I am writing a code that gives me tensor product of the required Pauli matrices, and I am doing this using kron function.

using StaticArrays
#%% We are definig the Pauli matrices here.
σˣ=SA[0 1.0; 1 0] #\sigma^x
σᶻ=SA[1.0 0; 0 -1] #\sigma^y
σʸ=-im*σᶻ*σˣ #\sigma^z
#println(σʸ,"\n",σᶻ,"\n",σˣ)
σvec=[σˣ,σʸ,σᶻ]

the above code stores the Pauli matrices

function σTensor(A,n::Int)
include("Pauli_matrices.jl")
#= Inputs: number of two level systems n (= number of tensor products + 1)
size of the input array is N and is of the form [1,1,3,2] for N=4
1 represents x, 2 represents y and 3 represents z
=#
ret=σvec[A[1]]
for i in 2:n
ret=kron(ret,σvec[A[i]])
end
return ret
end

As usual, the question with Kronecker products is: do you really need to build the matrix explicitly? Or can you consider it as a lazy operator in your code?

here A=[1,2,3]
I have changed the code in the following way as you have suggested

include("PauliMatrices.jl")
function σTensor(A::Array,n::Int)
#= Inputs: number of two level systems n (= number of tensor products + 1)
size of the input array is N and is of the form [1,1,3,2] for N=4
1 represents x, 2 represents y and 3 represents z
=#
ret=σvec[A[1]]
for i in 2:n
ret=kron(ret,σvec[A[i]])
end
return ret
end

and I am getting the following results for different A

The usual approach is to just use SparseArrays.jl. I don’t think that any further compiler optimization (i.e. StaticArrays.jl, type-level information) is going to help if you consider > 7 sites.

julia> using SparseArrays, LinearAlgebra, BenchmarkTools
julia> σx = sparse([0 1; 1.0 0])
2×2 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
⋅ 1.0
1.0 ⋅
julia> σy = sparse([0 -im; im 0.0])
2×2 SparseMatrixCSC{ComplexF64, Int64} with 2 stored entries:
⋅ 0.0-1.0im
0.0+1.0im ⋅
julia> σz = sparse([1 0; 0 -1.0])
2×2 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
1.0 ⋅
⋅ -1.0
julia> const σvec = [σx, σy, σz] # note the const! It is needed for performance
3-element Vector{SparseMatrixCSC{Tv, Int64} where Tv}:
sparse([2, 1], [1, 2], [1.0, 1.0], 2, 2)
sparse([2, 1], [1, 2], ComplexF64[0.0 + 1.0im, 0.0 - 1.0im], 2, 2)
sparse([1, 2], [1, 2], [1.0, -1.0], 2, 2)
julia> function σTensor(A)
n = length(A) # why did you pass it as parameter?
ret=σvec[A[1]]
for i in 2:n
ret=kron(ret,σvec[A[i]])
end
return ret
end
julia> @btime σTensor([1,2,3], 3)
539.524 ns (13 allocations: 1.08 KiB)
8×8 SparseMatrixCSC{ComplexF64, Int64} with 8 stored entries:
⋅ ⋅ ⋅ … ⋅ 0.0-1.0im ⋅
⋅ ⋅ ⋅ ⋅ ⋅ -0.0+1.0im
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ -0.0-1.0im ⋅ ⋅
⋅ ⋅ 0.0-1.0im ⋅ ⋅ ⋅
⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
0.0+1.0im ⋅ ⋅ ⋅ ⋅ ⋅
⋅ -0.0-1.0im ⋅ ⋅ ⋅ ⋅

The thing is I don’t know and I am new to this lazy matrices. How will I determine this?
I was using the Static Arrays because I read that they are quicker than normal Array. My end goal is to implement time evolution of a quantum system, as efficiently as possible.
the problem with quantum systems if there are n qubits in the system, the operators in the system are 2^nx2^n complex matrix.

If you want to do time evolution of quantum systems, then a lazy approach will not be helpful. In the simplest case you want to construct your Hamiltonian and then use a eigen decomposition to compute the matrix exponentials efficiently. This will work for up to ~12-14 spins reasonably fast and easy. Beyond that you will need to employ more and more sophisticated tricks

Shameless plug: If you want to study simple spin models, you might want to have a look at my library SpinModels.jl with which you can construct Hamiltonians with single and two-body terms rather efficiently (It is not in the registry for now, so you need to add it using Pkg.add(;url="https://github.com/abraemer/SpinModels.jl")).

Thank you for your answer! . I have implemented your code and it works great it brought down the allocations from 1000 to 10. I definitely will check out your SpinModels.jl. Thanks again.