Circuit identities Let U be a 2 2 unitary matrix. The controlled-U is a two-qubit gate, written C(U), which when applied to qubit registers 91, 92, is defined by: C(U) [91, 92] |k1)... | kq)... | kq2) ... kn) = k1)...kq)... (Uka1 |kq)) ... | kn) where qi is the control qubit and q2 is the target qubit, and where every k = {0,1}. The matrix representation of C(U) for application to two qubits is: I O C(U) = 0 U where I is the 2 2 identity matrix and 0 is the 2 2 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 [91, 92] 1...kq...kq2. .... kn) = 1. |k1 kqkq.. kn) where every k; {0, 1}. The assignment: Prove the following properties of controlled gates: 1. SWAP [91, 92] = C(X)[91, 92] C(X)[92, 91] C(X)[91, 92]. 2. C(X)p, q] = H[q] C(Z)[p,q] H[q. 3. C(Z)p, q] = C(Z)[q, p]. 4. Hp H[q] C(X)[p,q] H[p] H[q] = C(X)[q,p]. 5. C(X)p, q] Xp] C(X)[p,q] = X[p]Xq. 6. C(X)p, q] Yp] C(X)[p, q] = YpXq. 7. C(X)p, q] Zp] C(X)[p,q] = Z[p]. 8. C(X)[p, q] X[q] C(X)[p,q] = X[q]. 9. C(X)p, q] Yq] C(X)[p,q] = Z[p] Xq. 10. C(X)p, q] Z[q] C(X)[p, q] = Z[p] Zq.