Utilities

convert_ITensor_to_matrix(tens, sinit, sterm)

Converts an ITensor with two indices to a matrix. The index sinit is mapped to the column and sterm is mapped to the row.

identity_MPO(sites)

Returns the identity MPO based on the given sites.

MPO_to_MPS(ρ::MPO, fbcombiner)

Convert a given MPO to an MPS by combining the two site indices on every tensor into a single one using the vector of combiner tensors.

MPS_to_MPO(ρ::MPS, fbcombiner)

Split a given MPS to an MPO by using the vector of combiner tensors.

ITensorMPS.expect(ρ::MPO, ops; kwargs...)

Extends the ITensorMPS expect function to handle density matrices in the form of MPOs.

operator_times_identity(osum::Opsum, space, iden_space, combiners)

Calculates as an MPO the the operator corresponding to $osum \otimes 1$ where the first space is given by the indices in space and the second space is given by the indices in iden_space. Finally the MPO is put together in a single forward-backward MPO using the combiners .

forward_backward_combiner(sites1::Vector, sites2::Vector)

Generates combiners for generating forward-backward paths in the Liouville space from separate forward and backward path indices.

FBSites

Stores site information for density matrix simulations.

• sites_forward: Forward path sites tagged with +
• sites_backward: Backward path sites tagged with -
• fb_combiners: ITensor combiners for combining the forward and backward sites
• sites_fb: Forward-backward sites tagged FBSite

fb_siteinds(sitetype, Nsites::Int; kwargs...)

Generates sites necessary for simulations in the forward-backward Liouville space. This function returns variables of the FBSites type, with forward sites being tagged with + and backward sites being tagged with -. The forward-backward sites are tagged as FBSite, and a combiner is also provided.

Arguments:

• sitetype: Type of the individual sites. Supports all standard ITensor site types and some custom ones defined here
• Nsites: Number of sites
• kwargs...: Other keyword arguments, most important among which are whether quantum numbers should be used

density_matrix_mps(ψ::MPS, fbsites::FBSites)

Convert a given MPS wave function defined on fbsites.sites_forward into a density matrix represented by an MPS over the forward-backward sites.

create_and_select_group(base, new_group)

Checks if new_group exists in base. Selects it if it does, else creates it.

check_or_insert_value(base, variable, value)

Inserts value into base[variable]. If it already exists, checks if the value is correct, and throws if not.

merge_into(source::String, destination::String)

Merge data from the HDF5 file at source to the one at destination.

get_BLAS_implementation()

Reports the BLAS implementation under use. The default implementation used by Julia is OpenBlas. MKL is supported through the external package MKL.jl, which needs to be installed and loaded before the loading of QuantumDynamics.jl

trapezoid(x, y; discrete::Bool=false, exec=ThreadedEx())

Returns the trapezoidal integration of y with respect to x. If discrete is set to true, then returns sum of y. exec encodes the execution paradigm and is one of seq or par.

fourier_transform(x::AbstractArray{<:Real}, f::AbstractArray; neg::Bool=true, unitary::Bool=true)

Returns the Fourier transform of f:

$F(k) = \mathcal{N}\,\int_{x_\text{min}}^{x_\text{max}} f(x)\,e^{\pm i k x}\,dx$

Arguments:

• x: range over which to do the Riemann sum
• f: function whose Fourier transform is required
• neg [Default: true]: pick the negative sign in the exponent if neg==true
• unitary [Default: true]: choose $\mathcal{N}=\frac{1}{\sqrt{2\pi}}$ if unitary==true, or $\mathcal{N}=1$ otherwise

inverse_fourier_transform(k::AbstractArray{<:Real}, F::AbstractArray; x0::Float64=0.0, neg::Bool=true, unitary::Bool=true)

Returns the Fourier transform of f:

$f(x) = \mathcal{N}\,\int_{k_\text{min}}^{k_\text{max}} F(k)\,e^{\mp i k x}\,dk$

Arguments:

• k: range over which to do the Riemann sum
• F: function whose inverse Fourier transform is required
• x0 [Default: 0.0]: the first value of the output coordinate
• neg [Default: true]: pick the positive sign in the exponent if neg==true. Notice the reversed convention because this is the inverse Fourier transform.
• unitary [Default: true]: choose $\mathcal{N}=\frac{1}{\sqrt{2\pi}}$ if unitary==true, or $\mathcal{N}=\frac{1}{2\pi}$ otherwise

commutator(A, B)

Returns the commutator A and B: AB - BA.

nh_commutator(A, B)

Returns the commutator A and B: AB - BA'.

calculate_Liouvillian(H::OpSum, forward_sites, backward_sites, combiners)

Returns the Liouvillian MPO corresponding to the Hamiltonian provided as an ITensor OpSum, to be built on the forward_sites and backward_sites, and then combined using the combiners provided.

calculate_Liouvillian(Hamiltonian::AbstractMatrix{Complex})

Returns the Liouvillian corresponding to the given Hamiltonian.

ExternalField provides an abstract interface for encoding an external field, V(t), interacting with the system through the operator, coupling_op.

density_matrix_to_vector(ρ::AbstractMatrix{<:Complex})

Returns the vector representation of the density matrix ρ compatible with the forward-backward propagators.

density_matrix_vector_to_matrix(ρvec::AbstractVector{<:Complex})

Returns the matrix form of the vector ρvec.

hash_path(states, sdim)

Returns the hashed location of a path for a system with sdim dimensions.

unhash_path(path_num::Int, ntimes::Int, sdim::Int)

Construct a path for a system with sdim dimensions, corresponding to the number path_num, with ntimes time steps.

unhash_path_blips(ntimes::Int, sdim::Int, nblips::Int)

Construct all the paths for a system with sdim dimensions with ntimes time steps and nblips blips.

apply_propagator(; propagators, ρ0, ntimes, dt)

Apply a series of ntimes propagators to an initial reduced density matrix ρ0 and return the result as a tuple of (time, ρs).

create_tls_hamiltonian(; ϵ::AbstractFloat, Δ::AbstractFloat)

Creates a two-level system Hamiltonian:

$H = \frac{ϵ}{2}σ_z - \frac{Δ}{2}σ_x$

create_nn_hamiltonian(; site_energies::AbstractVector{AbstractFloat}, couplings::AbstractVector{AbstractFloat}, periodic::Bool)

Creates a nearest neighbour Hamiltonian with the given site_energies and couplings. Periodic boundary conditions can also be used by passing true into the periodic argument.

Abstract type for encoding all the method specific numerical parameters.

Extra parameters for solving differential equations. Currently has a threshold for magnitude-based filtering. The default values are:

• reltol = 1e-10
• abstol = 1e-10
• solver = Tsit5()

Extra parameters for tensor network algorithms. Currently has an SVD cutoff, a maximum bond dimension maxdim, and the contraction algorithm. The default values are as follows:

• cutoff = 1e-8
• maxdim = 500
• algorithm = "naive"

Other options for algorithm are "densitymatrix" and "fit".