MPS and MPO

MPS

A finite size matrix product state type. Keeps track of the orthogonality center.

MPO

A finite size matrix product operator type. Keeps track of the orthogonality center.

MPS(N::Int)

Construct an MPS with N sites with default constructed ITensors.

MPS([::Type{ElT} = Float64, ]sites; linkdims=1)

Construct an MPS filled with Empty ITensors of type ElT from a collection of indices.

Optionally specify the link dimension with the keyword argument linkdims, which by default is 1.

In the future we may generalize linkdims to allow specifying each individual link dimension as a vector, and additionally allow specifying quantum numbers.

randomMPS(sites::Vector{<:Index}; linkdims=1)

Construct a random MPS with link dimension linkdim of type Float64.

randomMPS(::Type{ElT<:Number}, sites::Vector{<:Index}; linkdims=1)

Construct a random MPS with link dimension linkdims of type ElT.

randomMPS(sites::Vector{<:Index}, state; linkdims=1)

Construct a real, random MPS with link dimension linkdims, made by randomizing an initial product state specified by state. This version of randomMPS is necessary when creating QN-conserving random MPS (consisting of QNITensors). The initial state array provided determines the total QN of the resulting random MPS.

MPS(sites::Vector{<:Index},states)

Construct a product state MPS having site indices sites, and which corresponds to the initial state given by the array states. The states array may consist of either an array of integers or strings, as recognized by the state function defined for the relevant Index tag type.

Examples

N = 10
sites = siteinds("S=1/2",N)
states = [isodd(n) ? "Up" : "Dn" for n=1:N]
psi = MPS(sites,states)

MPS(::Type{T},
    sites::Vector{<:Index},
    states::Union{Vector{String},
                  Vector{Int},
                  String,
                  Int})

Construct a product state MPS of element type T, having site indices sites, and which corresponds to the initial state given by the array states. The input states may be an array of strings or an array of ints recognized by the state function defined for the relevant Index tag type. In addition, a single string or int can be input to create a uniform state.

Examples

N = 10
sites = siteinds("S=1/2", N)
states = [isodd(n) ? "Up" : "Dn" for n=1:N]
psi = MPS(ComplexF64, sites, states)
phi = MPS(sites, "Up")

MPS(ivals::Vector{<:Pair{<:Index}})

Construct a product state MPS with element type Float64 and nonzero values determined from the input IndexVals.

MPS(::Type{T<:Number}, ivals::Vector{<:Pair{<:Index}})

Construct a product state MPS with element type T and nonzero values determined from the input IndexVals.

MPO(N::Int)

Make an MPO of length N filled with default ITensors.

MPO([::Type{ElT} = Float64}, ]sites, ops::Vector{String})

Make an MPO with pairs of sites s[i] and s[i]' and operators ops on each site.

MPO([::Type{ElT} = Float64, ]sites, op::String)

Make an MPO with pairs of sites s[i] and s[i]' and operator op on every site.

copy(::MPS)
copy(::MPO)

Make a shallow copy of an MPS or MPO. By shallow copy, it means that a new MPS/MPO is returned, but the data of the tensors are still shared between the returned MPS/MPO and the original MPS/MPO.

Therefore, replacing an entire tensor of the returned MPS/MPO will not modify the input MPS/MPO, but modifying the data of the returned MPS/MPO will modify the input MPS/MPO.

Use deepcopy for an alternative that copies the ITensors as well.

Examples

julia> using ITensors

julia> s = siteinds("S=1/2", 3);

julia> M1 = randomMPS(s; linkdims=3);

julia> norm(M1)
0.9999999999999999

julia> M2 = copy(M1);

julia> M2[1] *= 2;

julia> norm(M1)
0.9999999999999999

julia> norm(M2)
1.9999999999999998

julia> M3 = copy(M1);

julia> M3[1] .*= 3; # Modifies the tensor data

julia> norm(M1)
3.0000000000000004

julia> norm(M3)
3.0000000000000004

deepcopy(::MPS)
deepcopy(::MPO)

Make a deep copy of an MPS or MPO. By deep copy, it means that a new MPS/MPO is returned that doesn't share any data with the input MPS/MPO.

Therefore, modifying the resulting MPS/MPO will note modify the original MPS/MPO.

Use copy for an alternative that performs a shallow copy that avoids copying the ITensor data.

