# Please do not post
# Omar Aceval Garcia, due 2/09/2025. Assignment 5


# Gives a random state vector:
RandomState := proc(n,K) local ra,m,i,V :
ra := rand(-K..K):
V := [seq(ra() + I*ra(), i=1..n)]:
m := sqrt(add(V[i]^2, i=1..n)):
for i from 1 to n do
  V[i] := V[i]/m:
od:
return V:
end:

# I tried verifying that RHM(3,10) gives real eigenvalues by using EV() but the outputs are algebraically complicated 
# and simplify() and rationalize() didn't seem to help maple properly show these are real values. But evalf() showed that 
# numerically, each component had imaginary value of order around 10^-9, hence these are probably real.

# For a matrix A made by RandomState, the following code takes every pair of two vectors in EVC(A)[2] (the list of eigenvectors)
# and verifies that their inner product is zero. 
VectorCheck := proc(evc) local i,j,n,vecs:
vecs := evc[2]:
n := nops(vecs):
return [seq(seq(IP(vecs[i],vecs[j]) , j=(i+1)..n), i=1..n)]
end:

# Using the above I tried evalf(VectorCheck(EVC(RHM(3,10)))) and got a vector where each entry has real and imaginary components
# which are of order 10^-10, aka a zero vector for every inner product of eigenvectors.


GeneralPauli:= proc(n) local nx,ny,nz:
nx:= n[1]:
ny:= n[2]:
nz:= n[3]:
return [[nz, nx - I*ny],[nx + I*ny, -nz]]:
end:

# Experimenting with this function and using EVC() I keep getting Unitary matrices with eigenvalues 1 and -1.