Consider the 2 x 2 Hadamard matrix together with its three rotated formsfor fun, let's call them all the Damhard matrices: 1 H = [1 -2] : H2 - 6 1] 1] H3 = [1'1 H4 = Begin a quantum circuit C with 2 qubits by placing an Hadamard gate on line 1 followed by CNOT with control on 1 and target on line 2. Now add to line 1 a black box in which Alice has placed one of the four Damhard matrices. Your task is to finish C with some gates so that by measuring both qubits, Bob can learn exactly which one Alice used. For a footnote relating to lecture, it is not possible to learn exactly in the case of the four matrices in Deutsch's problem, even if we added a third qubit to the circuit that could be entangled with the others and kept by Bob. The reason is that those four matrices are not linearly independent: U1 + Ux = UT + Uf. Thus if you have any vector u, the four vectors V1 = Uju, v2 = Uxu, vz = Uru, and v4 = Ufu resulting from them are linearly dependent. Hence the vectors W1, W2, W3, W4 you would get from later stages of the circuit are also linearly dependent. This means in particular that their nonzero entries must overlap in some indices, and any such overlap prevents 100% certainty that a single measurement will distinguish them. However, the Damhard matrices are linearly independent. Try combining them with H and/or the Pauli matrices, remembering also that multiplication by a scalar unit constant such as 1 or i never changes any measurement, so you can disregard it. =