Matrix Product States (MPS)

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).

fig = Figure() # 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], 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

Matrix Product Operators (MPO)

Matrix Product Operators (MPO) are the operator version of Matrix Product State (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).

fig = Figure() # 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], 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).