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Answer to the second question: Here is the circuit for measuring in $M{{({{\theta }_{k}})}_{\pm }}=\left\{ 1/\sqrt{2}\left( |0\rangle \pm {{e}^{-i{{\theta }_{k}}}}|1\rangle \right) \right\}$ basis. I assume here that $\theta_k$ is given: circuit.u1(theta_k, q[0]) # q[0] is one of the qubits circuit.h(q[0]) circuit.measure(q[0], c[0]) #c[0] is a ...