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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.

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1 Answer 1

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Since Pauli strings form an (orthonormal) basis, and thus are linearly independent, the best approximation by other Pauli strings is to cut the Pauli strings with the smallest weight.

(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.)

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