We develop a fourth-order Magnus expansion based quantum algorithm for the simulation of many-body problems involving two-level quantum systems with time-dependent Hamiltonians, $\mathcal{H}(t)$. A major hurdle in the utilization of the Magnus expansion is the appearance of a commutator term which leads to prohibitively long circuits. We present a technique for eliminating this commutator and find that a single time-step of the resulting algorithm is only marginally costlier than that required for time-stepping with a time-independent Hamiltonian, requiring only three additional single-qubit layers. For a large class of Hamiltonians appearing in liquid-state nuclear magnetic resonance (NMR) applications, we further exploit symmetries of the Hamiltonian and achieve a surprising reduction in the expansion, whereby a single time-step of our fourth-order method has a circuit structure and cost that is identical to that required for a fourth-order Trotterized time-stepping procedure for time-independent Hamiltonians. Moreover, our algorithms are able to take time-steps that are larger than the wavelength of oscillation of the time-dependent Hamiltonian, making them particularly suited for highly-oscillatory controls. The resulting quantum circuits have shorter depths for all levels of accuracy when compared to first and second-order Trotterized methods, as well as other fourth-order Trotterized methods, making the proposed algorithm a suitable candidate for simulation of time-dependent Hamiltonians on near-term quantum devices.

翻译：我们开发了一种基于四阶Magnus展开的量子算法，用于模拟涉及含时哈密顿量$\mathcal{H}(t)$的两能级量子系统的多体问题。Magnus展开应用的主要障碍是对易子项的出现，这会导致电路长度过长。我们提出了一种消除该对易子的技术，并发现所得算法的单时间步计算成本仅略高于含时哈密顿量时间步进方法，仅需额外三个单量子比特层。针对液态核磁共振（NMR）应用中出现的多类哈密顿量，我们进一步利用哈密顿量的对称性实现了展开的显著简化，使得四阶方法的单时间步具有与时间无关哈密顿量的四阶Trotter化时间步进过程相同的电路结构和成本。此外，我们的算法能够采用比含时哈密顿量振荡波长更大的时间步长，因此特别适用于高度振荡控制。与一阶、二阶Trotter化方法及其他四阶Trotter化方法相比，所得量子电路在所有精度水平下均具有更浅的深度，这使得所提算法成为近期量子设备上模拟含时哈密顿量的理想候选方案。