Revert "Update MPS manual"

This reverts commit 5d81735.

docs/src/manual/ansatz/mps.md

@@ -1,9 +1,8 @@
 # Matrix Product States (MPS)
 
-Matrix Product States ([`MPS`](@ref)) are a Quantum Tensor Network ansatz whose tensors are laid out in a 1D chain.
-Due to this, these networks are also known as _Tensor Trains_ in other scientific fields.
-Depending on the boundary conditions, the chains can be open or closed (i.e. periodic boundary conditions), currently
-only `Open` boundary conditions are supported in `Tenet`.
+Matrix Product States (MPS) are a Quantum Tensor Network ansatz whose tensors are laid out in a 1D chain.
+Due to this, these networks are also known as _Tensor Trains_ in other mathematical fields.
+Depending on the boundary conditions, the chains can be open or closed (i.e. periodic boundary conditions).
 
 ```@setup viz
 using Makie
@@ -19,68 +18,36 @@ using NetworkLayout
 ```@example viz
 fig = Figure() # hide
 
-open_mps = rand(MPS, n=10, χ=4) # hide
+tn_open = rand(MatrixProduct{State,Open}, n=10, χ=4) # hide
+tn_periodic = rand(MatrixProduct{State,Periodic}, n=10, χ=4) # hide
 
-plot!(fig[1,1], open_mps, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
+plot!(fig[1,1], tn_open, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
+plot!(fig[1,2], tn_periodic, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
 
 Label(fig[1,1, Bottom()], "Open") # hide
+Label(fig[1,2, Bottom()], "Periodic") # hide
 
 fig # hide
 ```
 
-### Canonical Forms
-
-A Matrix Product State ([`MPS`](@ref)) representation is not unique. Instead, a single `MPS` can be represented in different canonical forms. We can check the canonical form of an `MPS` by calling the [`form`](@ref) function.
-
-```@example
-mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
-
-form(mps)
-```
-
-Each canonical form can be useful in different situations, and the choice of the canonical form can affect the efficiency of the algorithms used to manipulate the `MPS`. Currently, `Tenet` supports the [`NonCanonical`](@ref), [`CanonicalForm`](@ref) and [`MixedCanonical`](@ref) forms.
-
-#### `NonCanonical` Form
-The default form of an `MPS` when we do not specify a canonical form.
-
-#### `CanonicalForm`
-Also known as Vidal's form. This form stores each `Tensor` of the `MPS` as a sequence of $\Gamma$ unitary tensors and $\lambda$ vectors:
-
-```math
-| \psi \rangle = \sum_{i_1, \dots, i_N} \Gamma_1^{i_1} \lambda_2^{i_2} \Gamma_2^{i_2} \dots \lambda_{N-1}^{i_{N-1}} \Gamma_{N-1}^{i_{N-1}} \lambda_N^{i_N} \Gamma_N^{i_N} | i_1, \dots, i_N \rangle \, .
-```
-This form can be obtained by calling [`canonize!`](@ref) on an `MPS`:
-
-```@example
-mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
-canonize!(mps)
-
-form(mps)
-```
-
-#### `MixedCanonical` Form
-This form stores the `Tensor`s in an `MPS` as left or right canonical wether the `Tensor` is on the left or right of the ortogonality center, which is stored in the field `orthog_center` of the `MixedCanonical` form.
-
-```@example
-mps = MPS([rand(2, 2), rand(2, 2, 2), rand(2, 2)])
-mixed_canonize!(mps, Site(2))
-
-form(mps)
-```
-
 ## Matrix Product Operators (MPO)
 
-Matrix Product Operators ([`MPO`](@ref)) are the operator version of [Matrix Product State (MPS)](#matrix-product-states-mps).
-The major difference between them is that MPOs have 2 indices per site (1 input and 1 output) while MPSs only have 1 index per site (i.e. an output). Currently, only `Open` boundary conditions are supported in `Tenet`.
+Matrix Product Operators (MPO) are the operator version of [Matrix Product State (MPS)](#matrix-product-states-mps).
+The major difference between them is that MPOs have 2 indices per site (1 input and 1 output) while MPSs only have 1 index per site (i.e. an output).
 
 ```@example viz
 fig = Figure() # hide
 
-open_mpo = rand(MatrixProduct{Operator,Open}, n=10, χ=4) # hide
+tn_open = rand(MatrixProduct{Operator,Open}, n=10, χ=4) # hide
+tn_periodic = rand(MatrixProduct{Operator,Periodic}, n=10, χ=4) # hide
 
-plot!(fig[1,1], open_mpo, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
+plot!(fig[1,1], tn_open, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
+plot!(fig[1,2], tn_periodic, layout=Spring(iterations=1000, C=0.5, seed=100)) # hide
 
 Label(fig[1,1, Bottom()], "Open") # hide
+Label(fig[1,2, Bottom()], "Periodic") # hide
 
 fig # hide
 ```
+
+In `Tenet`, the generic `MatrixProduct` ansatz implements this topology. Type variables are used to address their functionality (`State` or `Operator`) and their boundary conditions (`Open` or `Periodic`).