Package 'qwalkr'

Apply a Function to an Operator

Description

Apply a Function to an Operator

Usage

act_eigfun(object, ...)

Arguments

object	a representation of the operator.
...	further arguments passed to or from other methods.

Value

The resulting operator from the application of the function.

See Also

act_eigfun.spectral()

Examples

s <- spectral(rbind(c(0.5, 0.3), c(0.3,0.7)))

act_eigfun(s, function(x) x^2) #-> act_eigfun.spectral(...)

---

Apply a Function to a Hermitian Matrix

Description

Apply a function to a Hermitian matrix based on the representation given by class spectral.

Usage

## S3 method for class 'spectral'
act_eigfun(object, FUN, ...)

Arguments

object	an instance of class spectral.
FUN	the function to be applied to the matrix.
...	further arguments passed on to FUN.

Value

The matrix resulting from the application of FUN.

A Hermitian Matrix admits the spectral decomposition

$H = \sum_k \lambda_k E_k$

where $\lambda_k$ are its eigenvalues and $E_k$ the orthogonal projector onto the $\lambda_k$-eigenspace.

If $f$=FUN is defined on the eigenvalues of H, then act_eigfun performs the following calculation

$f(H) = \sum_k f(\lambda_k) E_k$

See Also

spectral(), act_eigfun()

Examples

H <- matrix(c(0,1,1,1,0,1,1,1,0), nrow=3)
decomp <- spectral(H)

# Calculates H^2.
act_eigfun(decomp, FUN = function(x) x^2)

# Calculates sin(H).
act_eigfun(decomp, FUN = function(x) sin(x))

# Calculates H^3.
act_eigfun(decomp, FUN = function(x, y) x^y, 3)

---

The Average Mixing Matrix of a Quantum Walk

Description

The Average Mixing Matrix of a Quantum Walk

Usage

avg_matrix(object, ...)

Arguments

object	a representation of the quantum walk.
...	further arguments passed to or from other methods.

Value

The average mixing matrix.

See Also

mixing_matrix(), gavg_matrix(), avg_matrix.ctqwalk()

Examples

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

avg_matrix(w) #-> avg_matrix.ctqwalk(...)

---

The Average Mixing Matrix of a Continuous-Time Quantum Walk

Description

The Average Mixing Matrix of a Continuous-Time Quantum Walk

Usage

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

Arguments

object	a representation of the quantum walk.
...	further arguments passed to or from other methods.

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

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

Value

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

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)

---

Adjacency Matrix of the Cartesian Product

Description

Returns the adjacency matrix of the cartesian product of two graphs given the adjacency matrix of each one, $G$ and $H$.

Usage

cartesian(G, H = NULL)

Arguments

G	adjacency matrix of the first graph.
H	adjacency matrix of the second graph. If not provided, it takes the same value as G.

Value

Let $A(G),\ A(H)$ be the adjacency matrices of the graphs $G,\ H$ such that $|V(G)| = n$ and $|V(H)| = m$, then the adjacency matrix of the cartesian product $G \times H$ is given by

$A(G \times H) = A(G) \otimes I_{m\ x\ m} + I_{n\ x\ n} \otimes A(H)$

See Also

J(), tr(), trdot()

Examples

P3 <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
K3 <- matrix(c(0,1,1,1,0,1,1,1,0), nrow=3)

# Return the adjacency matrix of P3 X K3
cartesian(P3, K3)

# Return the adjacency matrix of P3 X P3
cartesian(P3)

---

Create a Continuous-time Quantum Walk

Description

ctqwalk() creates a quantum walk object from a hamiltonian.

Usage

ctqwalk(hamiltonian, ...)

