version, you can install from Github like so:
# install.packages("devtools")
devtools::install_github("vitormarquesr/qwalkr")Creating a Quantum Walk
At the moment, qwalkr only supports continuous-time
quantum walks.
You can create a continuous-time quantum walk by calling
qwalkr::ctqwalk() with an additional argument representing
the Hamiltonian of the system i.e. a hermitian matrix. For instance,
let’s create a walk on the complete graph of order three (\(K_3\)), the Hamiltonian is going to be its
adjacency matrix.
The ctqwalk object stores the spectral decomposition of
the Hamiltonian for future usage by other methods. We can get a glimpse
of the eigenvalues and their multiplicities by printing the object.
w
#> Continuous-Time Quantum Walk
#>
#> [+]Order: 3
#>
#> [+]Spectrum of the Hamiltonian:
#>
#> Eigenvalue: 2 -1
#> Multiplicity: 1 2qwalkr::ctqwalk() runs qwalkr::spectral()
under the hood and tries to infer eigenvalue multiplicity, you can
disable this feature with multiplicity=FALSE or set the
tolerance for numerical equality with tol.
Let’s talk a bit about spectral, the linear algebra
system used by qwalkr.
Linear Algebra System - Spectral
qwalkr::spectral() is a constructor for class
spectral which enhances base R spectral decomposition
function base::eigen(), providing specialized treatment for
hermitian matrices. Its constructor and methods can handle repeated
eigenvalues, orthogonal projectors, matrix functions, and so on.
You can create an instance by providing a hermitian matrix to
qwalkr::spectral():
The object stores the ordered eigenvalues (descending) and their multiplicities alongside their eigenvectors:
s
#> $eigvals
#> [1] 2 -1
#>
#> $multiplicity
#> [1] 1 2
#>
#> $eigvectors
#> [,1] [,2] [,3]
#> [1,] 0.5773503 0.8164966 0.0000000
#> [2,] 0.5773503 -0.4082483 -0.7071068
#> [3,] 0.5773503 -0.4082483 0.7071068
#>
#> attr(,"class")
#> [1] "spectral"You can turn off eigenvalue multiplicity inference with
multiplicity=FALSE and set the tolerance for numerical
equality with tol=1e-10.
Methods
With a spectral instance in hand, we can extract several
matrices of interest. Use qwalkr::get_eigspace() to extract
the eigenspace associated with an eigenvalue. For instance, let’s
extract the eigenvectors corresponding to the eigenvalue -1 by providing
its index in the ordered spectra (two in this case)
get_eigspace(s, id = 2)
#> [,1] [,2]
#> [1,] 0.8164966 0.0000000
#> [2,] -0.4082483 -0.7071068
#> [3,] -0.4082483 0.7071068Since the eigenvalue -1 has multiplicity two, its eigenspace has dimension two as well, thus we get two eigenvectors. If we extract the 2-eigenspace, we only get one vector.
get_eigspace(s, id=1)
#> [,1]
#> [1,] 0.5773503
#> [2,] 0.5773503
#> [3,] 0.5773503We can extract the orthogonal projector onto the -1-eigenspace by
calling qwalkr::get_eigproj() in the same way as the
previous function:
E2 <- get_eigproj(s, id=2)
E2
#> [,1] [,2] [,3]
#> [1,] 0.6666667 -0.3333333 -0.3333333
#> [2,] -0.3333333 0.6666667 -0.3333333
#> [3,] -0.3333333 -0.3333333 0.6666667Since the eigenspace has dimension two, its trace
(qwalkr::tr()) is going to take such a value
tr(E2)
#> [1] 2As to the 2-eigenspace we get
E1 <- get_eigproj(s, id=1)
E1
#> [,1] [,2] [,3]
#> [1,] 0.3333333 0.3333333 0.3333333
#> [2,] 0.3333333 0.3333333 0.3333333
#> [3,] 0.3333333 0.3333333 0.3333333
tr(E1)
#> [1] 1For additional methods for class spectral check its help page.
Quantum Walks of class spectral
As with continuous-time quantum walks, some walks have their
evolution influenced by the spectrum of the Hamiltonian. In such a case,
the class representing the walk is also a child of spectral
and thus inherits its methods. For instance, a ctqwalk
object is also an instance of spectral, and hence the
methods of the previous section apply:
w
#> Continuous-Time Quantum Walk
#>
#> [+]Order: 3
#>
#> [+]Spectrum of the Hamiltonian:
#>
#> Eigenvalue: 2 -1
#> Multiplicity: 1 2
get_eigspace(w, id=2)
#> [,1] [,2]
#> [1,] 0.8164966 0.0000000
#> [2,] -0.4082483 -0.7071068
#> [3,] -0.4082483 0.7071068
class(w)
#> [1] "ctqwalk" "spectral"Time-Evolution
You can get the unitary time evolution operator of the walk by
calling qwalkr::unitary_matrix() with the time
t you want to evaluate it at. Let’s take a look at the
operator of the walk on \(K_3\) at time
\(t=\frac{\pi}{3}\):
unitary_matrix(w, t=pi/3)
#> [,1] [,2] [,3]
#> [1,] 0.1666667-0.2886751i -0.3333333+0.5773503i -0.3333333+0.5773503i
#> [2,] -0.3333333+0.5773503i 0.1666667-0.2886751i -0.3333333+0.5773503i
#> [3,] -0.3333333+0.5773503i -0.3333333+0.5773503i 0.1666667-0.2886751iAnalogously, the mixing matrix of the walk can be obtained with
qwalkr::mixing_matrix():
mixing_matrix(w, t=pi/3)
#> [,1] [,2] [,3]
#> [1,] 0.1111111 0.4444444 0.4444444
#> [2,] 0.4444444 0.1111111 0.4444444
#> [3,] 0.4444444 0.4444444 0.1111111Average Evolution
Use qwalkr::avg_matrix() to obtain the (standard)
average mixing matrix:
avg_matrix(w)
#> [,1] [,2] [,3]
#> [1,] 0.5555556 0.2222222 0.2222222
#> [2,] 0.2222222 0.5555556 0.2222222
#> [3,] 0.2222222 0.2222222 0.5555556To instead calculate the (generalized) average mixing matrix under an
arbitrary probability distribution, call
qwalkr::gavg_matrix() with an additional parameter
R containing samples of the desired distribution. For
instance, the average mixing matrix of \(K_3\) under a standard Exponential random
variable can be obtained with:
set.seed(10)
gavg_matrix(w, R=rexp(1000))
#> [,1] [,2] [,3]
#> [1,] 0.6017741 0.1991130 0.1991130
#> [2,] 0.1991130 0.6017741 0.1991130
#> [3,] 0.1991130 0.1991130 0.6017741