Find a basis for a two-dimensional subspace of two qubits that is not affected by the noise operation rho₇ → (X ⊗ X)rho(X ⊗ X). In other words,
(a)find two orthogonal 2-qubit states |φ₀⟩ and |φ₁⟩ such that, for j = 0, 1, X ⊗ X|φⱼ⟩ = eiα|φⱼ⟩ where α is a constant.
(b) Describe how to implement the logical X gate in the basis where |φ₀⟩ is the logical '0' and |φ₁⟩ is the logical '1'.
(c) Describe how to implement a logical Z gate.
(d) Describe how to implement a logical H gate. (Use standard 1- and 2-qubit gates.)