Examples

julia> using ITensors

julia> s = siteinds("S=1/2", 3);

julia> M1 = randomMPS(s; linkdims=3);

julia> norm(M1)
1.0

julia> M2 = deepcopy(M1);

julia> M2[1] .*= 2; # Modifies the tensor data

julia> norm(M1)
1.0

julia> norm(M2)
2.0

julia> M3 = copy(M1);

julia> M3[1] .*= 3; # Modifies the tensor data

julia> norm(M1)
3.0

julia> norm(M3)
3.0

eltype(m::MPS)
eltype(m::MPO)

The element type of the MPS/MPO. Always returns ITensor.

For the element type of the ITensors of the MPS/MPO, use promote_itensor_eltype.

flux(M::MPS)

flux(M::MPO)

totalqn(M::MPS)

totalqn(M::MPO)

For an MPS or MPO which conserves quantum numbers, compute the total QN flux. For a tensor network such as an MPS or MPO, the flux is the sum of fluxes of each of the tensors in the network. The name totalqn is an alias for flux.

hasqns(M::MPS)

hasqns(M::MPO)

Return true if the MPS or MPO has tensors which carry quantum numbers.

length(::MPS/MPO)

The number of sites of an MPS/MPO.

maxlinkdim(M::MPS)
maxlinkdim(M::MPO)

Get the maximum link dimension of the MPS or MPO.

The minimum this will return is 1, even if there are no link indices.

siteinds(commoninds, A::MPO, B::MPS, j::Integer; kwargs...)
siteinds(commonind, A::MPO, B::MPO, j::Integer; kwargs...)

Get the site index (or indices) of the jth MPO tensor of A that is shared with MPS/MPO B.

siteinds(uniqueinds, A::MPO, B::MPS, j::Integer; kwargs...)
siteinds(uniqueind, A::MPO, B::MPS, j::Integer; kwargs...)

Get the site index (or indices) of MPO A that is unique to A (not shared with MPS/MPO B).

findsite(M::Union{MPS, MPO}, is)

Return the first site of the MPS or MPO that has at least one Index in common with the Index or collection of indices is.

To find all sites with common indices with is, use the findsites function.

Examples

s = siteinds("S=1/2", 5)
ψ = randomMPS(s)
findsite(ψ, s[3]) == 3
findsite(ψ, (s[3], s[4])) == 3

M = MPO(s)
findsite(M, s[4]) == 4
findsite(M, s[4]') == 4
findsite(M, (s[4]', s[4])) == 4
findsite(M, (s[4]', s[3])) == 3

findsites(M::Union{MPS, MPO}, is)

Return the sites of the MPS or MPO that have indices in common with the collection of site indices is.

Examples

s = siteinds("S=1/2", 5)
ψ = randomMPS(s)
findsites(ψ, s[3]) == [3]
findsites(ψ, (s[4], s[1])) == [1, 4]

M = MPO(s)
findsites(M, s[4]) == [4]
findsites(M, s[4]') == [4]
findsites(M, (s[4]', s[4])) == [4]
findsites(M, (s[4]', s[3])) == [3, 4]

firstsiteinds(M::MPO; kwargs...)

Get a Vector of the first site Index found on each site of M.

By default, it finds the first site Index with prime level 0.

linkind(M::MPS, j::Integer)
linkind(M::MPO, j::Integer)

Get the link or bond Index connecting the MPS or MPO tensor on site j to site j+1.

If there is no link Index, return nothing.

siteind(M::MPS, j::Int; kwargs...)

Get the first site Index of the MPS. Return nothing if none is found.

siteind(::typeof(first), M::Union{MPS,MPO}, j::Integer; kwargs...)

Return the first site Index found on the MPS or MPO (the first Index unique to the jth MPS/MPO tensor).

You can choose different filters, like prime level and tags, with the kwargs.

siteinds(M::MPS)
siteinds(::typeof(first), M::MPS)

Get a vector of the first site Index found on each tensor of the MPS.

siteinds(::typeof(only), M::MPS)

Get a vector of the only site Index found on each tensor of the MPS. Errors if more than one is found.

siteinds(::typeof(all), M::MPS)

Get a vector of the all site Indices found on each tensor of the MPS. Returns a Vector of IndexSets.

siteind(M::MPO, j::Int; plev = 0, kwargs...)

Get the first site Index of the MPO found, by default with prime level 0.

siteinds(M::MPO; kwargs...)

Get a Vector of IndexSets of all the site indices of M.

siteinds(M::Union{MPS, MPO}}, j::Integer; kwargs...)

Return the site Indices found of the MPO or MPO at the site j as an IndexSet.

