Huffman-Code-based Ternary Tree Transformation

Using a quantum computer to simulate fermionic systems requires fermion-to-qubit transformations. Usually, lower Pauli weight of transformations means shallower quantum circuits. Therefore, most existing transformations aim for lower Pauli weight. However, in some cases, the circuit depth depends not only on the Pauli weight but also on the coeffcients of the Hamiltonian terms. In order to characterize the circuit depth of these algorithms, we propose a new metric called weighted Pauli weight, which depends on Pauli weight and coeffcients of Hamiltonian terms. To achieve smaller weighted Pauli weight, we introduce a novel transformation, Huffman-code-based ternary tree (HTT) transformation, which is built upon the classical Huffman code and tailored to different Hamiltonians. We tested various molecular Hamiltonians and the results show that the weighted Pauli weight of the HTT transformation is smaller than that of commonly used mappings. At the same time, the HTT transformation also maintains a relatively small Pauli weight. The mapping we designed reduces the circuit depth of certain Hamiltonian simulation algorithms, facilitating faster simulation of fermionic systems.