Find a $2 \times 2$ Hamiltonian $\mathcal H_H$ such that $e^{i\mathcal H_H}$ equals the Hadamard matrix and a $4 \times 4$ Hamiltonian $\mathcal H_{CNOT}$ such that $e^{-i\mathcal H_{CNOT}}$ equals the matrix of the CNOT gate.

I have been trying to solve this but couldn't come to any conclusion. Any help would be really appreciated.

How can I find a $2 \times 2$ Hamiltonian $\mathcal H_H$ such that $e^{i\mathcal H_H}$ equals the Hadamard matrix?

On a similar note, how to find a $4 \times 4$ Hamiltonian $\mathcal H_{CNOT}$ such that $e^{-i\mathcal H_{CNOT}}$ equals the matrix of the CNOT gate?

I have been trying to solve this but couldn't come to any conclusion.

Find a $2 \times 2$ Hamiltonian $H_H$ such that $e^{iH_H}$ equals the Hadamard matrix and a $4 \times 4$ Hamiltonian $H_{CNOT}$ such that $e^{-iH_{CNOT}}$ equals the matrix of the CNOT gate.

I have been trying to solve this but couldn't come to any conclusion. Any help would be really appreciated.

Find a $2 \times 2$ Hamiltonian $\mathcal H_H$ such that $e^{i\mathcal H_H}$ equals the Hadamard matrix and a $4 \times 4$ Hamiltonian $\mathcal H_{CNOT}$ such that $e^{-i\mathcal H_{CNOT}}$ equals the matrix of the CNOT gate.

I have been trying to solve this but couldn't come to any conclusion. Any help would be really appreciated.

I have been trying to solve this but couldn't come to any conclusion. Any help would be really appreciated.

Find a $2 \times 2$ Hamiltonian $H_H$ such that $e^{iH_H}$ equals the Hadamard matrix and a $4 \times 4$ Hamiltonian $H_{CNOT}$ such that $e^{-iH_{CNOT}}$ equals the matrix of the CNOT gate.

I have been trying to solve this but couldn't come to any conclusion. Any help would be really appreciated.