Optionally filter prime tags and prime levels with keyword arguments like plev and tags.

prime[!](M::MPS, args...; kwargs...)
prime[!](M::MPO, args...; kwargs...)

Apply prime to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

prime[!](siteinds, M::MPS, args...; kwargs...)
prime[!](siteinds, M::MPO, args...; kwargs...)

Apply prime to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

prime[!](linkinds, M::MPS, args...; kwargs...)
prime[!](linkinds, M::MPO, args...; kwargs...)

Apply prime to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

prime[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
prime[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply prime to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

prime[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply prime to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

setprime[!](M::MPS, args...; kwargs...)
setprime[!](M::MPO, args...; kwargs...)

Apply setprime to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

setprime[!](siteinds, M::MPS, args...; kwargs...)
setprime[!](siteinds, M::MPO, args...; kwargs...)

Apply setprime to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

setprime[!](linkinds, M::MPS, args...; kwargs...)
setprime[!](linkinds, M::MPO, args...; kwargs...)

Apply setprime to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

setprime[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
setprime[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply setprime to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

setprime[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply setprime to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

noprime[!](M::MPS, args...; kwargs...)
noprime[!](M::MPO, args...; kwargs...)

Apply noprime to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

noprime[!](siteinds, M::MPS, args...; kwargs...)
noprime[!](siteinds, M::MPO, args...; kwargs...)

Apply noprime to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

noprime[!](linkinds, M::MPS, args...; kwargs...)
noprime[!](linkinds, M::MPO, args...; kwargs...)

Apply noprime to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

noprime[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
noprime[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply noprime to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

noprime[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply noprime to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

addtags[!](M::MPS, args...; kwargs...)
addtags[!](M::MPO, args...; kwargs...)

Apply addtags to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

addtags[!](siteinds, M::MPS, args...; kwargs...)
addtags[!](siteinds, M::MPO, args...; kwargs...)

Apply addtags to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

addtags[!](linkinds, M::MPS, args...; kwargs...)
addtags[!](linkinds, M::MPO, args...; kwargs...)

Apply addtags to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

addtags[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
addtags[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply addtags to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

addtags[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply addtags to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

removetags[!](M::MPS, args...; kwargs...)
removetags[!](M::MPO, args...; kwargs...)

Apply removetags to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

removetags[!](siteinds, M::MPS, args...; kwargs...)
removetags[!](siteinds, M::MPO, args...; kwargs...)

Apply removetags to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

removetags[!](linkinds, M::MPS, args...; kwargs...)
removetags[!](linkinds, M::MPO, args...; kwargs...)

Apply removetags to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

removetags[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
removetags[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply removetags to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

removetags[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply removetags to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

replacetags[!](M::MPS, args...; kwargs...)
replacetags[!](M::MPO, args...; kwargs...)

Apply replacetags to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

replacetags[!](siteinds, M::MPS, args...; kwargs...)
replacetags[!](siteinds, M::MPO, args...; kwargs...)

Apply replacetags to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

replacetags[!](linkinds, M::MPS, args...; kwargs...)
replacetags[!](linkinds, M::MPO, args...; kwargs...)

Apply replacetags to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

replacetags[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
replacetags[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply replacetags to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

replacetags[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply replacetags to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

settags[!](M::MPS, args...; kwargs...)
settags[!](M::MPO, args...; kwargs...)

Apply settags to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

settags[!](siteinds, M::MPS, args...; kwargs...)
settags[!](siteinds, M::MPO, args...; kwargs...)

Apply settags to all site indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

settags[!](linkinds, M::MPS, args...; kwargs...)
settags[!](linkinds, M::MPO, args...; kwargs...)

Apply settags to all link indices of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors.

settags[!](siteinds, commoninds, M1::MPO, M2::MPS, args...; kwargs...)
settags[!](siteinds, commoninds, M1::MPO, M2::MPO, args...; kwargs...)

Apply settags to the site indices that are shared by M1 and M2.

Returns new MPSs/MPOs. The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

settags[!](siteinds, uniqueinds, M1::MPO, M2::MPS, args...; kwargs...)

Apply settags to the site indices of M1 that are not shared with M2. Returns new MPSs/MPOs.

The ITensors of the MPSs/MPOs will be a view of the storage of the original ITensors.

expect(psi::MPS,ops::AbstractString...;kwargs...)

Given an MPS psi and an operator name, returns a vector of the expected value of the operator on each site of the MPS. If multiple operator names are provided, returns a tuple of expectation value vectors.

