论文标题
部分可观测时空混沌系统的无模型预测
On the complexity of implementing Trotter steps
论文作者
论文摘要
可以通过依次从哈密顿量从哈密顿式的基本术语来模拟量子动力学。但是，这种实现猪跑步步骤的复杂性取决于汉密尔顿的总学期数，与使用更高级技术的算法进行了不利的比较。我们开发了用术语数进行复杂性sublinear执行更快的猪棍步骤的方法。我们为一类哈密顿人实现了这一目标，其相互作用的强度根据权力法与距离衰减。我们的方法包括基于递归块编码的一种方法和一种基于平均成本模拟的方法，从而克服了这些高级量子模拟技术的归一化因子屏障。当某些哈密顿系数的排名较低时，我们还会意识到更快的猪跑步。结合更严格的错误分析，我们表明使用$ \ left（η^{1/3} n^{1/3}+\ frac {n^{2/3}} {η^η^η^{2/3}}} \ right）实际空间中的量化，渐近地改善了以前的最佳工作。当核的外部电势被引入在Born-Oppenheimer近似下时，我们会获得类似的结果。当汉密尔顿系数采用连续值时，我们证明了电路的下限，这表明具有通勤项的通用$ n $ n $ 2 $ 2 $ - 局部汉密尔顿人需要至少$ω（n^2）$ gates $ chates $ω（1/poly（1/poly（1/poly（n））$ $ time $ $ t =ω（ε（ε）$）。我们的证明是基于闸门有效的降低，从锤击权重-2 $子空间内的对角线单位的近似合成，这可能具有独立的利益。因此，我们的结果表明，使用哈密顿的结构特性是实施较低栅极复杂性的猪跑步骤所必需的和足够的。
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 $\left(η^{1/3}n^{1/3}+\frac{n^{2/3}}{η^{2/3}}\right)n^{1+o(1)}$ gates to simulate uniform electron gas with $n$ spin orbitals and $η$ 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 $Ω(n^2)$ gates to evolve with accuracy $ε=Ω(1/poly(n))$ for time $t=Ω(ε)$. 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.
