Is there a good book or paper where they describe how to get the ansatz and gates from a Hamiltonian?


I will do my best to answer from my understanding of your question:

I am assuming you mean something like Unitary Coupled Cluster variational form where it is implemented the exponential to the initial state $|\psi \rangle$ $$ e^{T - T^\dagger} $$ here $T$ is the cluster operator, which, when acting on $|\psi$, produces a linear combination of excited determinants from the reference state. It can be written as $$ T = \sum T_i$$ where $T_1$ is the operator for all single excitation, $T_2$ is the operator for all double excitation, and so forth.

$$T_1 = \sum_{i \in occupied} \sum_{\alpha \in Virtual} t_{\alpha}^i a^\dagger_{\alpha} a_i $$

$$T_2 = \dfrac{1}{4}\sum_{i,j \in occupied} \sum_{\alpha,\beta \in Virtual} t_{\alpha \beta}^{ij} a^\dagger_{\alpha} a^\dagger_{\beta} a_ia_j $$ $$ \vdots $$ $$T_n = \dfrac{1}{(n!)^2}\sum_{i_1, i_2 \cdots ,i_n \in occupied} \sum_{\alpha_1, \alpha_2 \cdots ,\alpha_n \in Virtual} t_{\alpha_1 \alpha_2, \cdots, \alpha_n}^{i_1, i_2, \cdots, i_n} a^\dagger_{\alpha_1} a^\dagger_{\alpha_2} \cdots a^\dagger_{\alpha_n} a_{i_1}a_{i_2}\cdots a_{i_n} $$

If you only pick $T_1$ and $T_2$ then you have what called as UCCSD variational form, which can be implemented very easily through Qiskit.

To understand how to convert exponential of matrix into quantum gates through Trotter, I would suggest you to look at this answer. In that answer, there are two very good references that you should look into.

I also suggest this paper:

Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz.

If this is not what you are looking for then edit your question or leave a comment so I understand what you are looking for. This way I might be able to give you some better references.


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