# Are there techniques to reduce the number of Pauli strings in a Hamiltonian?

Suppose $$H=\sum_{i \in n} c_i h_i$$ is a generic Hamiltonian. Is there a way to find $$\tilde{H} = \sum_{j \in m} d_j h_j$$ where $$m < n$$ is a user defined cut-off parameter that would accurately represent the ground state of the Hamiltonian $$H$$, i.e., $$H \Psi \approx \tilde{H} \Psi$$ where $$\Psi$$ is the ground state of $$H$$. $$c_i,d_j$$ are the coefficients of the Pauli strings $$h_i$$.

Context : I'm planning to evaluate the ground state of $$H$$ via Quantum Approximate Optimisation Algorithm (QAOA). The number of Pauli strings that constitutes $$H$$ grows exponentially with the system size, and I would like to truncate the Hamiltonian such that the QAOA ansatz is not too deep. A trivial answer is to keep the $$m$$ strings with the highest coefficients, but I am curious to know whether there is a more elegant solution.

(Linear independence implies that there is a unique way of writing $$H$$ and $$\tilde H$$ as a linear combination of Paulis, and thus, the error is simply $$\sum |c_i-d_i|$$. Since you want to set as many $$d_i$$ as possible to zero, dropping the smallest ones -- and choosing the others equal to the $$c_i$$ -- will result in the smallest error.)