Arguments

hamiltonian	a Hermitian Matrix representing the Hamiltonian of the system.
...	further arguments passed on to spectral()

Value

A list with the walk related objects, i.e the hamiltonian and its spectral decomposition (See spectral() for further details)

See Also

spectral(), unitary_matrix.ctqwalk(), mixing_matrix.ctqwalk(), avg_matrix.ctqwalk(), gavg_matrix.ctqwalk()

Examples

# Creates a walk from the adjacency matrix of the graph P3.
ctqwalk(matrix(c(0,1,0,1,0,1,0,1,0), nrow=3))

---

The Generalized Average Mixing Matrix of a Quantum Walk

Description

The Generalized Average Mixing Matrix of a Quantum Walk

Usage

gavg_matrix(object, ...)

Arguments

object	a representation of the quantum walk.
...	further arguments passed to or from other methods.

Value

The generalized average mixing matrix.

See Also

mixing_matrix(), avg_matrix(), gavg_matrix.ctqwalk()

Examples

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

gavg_matrix(w, rnorm(100)) #-> gavg_matrix.ctqwalk(...)

---

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

Description

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

Usage

## S3 method for class '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.

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

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

Value

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

See Also

ctqwalk(), gavg_matrix()

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))

---

Extract an Eigen-Projector from an operator

Description

Extract an Eigen-Projector from an operator

Usage

get_eigproj(object, ...)

Arguments

object	a representation of the operator.
...	further arguments passed to or from other methods.

Value

A representation of the requested eigen-projector.

See Also

get_eigspace(), get_eigschur(), get_eigproj.spectral()

Examples

s <- spectral(rbind(c(0.5, 0.3), c(0.3,0.7)))

get_eigproj(s, 1) #-> get_eigproj.spectral(...)

---

Extract an Eigen-Projector from a Hermitian Matrix

Description

Get the orthogonal projector associated with an eigenspace based on the representation of a Hermitian Matrix given by class spectral.

Usage

## S3 method for class 'spectral'
get_eigproj(object, id, ...)

Arguments

object	an instance of class spectral.
id	index for the desired eigenspace according to the ordered (decreasing) spectra.
...	further arguments passed to or from other methods.

Value

The orthogonal projector of the desired eigenspace.

A Hermitian matrix S admits the spectral decomposition $S = \sum_{r}\lambda_r E_r$ such that $E_r$ is the orthogonal projector onto the $\lambda_r$-eigenspace. If $V_{id}$ is the matrix associated to the eigenspace, then

$E_{id} = V_{id}V_{id}^*$

See Also

spectral(), get_eigproj()

Examples

# Spectra is {2, -1} with multiplicities one and two respectively.
decomp <- spectral(matrix(c(0,1,1,1,0,1,1,1,0), nrow=3))

# Returns the projector associated to the eigenvalue -1.
get_eigproj(decomp, id=2)

# Returns the projector associated to the eigenvalue 2.
get_eigproj(decomp, id=1)

---

Extract a Schur Cross-Product from an Operator

Description

Extract a Schur Cross-Product from an Operator

Usage

get_eigschur(object, ...)

Arguments

object	a representation of the operator.
...	further arguments passed to or from other methods.

Value

A representation of the requested Schur cross-product.

See Also

get_eigspace(), get_eigproj(), get_eigschur.spectral()

Examples

s <- spectral(rbind(c(0.5, 0.3), c(0.3,0.7)))

get_eigschur(s, 1, 2) #-> get_eigschur.spectral(...)

---

Extract a Schur Cross-Product from a Hermitian Matrix

Description

Get the Schur product between eigen-projectors based on the representation of a Hermitian Matrix given by class spectral.

Usage

## S3 method for class 'spectral'
get_eigschur(object, id1, id2 = NULL, ...)

Arguments

object	an instance of class spectral.
id1	index for the first eigenspace according to the ordered (decreasing) spectra.
id2	index for the second eigenspace according to the ordered (decreasing) spectra. If not provided, it takes the same value as id1.
...	further arguments passed to or from other methods.

Value

The Schur product of the corresponding eigenprojectors, $E_{id_1} \circ E_{id_2}$.

