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-toqubit 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-toqubit mapping and reduce the complexity of our algorithm from $O\left(N^{4}\right)$ to $O\left(N^{3}\right)$. Evaluations on various Fermionic systems demonstrate $5 \sim 25 \%$ reduction in Pauli weight, gate count, and circuit depth, alongside excellent scalability to larger systems. Experiments on the Ionq device also show the advantages of HATT in noise resistance in quantum simulations.

Although estrogen-receptor-positive (ER+) breast cancer is generally associated with favorable prognosis, clinical outcome varies substantially among patients. Genomic assays have been developed and applied to predict patient prognosis for personalized treatment. We hypothesize that the recurrence risk of ER+ breast cancer patients is determined by both genomic mutations intrinsic to tumor cells and extrinsic immunological features in the tumor microenvironment. Based on the Cancer Genome Atlas (TCGA) breast cancer data, we identified the 72 most common genomic aberrations (including gene mutations and indels) in ER+ breast cancer and defined sample-specific scores that systematically characterized the deregulated pathways intrinsic to tumor cells. To further consider tumor cell extrinsic features, we calculated immune infiltration scores for six major immune cell types. Many individual intrinsic features are predictive of patient prognosis in ER+ breast cancer, and some of them achieved comparable accuracy with the Oncotype DX assay. In addition, statistical learning models that integrated these features predicts the recurrence risk of patients with significantly better performance than the Oncotype DX assay (our optimized random forest model AUC = 0.841, Oncotype DX model AUC = 0.792, p = 0.04). As a proof-of-concept, our study indicates the great potential of genomic and immunological features in prognostic prediction for improving breast cancer precision medicine. The framework introduced in this work can be readily applied to other cancers.