Optional Keyword Arguments

• site_range = 1:length(psi): compute expected values only for sites in the given range

Examples

N = 10

s = siteinds("S=1/2",N)
psi = randomMPS(s; linkdims=8)
Z = expect(psi,"Sz";site_range=2:6)

s = siteinds("Electron",N)
psi = randomMPS(s; linkdims=8)
dens = expect(psi,"Ntot")
updens,dndens = expect(psi,"Nup","Ndn")

correlation_matrix(psi::MPS,
                   Op1::AbstractString,
                   Op2::AbstractString;
                   kwargs...)

Given an MPS psi and two strings denoting operators (as recognized by the op function), computes the two-point correlation function matrix C[i,j] = <psi| Op1i Op2j |psi> using efficient MPS techniques. Returns the matrix C.

Optional Keyword Arguments

• site_range = 1:length(psi): compute correlations only for sites in the given range

For a correlation matrix of size NxN and an MPS of typical bond dimension m, the scaling of this algorithm is N^2*m^3.

Examples

N = 30
m = 4

s = siteinds("S=1/2",N)
psi = randomMPS(s; linkdims=m)
Czz = correlation_matrix(psi,"Sz","Sz")

s = siteinds("Electron",N; conserve_qns=true)
psi = randomMPS(s, n->isodd(n) ? "Up" : "Dn"; linkdims=m)
Cuu = correlation_matrix(psi,"Cdagup","Cup";site_range=2:8)

dag[!](M::MPS, args...; kwargs...)
dag[!](M::MPO, args...; kwargs...)

Apply dag to all ITensors of an MPS/MPO, returning a new MPS/MPO.

The ITensors of the MPS/MPO will be a view of the storage of the original ITensors. Alternatively apply the function in-place.

dense(::MPS/MPO)

Given an MPS (or MPO), return a new MPS (or MPO) having called dense on each ITensor to convert each tensor to use dense storage and remove any QN or other sparse structure information, if it is not dense already.

movesite(::Union{MPS, MPO}, n1n2::Pair{Int, Int})

Create a new MPS/MPO where the site at n1 is moved to n2, for a pair n1n2 = n1 => n2.

This is done with a series a pairwise swaps, and can introduce a lot of entanglement into your state, so use with caution.

orthogonalize!(M::MPS, j::Int; kwargs...)
orthogonalize(M::MPS, j::Int; kwargs...)

orthogonalize!(M::MPO, j::Int; kwargs...)
orthogonalize(M::MPO, j::Int; kwargs...)

Move the orthogonality center of the MPS to site j. No observable property of the MPS will be changed, and no truncation of the bond indices is performed. Afterward, tensors 1,2,...,j-1 will be left-orthogonal and tensors j+1,j+2,...,N will be right-orthogonal.

Either modify in-place with orthogonalize! or out-of-place with orthogonalize.

replacebond!(M::MPS, b::Int, phi::ITensor; kwargs...)

Factorize the ITensor phi and replace the ITensors b and b+1 of MPS M with the factors. Choose the orthogonality with ortho="left"/"right".

sample(m::MPS)

Given a normalized MPS m with orthocenter(m)==1, returns a Vector{Int} of length(m) corresponding to one sample of the probability distribution defined by squaring the components of the tensor that the MPS represents

sample!(m::MPS)

Given a normalized MPS m, returns a Vector{Int} of length(m) corresponding to one sample of the probability distribution defined by squaring the components of the tensor that the MPS represents. If the MPS does not have an orthogonality center, orthogonalize!(m,1) will be called before computing the sample.

sample(M::MPO)

Given a normalized MPO M, returns a Vector{Int} of length(M) corresponding to one sample of the probability distribution defined by the MPO, treating the MPO as a density matrix.

The MPO M should have an (approximately) positive spectrum.

swapbondsites(ψ::Union{MPS, MPO}, b::Integer; kwargs...)

Swap the sites b and b+1.

truncate!(M::MPS; kwargs...)
truncate!(M::MPO; kwargs...)

Perform a truncation of all bonds of an MPS/MPO, using the truncation parameters (cutoff,maxdim, etc.) provided as keyword arguments.

product(As::Vector{<:ITensor}, M::Union{MPS, MPO}; <keyword arguments>)
apply([...])

Apply the ITensors As to the MPS or MPO M, treating them as gates or matrices from pairs of prime or unprimed indices.

