Replace Gram-Schmidt method by QR for random Chain initialization

[jofrevalles]

In this PR, we replace the former Gram-Schmidt method for random chain initialization by a QR method, which applies the qr decomposition to some random tensors in order to get a randomly normalized right-canonized MPS.

Additionally, we benchmarked both methods and we conclude that the new method outscales the former in terms of running time, memory required and allocations produced.

Comparison

· Gram-Schmidt method:

julia> using Tenet; using BenchmarkTools

julia> @btime mps = rand(Chain, Open, State; n=50, χ=300)
  2.002 s (53812 allocations: 122.75 MiB)
MPS (inputs=0, outputs=50)


julia> @btime mpo = rand(Chain, Open, Operator; n=50, χ=200)
  1.537 s (63338 allocations: 120.70 MiB)
MPO (inputs=50, outputs=50)

· QR method:

julia> using Tenet; using BenchmarkTools

julia> @btime mps = rand(Chain, Open, State; n=50, χ=300)
  423.495 ms (10480 allocations: 101.18 MiB)
MPS (inputs=0, outputs=50)

julia> @btime mpo = rand(Chain, Open, Operator; n=50, χ=200)
  573.715 ms (28459 allocations: 109.77 MiB)
MPO (inputs=50, outputs=50)

[mofeing]

Can't you reuse the map(1:n) do i block but just changing the line A = gramschmidt!(rand(rng, T, χl, χr * p)) for A, _ = qr!(rand(rng, T, χl, χr * p)).

Is there another reason for refactoring to a for loop?

[mofeing]

Why lq! and not qr!?

[jofrevalles]

lq! is useful here since we want the Chain to be in the right-canonical form. We could do it with qr! too but it would involve some permutations and reshapes, which is much elegantly solved with lq!.

[jofrevalles]

If we changed lq! for qr! and we reshaped the physical leg with the left bond, we would get a left-canonical Chain.

[jofrevalles]

Can't you reuse the map(1:n) do i block but just changing the line A = gramschmidt!(rand(rng, T, χl, χr * p)) for A, _ = qr!(rand(rng, T, χl, χr * p)).

Is there another reason for refactoring to a for loop?

We can, but then we would have to put conditions for the first and last tensor, which seems more ugly.