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：量子动力学可通过顺序指数化哈密顿量中的基本项在量子计算机上模拟。然而，这种Trotter步的实现方法其量子门复杂度取决于总哈密顿项数，与采用更先进技术的算法相比表现不佳。我们开发了实现更快Trotter步的方法，其复杂度与项数呈亚线性关系。针对相互作用强度随距离呈幂律衰减的一类哈密顿量，我们实现了这一目标。所提方法包括基于递归块编码的方案和基于平均成本模拟的方案，克服了这些先进量子模拟技术的归一化因子障碍。当哈密顿系数特定区块具有低秩特性时，我们还能实现更快的Trotter步。结合更精确的误差分析，我们证明在实空间第二量子化框架下模拟含$n$个自旋轨道和$\eta$个电子的均匀电子气时，仅需使用$\left(\eta^{1/3}n^{1/3}+\frac{n^{2/3}}{\eta^{2/3}}\right)n^{1+o(1)}$个量子门，渐进性优于此前最佳结果。在Born-Oppenheimer近似下引入原子核外势场时，我们获得类似结论。当哈密顿系数取连续值范围时，我们证明了电路下界，表明对于通用$n$比特$2$-局域且项对易的哈密顿量，在时间$t=\Omega(\epsilon)$内达到精度$\epsilon=\Omega(1/poly(n))$至少需要$\Omega(n^2)$个量子门。该证明基于从Hamming权重-$2$子空间中对角酉算子的近似合成到量子门高效归约，该归约方法本身可能具有独立价值。我们的研究结果表明，利用哈密顿结构性质既是实现低量子门复杂度Trotter步的充分条件也是必要条件。