See Also

spectral(), get_eigschur()

Examples

# Spectra is {2, -1} with multiplicities one and two respectively.
decomp <- spectral(matrix(c(0,1,1,1,0,1,1,1,0), nrow=3))

# Returns the Schur product between the 2-projector and -1-projector.
get_eigschur(decomp, id1=2, id2=1)

# Returns the Schur square of the 2-projector.
get_eigschur(decomp, id1=1, id2=1)

# Also returns the Schur square of the 2-projector
get_eigschur(decomp, id1=1)

---

Extract an Eigenspace from an Operator

Description

Extract an Eigenspace from an Operator

Usage

get_eigspace(object, ...)

Arguments

object	a representation of the operator.
...	further arguments passed to or from other methods.

Value

A representation of the requested eigenspace.

See Also

get_eigproj(), get_eigschur(), get_eigspace.spectral()

Examples

s <- spectral(rbind(c(0.5, 0.3), c(0.3,0.7)))

get_eigspace(s, 1) #-> get_eigspace.spectral(...)

---

Extract an Eigenspace from a Hermitian Matrix

Description

Get the eigenbasis associated with an eigenvalue based on the representation of a Hermitian Matrix given by class spectral.

Usage

## S3 method for class 'spectral'
get_eigspace(object, id, ...)

Arguments

object	an instance of class spectral.
id	index for the desired eigenspace according to the ordered (decreasing) spectra.
...	further arguments passed to or from other methods.

Value

A matrix whose columns form the orthonormal eigenbasis.

If s <- spectral(A) and V <- s$eigvectors, then the extracted eigenspace $V_{id}$ is some submatrix ⁠V[, _]⁠.

See Also

spectral(), get_eigspace()

Examples

# Spectra is {2, -1} with multiplicities one and two respectively.
decomp <- spectral(matrix(c(0,1,1,1,0,1,1,1,0), nrow=3))

# Returns the two orthonormal eigenvectors corresponding to the eigenvalue -1.
get_eigspace(decomp, id=2)

# Returns the eigenvector corresponding to the eigenvalue 2.
get_eigspace(decomp, id=1)

---

The All-Ones Matrix

Description

Returns the all-ones matrix of order n.

Usage

J(n)

Arguments

n	the order of the matrix.

Value

A square matrix of order $n$ in which every entry is equal to 1. The all-ones matrix is given by $J_{n\ x\ n} = 1_{n\ x\ 1}1_{n\ x\ 1}^T$.

See Also

tr(), trdot(), cartesian()

Examples

# Return the all-ones matrix of order 5.
J(5)

---

The Mixing Matrix of a Quantum Walk

Description

The Mixing Matrix of a Quantum Walk

Usage

mixing_matrix(object, ...)

Arguments

object	a representation of the quantum walk.
...	further arguments passed to or from other methods.

Value

The mixing matrix of the quantum walk.

See Also

unitary_matrix(), avg_matrix(), gavg_matrix(), mixing_matrix.ctqwalk()

Examples

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

mixing_matrix(w, t = 2*pi) #-> mixing_matrix.ctqwalk(...)

---

The Mixing Matrix of a Continuous-Time Quantum Walk

Description

The Mixing Matrix of a Continuous-Time Quantum Walk

Usage

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

Arguments

object	an instance of class ctqwalk.
t	it will be returned the mixing matrix at time t.
...	further arguments passed to or from other methods.

Details

Let $U(t)$ be the time evolution operator of the quantum walk at time $t$, then the mixing matrix is given by

$M(t) = U(t) \circ \overline{U(t)}$

$M(t)$ is a doubly stochastic real symmetric matrix, which encodes the probability density of the quantum system at time $t$.

More precisely, the $(M(t))_{ab}$ entry gives us the probability of measuring the standard basis state $|b \rangle$ at time $t$, given that the quantum walk started at $|a \rangle$.

Value

mixing_matrix() returns the mixing matrix of the CTQW evaluated at time t.

