An Efficient Quantum Circuit for Block Encoding a Pairing Hamiltonian

We present an efficient quantum circuit for block encoding pairing Hamiltonians studied in nuclear physics. The new block encoding scheme does not require mapping the creation and annihilation operators to Pauli operators and representing the Hamiltonian as a linear combination of unitaries. Instead, we show how to encode these operators directly using controlled swaps. We analyze the gate complexity of the block encoding circuit and show that it scales polynomially with respect to the number of qubits required to represent a quantum state associated with the pairing Hamiltonian. We also show how the block encoding circuit can be combined with quantum singular value transformation to construct an efficient quantum circuit for approximating the density of state of a pairing Hamiltonian. Athough we focus on block encoding circuit for pair Hamiltonians in this paper, the techniques presented here can be extended to encode more general second quantized Hamiltonians.