Time-dependent unbounded Hamiltonian simulation with vector norm scaling
Time-dependent unbounded Hamiltonian simulation with vector norm scaling
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DOI：
10.22331/q-2021-05-26-459
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发表时间：
2021-05-26
期刊：
影响因子：
6.4
通讯作者：
Lin, Lin
中科院分区：
文献类型：
--
作者：
An, Dong;Fang, Di;Lin, Lin
量子动力学模拟的准确性通常由幺正演化算符在算符范数下的误差来衡量，而算符范数又依赖于哈密顿量的范数。对于无界算子，在适当的离散化之后，哈密尔顿算子的范数可以非常大，这显著地增加了模拟成本。然而，算子范数测量量子模拟的最坏情况的误差，而实际模拟关注相对于给定的初始向量的误差。我们证明，在适当的假设下的哈密顿量和初始向量，如果误差是衡量的向量范数，计算成本可能不会增加，在所有的哈密顿量的范数增加使用Trotter型方法。在这个意义上，我们的结果优于所有以前的量子模拟文献中的误差界。我们的结果推广了[Jahnke，Lubich，BIT Numer. Math.2000]到时间依赖性设置。我们还澄清了存在和重要性的换向器缩放的Trotter和广义Trotter方法的时间相关的哈密顿模拟。
The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.