u = [0. +0.1j 0.1+0.1j 0.2+0.1j 0.3+0.1j]
M = [[0. +0.j  0. +0.1j 0. +0.2j 0. +0.3j]
 [0.1+0.j  0.1+0.1j 0.1+0.2j 0.1+0.3j]
 [0.2+0.j  0.2+0.1j 0.2+0.2j 0.2+0.3j]
 [0.3+0.j  0.3+0.1j 0.3+0.2j 0.3+0.3j]]
v = [-0.06+0.14j  0.  +0.18j  0.06+0.22j  0.12+0.26j]
u.v = (-0.03200000000000002+0.15200000000000002j)
trace(M) = (0.6+0.6j)
transpose(M) = [[0. +0.j  0.1+0.j  0.2+0.j  0.3+0.j ]
 [0. +0.1j 0.1+0.1j 0.2+0.1j 0.3+0.1j]
 [0. +0.2j 0.1+0.2j 0.2+0.2j 0.3+0.2j]
 [0. +0.3j 0.1+0.3j 0.2+0.3j 0.3+0.3j]]
quantum circuit:
     ┌───┐      ░ ┌───┐ ░ ┌─┐   
q_0: ┤ H ├──■───░─┤ H ├─░─┤M├───
     └───┘┌─┴─┐ ░ ├───┤ ░ └╥┘┌─┐
q_1: ─────┤ X ├─░─┤ H ├─░──╫─┤M├
          └───┘ ░ └───┘ ░  ║ └╥┘
c: 2/══════════════════════╩══╩═
                           0  1 
circuit output:
{'counts': {'0x0': 53, '0x3': 47}}