AI 中文总结
研究哈密顿量模拟的最优下界，通过基于局部、有界度经典哈密顿量的基本证明得出渐近紧下界，其门数在\(1/\epsilon\)中多项式缩放，与相干预言查询复杂度不同，给出了复合qDRIFT匹配上界。
AI 中文摘要
对于哈密顿量\(H = \sum_j h_j\)，我们证明了在量子计算机上模拟时间演化的门复杂度和查询复杂度的渐近紧下界。这些界适用于任意项范数\(\|h_j\|\)、时间\(t\)和迹距离误差\(\epsilon\)。匹配的上界（称为复合qDRIFT）由大项的高阶 Trotter 化和小项的随机一阶 Trotter 化组成。与之前选择最坏情况\(\|h_j\|\)来编码时间演化中奇偶性或其他布尔函数计算的工作不同，我们的证明是基本的，基于局部、有界度的经典哈密顿量。我们的工作表明，对于许多物理系统（如幂律相互作用），门数必须在\(1/\epsilon\)中多项式缩放，这与通过计算相干预言查询（如块编码模型中的那些）所暗示的复杂度相反。
英文摘要
For Hamiltonian $H = \sum_j h_j$, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms $\|h_j\|$, time $t$, and trace-distance error $ε$. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case $\|h_j\|$ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in $1/ε$, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.
Comments20 pages
