时间周期系统的最优Hamilton模拟
Optimal Hamiltonian simulation for time-periodic systems
摘要：
时间演化算符$U(t)$的实现，称为哈密顿模拟，是量子计算机最有前途的应用之一。对于与时间无关的哈密尔顿算子，量子化最近建立了时间演化$U(t)=e^{-iHt}$的有效实现，在时间$t$和允许的误差$\varepsilon$两者上实现了最佳计算资源。相比之下，对于时间依赖性系统，由于难以处理时间依赖性，因此需要更大的成本。在本文中，我们建立了最佳/接近最佳的哈密顿模拟一般的时间依赖系统的时间周期性，称为Floquet系统。通过使用一个所谓的Floquet-Hilbert空间配备了辅助状态标记傅立叶指数，我们开发了一种方法，以确保获得目标的时间演化状态，而不依赖于时间顺序的产品或戴森级数展开。因此，查询的复杂性，它的措施，实现时间演化的成本，具有最佳和接近最佳的依赖性分别在时间$t$和逆误差$\varepsilon$，并变得足够接近qubitization。因此，我们的协议告诉我们，在一般的时间相关系统，时间周期系统提供了一个类访问的时间无关系统，尽管存在时间依赖性一样有效。由于我们还提供了应用程序的非平衡现象和绝热态制备的模拟，我们的结果将揭示在凝聚态物理和量子化学的非平衡现象，量子任务产生量子计算中的时间依赖性。
英文摘要：
The implementation of time-evolution operators $U(t)$, called Hamiltonian simulation, is one of the most promising usage of quantum computers. For time-independent Hamiltonians, qubitization has recently established efficient realization of time-evolution $U(t)=e^{-iHt}$, with achieving the optimal computational resource both in time $t$ and an allowable error $\varepsilon$. In contrast, those for time-dependent systems require larger cost due to the difficulty of handling time-dependency. In this paper, we establish optimal/nearly-optimal Hamiltonian simulation for generic time-dependent systems with time-periodicity, known as Floquet systems. By using a so-called Floquet-Hilbert space equipped with auxiliary states labeling Fourier indices, we develop a way to certainly obtain the target time-evolved state without relying on either time-ordered product or Dyson-series expansion. Consequently, the query complexity, which measures the cost for implementing the time-evolution, has optimal and nearly-optimal dependency respectively in time $t$ and inverse error $\varepsilon$, and becomes sufficiently close to that of qubitization. Thus, our protocol tells us that, among generic time-dependent systems, time-periodic systems provides a class accessible as efficiently as time-independent systems despite the existence of time-dependency. As we also provide applications to simulation of nonequilibrium phenomena and adiabatic state preparation, our results will shed light on nonequilibrium phenomena in condensed matter physics and quantum chemistry, and quantum tasks yielding time-dependency in quantum computation.
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