In [1]:
using LinearAlgebra
In [2]:
const m = 2
const M = 2^m  ;
In [3]:
QFT⁻¹(f ::Function, ξ ::Int) :: Complex{Float64} = sum( x -> exp(-2π*im * x*ξ/M) * f(x) , 0:M-1) / sqrt(M)  ;
In [4]:
T = [ 1 0  0  1
      1 0  0 -1
      0 1  1  0
      0 1 -1  0 ]' / sqrt(2)

H   =  T * [  0. 0.     0.          0.
              0. 1/4    0.          0.
              0. 0.     1/4+1/32    0.
              0. 0.     0.          1/13 ] * T'

eigvecs(H)
Out[4]:
4×4 Array{Float64,2}:
 0.707107   0.0        0.707107  0.0     
 0.0       -0.707107   0.0       0.707107
 0.0        0.707107   0.0       0.707107
 0.707107   0.0       -0.707107  0.0
In [14]:
Λ = eigvals(H)
println("Eigenvalues:")
println("λ₁ = ", bitstring(Int32(round( Λ[1] * 2^32 ))), "  = 0")
println("λ₂ = ", bitstring(Int32(round( Λ[2] * 2^32 ))), "  = 1/13")
println("λ₃ = ", bitstring(Int32(round( Λ[3] * 2^32 ))), "  = 1/4" )
println("λ₄ = ", bitstring(Int32(round( Λ[4] * 2^32 ))), "  = 1/4+1/32" )

Eigenvalues:
λ₁ = 00000000000000000000000000000000  = 0
λ₂ = 00010011101100010011101100010100  = 1/13
λ₃ = 01000000000000000000000000000000  = 1/4
λ₄ = 01001000000000000000000000000000  = 1/4+1/32