Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle $O_C$, and a direct sampling method for the amplitude oracle $O_A$ with SELECT-SWAP architecture, we achieve a T count that scales as $\mathcal{\tilde{O}}(\sqrt{L})$ with respect to the number of interaction terms $L$ in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the $\eta$-particle subspace, thereby reducing the subnormalization factor from $\mathcal{O}(L)$ to $\mathcal{O}(\sqrt{L})$, and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.

翻译：高效的多体哈密顿量块编码是科学计算中量子算法的核心需求，尤其是在早期容错时代。本工作中，我们为块编码二阶量子化哈密顿量引入了新的显式构造，显著降低了Clifford+T门的复杂度与辅助量子比特开销。通过采用基于SWAP架构的稀疏预言机$O_C$数据查找策略，以及采用SELECT-SWAP架构的振幅预言机$O_A$直接采样方法，我们实现了T门数量相对于一般二阶量子化哈密顿量中相互作用项数$L$按$\mathcal{\tilde{O}}(\sqrt{L})$标度增长。同时，我们在预言机的Clifford门数量上也获得了改进的常数因子。此外，我们设计了一种直接针对$\eta$粒子子空间的块编码，从而将次归一化因子从$\mathcal{O}(L)$降低至$\mathcal{O}(\sqrt{L})$，并在模拟固定粒子数系统时提高了容错效率。基于为一般多体哈密顿量开发的块编码框架，我们将方法扩展至系数张量具有平移不变性或衰减结构的电子哈密顿量。我们的结果为多体系统的早期容错量子模拟提供了一条实用路径，与先前方法相比显著降低了资源开销。