Implementing derivatives with periodic bc in QTT format.

[sakurairihito]

Pull request (PR) for the following implementations:

• First derivative using centered differences with periodic boundary conditions.
• Second-order derivative using centered differences with periodic boundary conditions.
• Linear function ( av + b ) on a uniform grid, which is for the test on derivative.

Note that the code does not depend on ITensors.jl. I would appreciate it if you could check it. Any comments and improvements on codes are welcome. If needed, other types of derivatives can be implemented. Thank you.

Reference Papers

1. Quantized Tensor Networks for Solving the Vlasov-Maxwell Equations:
   • arXiv:2311.07756
2. A Quantum-Inspired Method for Solving the Vlasov-Poisson Equations:
   • PhysRevE.106.035208

[sakurairihito]

I found an issue in my source and test code in this pull request, which is related to the mesh size (\Delta = 1 / 2^R), where (R) is the number of bits.

In the current pull requests, the derivative operator incorrectly uses (\Delta). Instead, we should apply (1/\Delta) to the derivative operator.

For instance, the function first_order_central_difference_periodic should be corrected as follows:

function first_order_central_difference_periodic(R::Int64)::Vector{Array{Float64,4}}
    c1 = 1 / 2
    t_left = zeros(1, 2, 2, 3)
    t_left[1, :, :, 1] = I() * 2^R
    t_left[1, :, :, 2] = (S(:plus) + S(:minus)) * 2^R
    t_left[1, :, :, 3] = (S(:plus) + S(:minus)) * 2^R

The function of second_order_derivative and test code should be corrected as the same way.
I would appreciate it if you consider this point.