Data From: Fixed depth Hamiltonian simulation via Cartan decomposition

Simulating quantum dynamics on classical computers is challenging for large systems due to the significant memory requirements. Simulation on quantum computers is a promising alternative, but fully optimizing quantum circuits to minimize limited quantum resources remains an open problem. We tackle this problem presenting a constructive algorithm, based on Cartan decomposition of the Lie algebra generated by the Hamiltonian, that generates quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model, where a O(n^2)-gate circuits naturally emerge. Compared to product formulas with significantly larger gate counts, our algorithm drastically improves simulation precision. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.

Methods
Data provided here consist of two different Hamiltonians, RMS position of the excitation for each timestep calculated via Trotterization, RMS position of the excitation for each timestep calculated via Cartan Decomposition, and errors of each approach for each Hamiltonian. Initial state is an excitation on the first site (rightmost site with the convention used in the dataset). All the data is obtained via simulating the state vector evolution by using a python code that can be found in https://github.com/kemperlab/cartan-quantum-synthesizer.

在经典计算机上模拟大型系统的量子动力学颇具挑战，这源于其极高的内存需求。量子计算机上的量子模拟是极具前景的替代方案，但如何全面优化量子电路以最小化有限的量子资源，仍是一个尚未解决的开放问题。我们针对该问题提出了一种构造性算法，该算法基于哈密顿量（Hamiltonian）生成的李代数（Lie algebra）的嘉当分解（Cartan decomposition），可生成深度与时间无关的量子电路。我们针对多类特殊模型展示了该算法的应用效果，其中包括一维横向场XY模型中的安德森局域化（Anderson localization）现象，此时可自然得到O(n²)规模的门电路。与门数量显著更多的乘积公式相比，我们的算法大幅提升了模拟精度。除了可为广泛的自旋与费米子模型提供精确电路外，我们的算法还为最优哈密顿量模拟提供了丰富的解析与数值视角。
方法
本文提供的数据包含两类不同的哈密顿量、通过特罗特分解（Trotterization）计算得到的每个时间步激发的均方根位置、通过嘉当分解计算得到的每个时间步激发的均方根位置，以及两种方法针对每个哈密顿量的误差。初始态为第一格点（按照本数据集采用的惯例，即最右侧格点）上的一个激发。所有数据均通过模拟态矢量演化的Python代码获得，该代码可在https://github.com/kemperlab/cartan-quantum-synthesizer处获取。