Digital quantum simulation has broad applications in approximating unitary evolutions of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller step complexities. This improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.

翻译：数字量子模拟在近似哈密顿量的酉演化中具有广泛应用。实践中，许多量子系统的模拟任务仅关注低能子空间中的量子态，而非整个希尔伯特空间。本文系统研究了基于乘积公式在低能子空间中进行数字量子模拟的复杂性。我们证明，对于各类数字量子模拟算法和量子系统，模拟误差取决于哈密顿量的有效低能范数，这使得即使在非理想态制备条件下，也能较全酉模拟的先前复杂度实现改进。特别地，在模拟低能子空间中的自旋模型时，我们证明了诸如qDRIFT和随机置换等随机乘积公式所需的步长复杂度更小。这一改进在对称性保护的数字量子模拟中依然成立。在模拟幂律量子相互作用动力学时，我们也证明了类似的改进。最后，我们给出了低能子空间中通用数字量子模拟的查询下界。