One of the main application of VQE is its application to find the approximation to the ground state energy (smallest eigenvalue) for a particular molecule through an iterative method.

To be able to do this, we first need to write the Hamiltonian of the molecule in the second quantization form: $$ H_{fer} = \sum h_{pq} a_p^\dagger a_q + \sum h_{pqrs} a_p^\dagger a_q^\dagger a_r a_s $$ then we map $H_{fer}$ to $H_{qubit}$ by one of the maps (JW, parity, BK) so it's easier to calculate the expectation value. That is

$$H_{fer} = \sum h_{pq} a_p^\dagger a_q + \sum h_{pqrs} a_p^\dagger a_q^\dagger a_r a_s \rightarrow H_{qubit} = \sum_{i\alpha} h^i_\alpha \sigma^i_\alpha + \sum_{ij\alpha \beta} h_{\alpha \beta}^{ij}\sigma_\alpha^i \sigma_\beta^j + ... $$

I understand that the set $\{\sigma^i\}^{\otimes n}$ formed a basis for an $n \times n$ Hermitian operator so it's reasonable to consider the map from $H_{fer}$ to $H_{qubit}$

However, the advantage of VQE is to be able to find the min energy in an efficient manner, and to be able to do that one need to evaluate the expectation value of $H_{qubit}$, that is, $\langle H_{qubit} \rangle$. To be able to do this, you must make sure that $H_{qubit}$ has an efficient decomposition. That is, you don't want to use all $4^n$ terms to describe the Hamiltonian... since this will kill off all the efficiency you want to achieve.

So my question is, how do we know that we can always write the Hamiltonian for a particular system in the pauli matrices basis using only polynomial terms? It turns out that this is true for electronic structure Hamiltonian for a molecule, but why?

Given some arbitrary physical system, how do I know whether I can write out a specific Hamiltonian for that system in polynomial number of terms for the Pauli decomposition? Can you give me example when this is not the case?


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