Pairing Hamiltonian is a simplified model Hamiltonian that can be used to study a number of interesting properties of nuclear structures. We discuss how to construct an efficient block encoding quantum circuit for such a Hamiltonian in a systematic fashion. In particular, we discuss the type of ancilla qubits we use to achieve an efficient block encoding scheme and how the circuit is structured to account for all terms in the Hamiltonian. We will also discuss how to extend this block encoding scheme to a more general Fermionic two-body Hamiltonian without resorting to the use of Jordan-Wigner transformation and a linear combination of unitaries.
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