Problem 4.3: (Alternate universality construction) Suppose U is a unitary matrix on n qubits. Define H = i ln(U). Show that (1) H is Hermitian, with eigenvalues in the range 0 to 27. (2) H can be written H = hgg, (4.117) 9 where hg are real numbers and the sum is over all n-fold tensor products g of the Pauli matrices {I, X, Y, Z}. (3) Let A = 1/k, for some positive integer k. Explain how the unitary operation exp(-ih gA) may be implemented using O(n) one and two qubit operations. (4) Show that exp(-iHA) = 1] exp(-ih994) + O(4A?), (4.118) 9 where the product is taken with respect to any fixed ordering of the n-fold tensor products of Pauli matrices, g. Problem 4.3: (Alternate universality construction) Suppose U is a unitary matrix on n qubits. Define H = i ln(U). Show that (1) H is Hermitian, with eigenvalues in the range 0 to 27. (2) H can be written H = hgg, (4.117) 9 where hg are real numbers and the sum is over all n-fold tensor products g of the Pauli matrices {I, X, Y, Z}. (3) Let A = 1/k, for some positive integer k. Explain how the unitary operation exp(-ih gA) may be implemented using O(n) one and two qubit operations. (4) Show that exp(-iHA) = 1] exp(-ih994) + O(4A?), (4.118) 9 where the product is taken with respect to any fixed ordering of the n-fold tensor products of Pauli matrices, g