近似最优格点模拟的乘积公式
Nearly optimal lattice simulation by product formulas
摘要：
我们考虑在量子计算机上模拟具有随时间演化的最近邻相互作用的$n$ -量子比特哈密顿量$t$。我们表明，这种模拟门的复杂性$(nt)^{1+o(1)}$使用产品的公式，一个简单的方法，已被证明由几个实验组。虽然这是合理的期望这种复杂性-特别是，这是声称没有严格的理由乔丹，李，和Preskill-我们不知道一个简单的证明。我们的方法是基于本地错误结构的产品公式的分析，介绍了Descombes和Thalhammer和进一步简化这里。我们证明了标准产品的公式，其中包括著名的建设，如Lie-Trotter-Suzuki公式的误差界。我们还开发了一个局部误差表示的时间相关的哈密顿模拟，我们讨论了概括周期性的边界条件，恒定范围的相互作用，和更高的维度。结合先前的下限，我们的结果意味着乘积公式可以模拟具有接近最佳门复杂度的格点哈密顿。
英文摘要：
We consider simulating an $n$-qubit Hamiltonian with nearest-neighbor interactions evolving for time $t$ on a quantum computer. We show that this simulation has gate complexity $(nt)^{1+o(1)}$ using product formulas, a straightforward approach that has been demonstrated by several experimental groups. While it is reasonable to expect this complexity---in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill---we are not aware of a straightforward proof. Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and further simplified here. We prove error bounds for canonical product formulas, which include well-known constructions such as the Lie-Trotter-Suzuki formulas. We also develop a local error representation for time-dependent Hamiltonian simulation, and we discuss generalizations to periodic boundary conditions, constant-range interactions, and higher dimensions. Combined with a previous lower bound, our result implies that product formulas can simulate lattice Hamiltonians with nearly optimal gate complexity.
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