qwalkr: Handle Continuous-Time Quantum Walks with R

Description

Functions and tools for creating, visualizing, and investigating properties of continuous-time quantum walks, including efficient calculation of matrices such as the mixing matrix, average mixing matrix, and spectral decomposition of the Hamiltonian. E. Farhi (1997): arXiv:quant-ph/9706062v2; C. Godsil (2011) arXiv:1103.2578v3.

Author(s)

Maintainer: Vitor Marques vmrodriguespro@gmail.com [copyright holder]

See Also

Useful links:

• https://github.com/vitormarquesr/qwalkr

• https://vitormarquesr.github.io/qwalkr/

• Report bugs at https://github.com/vitormarquesr/qwalkr/issues

The All-Ones Matrix

Description

Returns the all-ones matrix of order n.

Usage

J(n)

Arguments

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)

Apply a Function to an Operator

Description

Apply a Function to an Operator

Usage

act_eigfun(object, ...)

Arguments

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Mixing Matrix of a Quantum Walk

Description

The Mixing Matrix of a Quantum Walk

Usage

mixing_matrix(object, ...)

Arguments

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

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

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

Value

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

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

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

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

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

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)