Recall that the approximation error E(U, V ) of two unitaries U, V is defined as E(U, V ) = ||U V || = max ||(U V )||| | where the max above is over pure states | that is, unit vectors. 1. Prove that approximation error is subadditive that is, show that for any gates U1, U2, V1, V2, E(U2U1, V2V1) E(U2, V2) + E(U1, V1) You may use without proof two facts: the triangle inequality ||A + B|| ||A|| + ||B|| and ||UA|| = ||A|| = ||AU|| for any unitary U and complex valued matrix A. 2. Suppose you have a circuit U1 Uk consisting of k gates and you wish to approximate over some particular gate set to an error of . What approximation factor should you choose for each gate