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It is already known that if the Hamiltonian is a sum of Poly(N) Pauli terms, it has an efficient implementation as a quantum circuit. This should mean that the circuit can be implemented with Poly(N) gates. Is the reverse true as well? Can we show that an implementable quantum circuit has a Hamiltonian with Poly(N) terms in its decomposition?

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  • $\begingroup$ Hi and welcome to QCSE. Have you seen this question? $\endgroup$ Apr 11, 2023 at 17:42
  • $\begingroup$ See this question for a construction of such a Hamiltonian. I am not sure if your question is a duplicate, but I do think the top answer answers your question: yes, the reverse is true. $\endgroup$
    – Jacob
    Apr 11, 2023 at 19:44

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