HATT: Hamiltonian Adaptive Ternary Tree for Optimizing Fermion-to-Qubit Mapping

TL;DR

The paper tackles the challenge of efficiently simulating Fermionic systems on quantum hardware by introducing HATT, a Hamiltonian-Adaptive Ternary Tree that constructs Fermion-to-qubit mappings tailored to the input Hamiltonian. By growing a ternary tree bottom-up and enforcing vacuum-state preservation via controlled operator pairing, HATT reduces Pauli weight and circuit resources with a polynomial-time complexity of $O(N^3)$, improving over traditional and exhaustive-search methods. Extensive evaluations across electronic-structure, Fermi-Hubbard, and neutrino-oscillation models show consistent gains in Pauli weight, CNOT count, and circuit depth, including robustness to noise and favorable real-device performance on IonQ. The approach is compatible with existing quantum-compilation pipelines and notably enhances scalability for large quantum simulations, offering a practical path toward more efficient quantum chemistry and materials studies.

Abstract

This paper introduces the Hamiltonian-Adaptive Ternary Tree (HATT) framework to compile optimized Fermion-to-qubit mapping for specific Fermionic Hamiltonians. In the simulation of Fermionic quantum systems, efficient Fermion-to-qubit mapping plays a critical role in transforming the Fermionic system into a qubit system. HATT utilizes ternary tree mapping and a bottom-up construction procedure to generate Hamiltonian aware Fermion-to-qubit mapping to reduce the Pauli weight of the qubit Hamiltonian, resulting in lower quantum simulation circuit overhead. Additionally, our optimizations retain the important vacuum state preservation property in our Fermion-to-qubit mapping and reduce the complexity of our algorithm from $O(N^4)$ to $O(N^3)$. Evaluations and simulations of various Fermionic systems demonstrate $5\sim20\%$ reduction in Pauli weight, gate count, and circuit depth, alongside excellent scalability to larger systems. Experiments on the Ionq quantum computer also show the advantages of our approach in noise resistance in quantum simulations.

Figures (10)

• Figure 1: Overview of simulating Fermionic systems with quantum computers. The mapping on the upper half is better than that in the lower half because it generated a qubit Hamiltonian with fewer Pauli operators, and the circuit implementation cost is lower.
• Figure 2: Circuit snippet of operator $\exp(itc_j XYIZ/n)$. $q_0$ is selected as the target qubit.
• Figure 3: Example of a ternary tree and extracting Fermion-to-qubit mapping
• Figure 4: $3$-mode ternary tree Fermion-to-qubit examples. (a) Balanced ternary tree. (b) Unbalanced ternary tree.
• Figure 5: Algorithm for ternary tree construction