See Also

ctqwalk(), mixing_matrix()

Examples

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

# Returns the mixing matrix at time t = 2*pi, M(2pi)
mixing_matrix(walk, t = 2*pi)

---

Print the ctqwalk output

Description

Print the ctqwalk output

Usage

## S3 method for class 'ctqwalk'
print(x, ...)

Arguments

x	an object of the class ctqwalk.
...	further arguments passed to or from other methods.

Value

Called mainly for its side effects. However, also returns x invisibly.

---

Spectral Decomposition of a Hermitian Matrix

Description

spectral() is a wrapper around base::eigen() designed for Hermitian matrices, which can handle repeated eigenvalues.

Usage

spectral(S, multiplicity = TRUE, tol = .Machine$double.eps^0.5, ...)

Arguments

S	a Hermitian matrix. Obs: The matrix is always assumed to be Hermitian, and only its lower triangle (diagonal included) is used.
multiplicity	if TRUE (default), tries to infer eigenvalue multiplicity. If set to FALSE, each eigenvalue is considered unique with multiplicity one.
tol	two eigenvalues x, y are considered equal if abs(x-y) < tol. Defaults to tol=.Machine$double.eps^0.5.
...	further arguments passed on to base::eigen()

Value

The spectral decomposition of S is returned as a list with components

eigvals	vector containing the unique eigenvalues of S in decreasing order.
multiplicity	multiplicities of the eigenvalues in eigvals.
eigvectors	a ⁠nrow(S) x nrow(S)⁠ unitary matrix whose columns are eigenvectors ordered according to eigvals. Note that there may be more eigenvectors than eigenvalues if multiplicity=TRUE, however eigenvectors of the same eigenspace are next to each other.

The Spectral Theorem ensures the eigenvalues of S are real and that the vector space admits an orthonormal basis consisting of eigenvectors of S. Thus, if s <- spectral(S), and ⁠V <- s$eigvectors; lam <- s$eigvals⁠, then

$S = V \Lambda V^{*}$

where \Lambda =\diag(rep(lam, times=s$multiplicity))

See Also

base::eigen(), get_eigspace.spectral(), get_eigproj.spectral(), get_eigschur.spectral(), act_eigfun.spectral()

Examples

spectral(matrix(c(0,1,0,1,0,1,0,1,0), nrow=3))

# Use "tol" to set the tolerance for numerical equality
spectral(matrix(c(0,1,0,1,0,1,0,1,0), nrow=3), tol=10e-5)

# Use "multiplicity=FALSE" to force each eigenvalue to be considered unique
spectral(matrix(c(0,1,0,1,0,1,0,1,0), nrow=3), multiplicity = FALSE)

---

The Trace of a Matrix

Description

Computes the trace of a matrix A.

Usage

tr(A)

Arguments

A	a square matrix.

Value

If $A$ has order $n$, then $tr(A) = \sum_{i=1}^{n}a_{ii}$.

See Also

J(), trdot(), cartesian()

Examples

A <- rbind(1:5, 2:6, 3:7)

# Calculate the trace of A
tr(A)

---

The Trace Inner Product of Matrices

Description

Computes the trace inner product of two matrices A and B.

Usage

trdot(A, B)

Arguments

A, B	square matrices.

Value

The trace inner product on $Mat_{n\ x\ n}(\mathbb{C})$ is defined as

$\langle A, B \rangle := tr(A^*B)$

See Also

J(), tr(), cartesian()

Examples

A <- rbind(1:5, 2:6, 3:7)
B <- rbind(7:11, 8:12, 9:13)

# Compute the trace inner product of A and B
trdot(A, B)

---

The Unitary Time Evolution Operator of a Quantum Walk

Description

The Unitary Time Evolution Operator of a Quantum Walk

Usage

unitary_matrix(object, ...)

Arguments

object	a representation of the quantum walk.
...	further arguments passed to or from other methods.

Value

The unitary time evolution operator.

See Also

mixing_matrix(), unitary_matrix.ctqwalk()

Examples

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

unitary_matrix(w, t = 2*pi) #-> unitary_matrix.ctqwalk(...)

---

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)

---