SIAM Journal on Computing, Ahead of Print.
We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using ${\mathcal O}( nT {polylog} (nT/\epsilon))$ gates with depth ${\mathcal O}(T { polylog} (nT/\epsilon))$. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires ${\widetilde{\Omega}}(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.

中文翻译：

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模拟晶格哈密顿量实时演化的量子算法

《 SIAM计算杂志》，预印本。
我们研究了模拟晶格哈密顿量的时间演化问题，其中量子位被放置在晶格上，而哈密顿量仅包括几何局部相互作用（即，量子位只能与附近的量子位相互作用）。此类哈密顿量非常笼统，被认为可以捕捉物理学的基本相互作用。我们的算法使用$ {\ mathcal O}（nT {polylog}（nT / \ epsilon））$具有深度的门来模拟这样的哈密顿量在$ n $个比特上的时间演化，直到时间$ T $直到误差$ \ epsilon $ $ {\ mathcal O}（T {polylog}（nT / \ epsilon））$。我们的算法是第一个仿真算法，在$ nT $中实现门成本准线性，在$ 1 / \ epsilon $中实现多对数。我们的算法还可以很容易地推广到时间相关的哈密顿量，并针对任何分段缓慢变化的时间相关的有界哈密顿量得出具有相似门数的算法。我们还证明了这种模拟的门数上具有匹配的下限，表明任何能够将一维中的分段恒定有界局部哈密顿量模拟成恒定误差的量子算法都需要$ {\ widetilde {\ Omega}}（nT）$最坏情况下的盖茨。即使我们仅要求本地测量的输出状态正确，下限仍然成立。据我们所知，这是模拟问题的门复杂度的第一个非平凡的下界。我们的算法基于使用Lieb-Robinson边界将时间演化unit分解为小unit的乘积。在附录中，我们证明了为哈密顿量身定做的利布罗宾逊定界，在局部项之间具有小的换向器，在通勤的哈密顿量的极限中，零利布罗宾逊速度为零。当哈密顿量接近通勤时，这可以提高我们算法的性能。