I have a problem in my simulation. I need to simulate a Hadamard gate by integrating Schrödinger equation (time evolution).
This requires me to construct a Hamiltonian operator related to Hadamard gate and perform a time evolution, so I would also need eigenvalues.
My problem is, all resources I can find about quantum gates are in unitary formalism (discrete time), I don't manage to find useful resources for continuous time case, especially with eigenvalues.
Quantum Computation and Quantum Information by Nielsen and Chuang page 83
There is therefore one-to-one correspondence between discrete-time description of dynamics using unitary operators and the continuous time description using Hamiltonians
I just don't know how to find Hamiltonian related to Hadamard gate.
I found some examples such as paper A Sequence of Quantum Gates page 5, but it doesn't scale up for more than single qubit and doesn't mention eigendecomposition, which is required for efficient time evolution as H easily becomes a huge matrix and it is difficult to exponentiate.