Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner. However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques. We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms. We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law. Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques. We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank. Combining with a tighter error analysis, we show that it suffices to use $\left(\eta^{1/3}n^{1/3}+\frac{n^{2/3}}{\eta^{2/3}}\right)n^{1+o(1)}$ gates to simulate uniform electron gas with $n$ spin orbitals and $\eta$ electrons in second quantization in real space, asymptotically improving over the best previous work. We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation. We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic $n$-qubit $2$-local Hamiltonians with commuting terms require at least $\Omega(n^2)$ gates to evolve with accuracy $\epsilon=\Omega(1/poly(n))$ for time $t=\Omega(\epsilon)$. Our proof is based on a gate-efficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-$2$ subspace, which may be of independent interest. Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.

翻译：量子动力学可通过将哈密顿量中的基本项按顺序指数化在量子计算机上进行模拟。然而,这种实现Trotter步骤的电路复杂度取决于总哈密顿量项数,与使用更先进技术的算法相比不够有利。我们开发了能以亚线性于项数的复杂度实现更快Trotter步骤的方法。对于一类其相互作用强度按幂律随距离衰减的哈密顿量,我们实现了这一目标。我们的方法包括一种基于递归块编码的方法和一种基于平均成本模拟的方法,克服了这些先进量子模拟技术的归一化因子障碍。当哈密顿量系数的某些块具有低秩时,我们还能实现更快的Trotter步骤。结合更严格的误差分析,我们证明仅需使用$\left(\eta^{1/3}n^{1/3}+\frac{n^{2/3}}{\eta^{2/3}}\right)n^{1+o(1)}$个门即可模拟具有$n$个自旋轨道和$\eta$个电子的均匀电子气(采用二次量子化的实空间表示),其渐近性能优于此前最优工作。在Born-Oppenheimer近似下引入原子核外势时,我们得到了类似结果。当哈密顿量系数取连续取值范围时,我们证明了一个电路下界:对于具有对易项的通用$n$量子比特$2$-局域哈密顿量,在时间$t=\Omega(\epsilon)$内以精度$\epsilon=\Omega(1/poly(n))$演化至少需要$\Omega(n^2)$个门。我们的证明基于从Hamming权重-$2$子空间内的对角酉矩阵近似合成到电路高效的归约,这一方法可能具有独立意义。因此,我们的结果表明,哈密顿量结构特性的利用对于以更低电路复杂度实现Trotter步骤既是必要条件也是充分条件。