Prove the following:

Let U be a 22 unitary matrix. The controlled- U is a two-qubit gate, written C(U), which when applied to qubit registers q1,q2, is defined by: C(U)[q1,q2]k1kq1kq2kn=k1kq1(Ukq1kq2)kn where q1 is the control qubit and q2 is the target qubit, and where every ki{0,1}. The matrix representation of C(U) for application to two qubits is: C(U)=(I00U) where I is the 22 identity matrix and 0 is the 22 matrix in which every entry is 0 . Notice that CNOT =C(X), where X is one of the Pauli matrices. Define SWAP to be the two-qubit gate that swaps the states of two qubit registers: SWAP[q1,q2]k1kq1kq2kn=k1kq2kq1kn where every ki{0,1}. C(X)[p,q]X[p]C(X)[p,q]=X[p]X[q] Let U be a 22 unitary matrix. The controlled- U is a two-qubit gate, written C(U), which when applied to qubit registers q1,q2, is defined by: C(U)[q1,q2]k1kq1kq2kn=k1kq1(Ukq1kq2)kn where q1 is the control qubit and q2 is the target qubit, and where every ki{0,1}. The matrix representation of C(U) for application to two qubits is: C(U)=(I00U) where I is the 22 identity matrix and 0 is the 22 matrix in which every entry is 0 . Notice that CNOT =C(X), where X is one of the Pauli matrices. Define SWAP to be the two-qubit gate that swaps the states of two qubit registers: SWAP[q1,q2]k1kq1kq2kn=k1kq2kq1kn where every ki{0,1}. C(X)[p,q]X[p]C(X)[p,q]=X[p]X[q]