Examples

Apply one-site gates to an MPS:

N = 3

ITensors.op(::OpName"σx", ::SiteType"S=1/2", s::Index) =
  2*op("Sx", s)

ITensors.op(::OpName"σz", ::SiteType"S=1/2", s::Index) =
  2*op("Sz", s)

# Make the operator list.
os = [("σx", n) for n in 1:N]
append!(os, [("σz", n) for n in 1:N])

@show os

s = siteinds("S=1/2", N)
gates = ops(os, s)

# Starting state |↑↑↑⟩
ψ0 = MPS(s, "↑")

# Apply the gates.
ψ = apply(gates, ψ0; cutoff = 1e-15)

# Test against exact (full) wavefunction
prodψ = apply(gates, prod(ψ0))
@show prod(ψ) ≈ prodψ

# The result is:
# σz₃ σz₂ σz₁ σx₃ σx₂ σx₁ |↑↑↑⟩ = -|↓↓↓⟩
@show inner(ψ, MPS(s, "↓")) == -1

Apply nonlocal two-site gates and one-site gates to an MPS:

# 2-site gate
function ITensors.op(::OpName"CX", ::SiteType"S=1/2", s1::Index, s2::Index)
  mat = [1 0 0 0
         0 1 0 0
         0 0 0 1
         0 0 1 0]
  return itensor(mat, s2', s1', s2, s1)
end

os = [("CX", 1, 3), ("σz", 3)]

@show os

# Start with the state |↓↑↑⟩
ψ0 = MPS(s, n -> n == 1 ? "↓" : "↑")

# The result is:
# σz₃ CX₁₃ |↓↑↑⟩ = -|↓↑↓⟩
ψ = apply(ops(os, s), ψ0; cutoff = 1e-15)
@show inner(ψ, MPS(s, n -> n == 1 || n == 3 ? "↓" : "↑")) == -1

Perform TEBD-like time evolution:

# Define the nearest neighbor term `S⋅S` for the Heisenberg model
function ITensors.op(::OpName"expS⋅S", ::SiteType"S=1/2",
                     s1::Index, s2::Index; τ::Number)
  O = 0.5 * op("S+", s1) * op("S-", s2) +
      0.5 * op("S-", s1) * op("S+", s2) +
            op("Sz", s1) * op("Sz", s2)
  return exp(τ * O)
end

τ = -0.1im
os = [("expS⋅S", (1, 2), (τ = τ,)),
      ("expS⋅S", (2, 3), (τ = τ,))]
ψ0 = MPS(s, n -> n == 1 ? "↓" : "↑")
expτH = ops(os, s)
ψτ = apply(expτH, ψ0)

dot(A::MPS, B::MPS; make_inds_match = true)
inner(A::MPS, B::MPS; make_inds_match = true)

dot(A::MPO, B::MPO)
inner(A::MPO, B::MPO)

Compute the inner product <A|B>. If A and B are MPOs, computes the Frobenius inner product.

If make_inds_match = true, the function attempts to make the site indices match before contracting (so for example, the inputs can have different site indices, as long as they have the same dimensions or QN blocks).

For now, make_inds_match is only supported for MPSs.

See also logdot/loginner.

logdot(A::MPS, B::MPS; make_inds_match = true)
loginner(A::MPS, B::MPS; make_inds_match = true)

logdot(A::MPO, B::MPO)
loginner(A::MPO, B::MPO)

Compute the logarithm of the inner product <A|B>. If A and B are MPOs, computes the logarithm of the Frobenius inner product.

This is useful for larger MPS/MPO, where in the limit of large numbers of sites the inner product can diverge or approach zero.

If make_inds_match = true, the function attempts to make the site indices match before contracting (so for example, the inputs can have different site indices, as long as they have the same dimensions or QN blocks).

For now, make_inds_match is only supported for MPSs.

norm(A::MPS)
norm(A::MPO)

Compute the norm of the MPS or MPO.

If the MPS or MPO has a well defined orthogonality center, this reduces to the norm of the orthogonality center tensor. Otherwise, it computes the norm with the full inner product of the MPS/MPO with itself.

See also lognorm.

lognorm(A::MPS)
lognorm(A::MPO)

Compute the logarithm of the norm of the MPS or MPO.

This is useful for larger MPS/MPO that are not gauged, where in the limit of large numbers of sites the norm can diverge or approach zero.

See also norm, logdot.

+(A::MPS/MPO...; kwargs...)
add(A::MPS/MPO...; kwargs...)

Add arbitrary numbers of MPS/MPO with each other, with some optional truncation.

A cutoff of 1e-15 is used by default, and in general users should set their own cutoff for their particular application.

In the future we will give an interface for returning the truncation error.

Examples

N = 10

s = siteinds("S=1/2", N; conserve_qns = true)

state = n -> isodd(n) ? "↑" : "↓"
ψ₁ = randomMPS(s, state, 2)
ψ₂ = randomMPS(s, state, 2)
ψ₃ = randomMPS(s, state, 2)

ψ = +(ψ₁, ψ₂; cutoff = 1e-8)

# Can use:
#
# ψ = ψ₁ + ψ₂
#
# but generally you want to set a custom `cutoff` and `maxdim`.

println()
@show inner(ψ, ψ)
@show inner(ψ₁, ψ₂) + inner(ψ₁, ψ₂) + inner(ψ₂, ψ₁) + inner(ψ₂, ψ₂)

# Computes ψ₁ + 2ψ₂
ψ = ψ₁ + 2ψ₂

println()
@show inner(ψ, ψ)
@show inner(ψ₁, ψ₁) + 2 * inner(ψ₁, ψ₂) + 2 * inner(ψ₂, ψ₁) + 4 * inner(ψ₂, ψ₂)

# Computes ψ₁ + 2ψ₂ + ψ₃
ψ = ψ₁ + 2ψ₂ + ψ₃

println()
@show inner(ψ, ψ)
@show inner(ψ₁, ψ₁) + 2 * inner(ψ₁, ψ₂) + inner(ψ₁, ψ₃) +
      2 * inner(ψ₂, ψ₁) + 4 * inner(ψ₂, ψ₂) + 2 * inner(ψ₂, ψ₃) +
      inner(ψ₃, ψ₁) + 2 * inner(ψ₃, ψ₂) + inner(ψ₃, ψ₃)

contract(ψ::MPS, A::MPO; kwargs...) -> MPS
*(::MPS, ::MPO; kwargs...) -> MPS

contract(A::MPO, ψ::MPS; kwargs...) -> MPS
*(::MPO, ::MPS; kwargs...) -> MPS

Contract the MPO A with the MPS ψ, returning an MPS with the unique site indices of the MPO.

Choose the method with the method keyword, for example "densitymatrix" and "naive".

Keywords

• cutoff::Float64=1e-13: the cutoff value for truncating the density matrix eigenvalues. Note that the default is somewhat arbitrary and subject to change, in general you should set a cutoff value.
• maxdim::Int=maxlinkdim(A) * maxlinkdim(ψ)): the maximal bond dimension of the results MPS.
• mindim::Int=1: the minimal bond dimension of the resulting MPS.
• normalize::Bool=false: whether or not to normalize the resulting MPS.
• method::String="densitymatrix": the algorithm to use for the contraction.

contract(A::MPO, B::MPO; kwargs...) -> MPO
*(::MPO, ::MPO; kwargs...) -> MPO

Contract the MPO A with the MPO B, returning an MPO with the site indices that are not shared between A and B.

Keywords

• cutoff::Float64=1e-13: the cutoff value for truncating the density matrix eigenvalues. Note that the default is somewhat arbitrary and subject to change, in general you should set a cutoff value.
• maxdim::Int=maxlinkdim(A) * maxlinkdim(B)): the maximal bond dimension of the results MPS.
• mindim::Int=1: the minimal bond dimension of the resulting MPS.

outer(x::MPS, y::MPS; <keyword argument>) -> MPO

Compute the outer product of MPS x and MPS y, returning an MPO approximation.

Note that y will be conjugated, and the site indices of x will be primed.

In Dirac notation, this is the operation |x⟩⟨y|.

The keyword arguments determine the truncation, and accept the same arguments as contract(::MPO, ::MPO; kw...).

See also product, contract.

projector(x::MPS; <keyword argument>) -> MPO

Computes the projector onto the state x. In Dirac notation, this is the operation |x⟩⟨x|/|⟨x|x⟩|².

Use keyword arguments to control the level of truncation, which are the same as those accepted by contract(::MPO, ::MPO; kw...).

Keywords

• normalize::Bool=true: whether or not to normalize the input MPS before forming the projector. If normalize==false and the input MPS is not already normalized, this function will not output a proper project, and simply outputs outer(x, x) = |x⟩⟨x|, i.e. the projector scaled by norm(x)^2.
• truncation keyword arguments accepted by contract(::MPO, ::MPO; kw...).

See also outer, contract.