Digital quantum simulation has broad applications in approximating unitary evolution 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 {due to thermalization}. 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 Trotter numbers. Such 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：数字量子模拟在哈密顿量的酉演化近似中具有广泛应用。实践中，许多量子系统的模拟任务聚焦于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的低能子空间数字量子模拟的复杂性。我们证明，对于多种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，即使对于因热化导致的不完美态制备，也能改进以往完整酉模拟的复杂度。特别地，对于低能子空间中的自旋模型模拟，我们证明随机化乘积公式（如qDRIFT和随机置换）需要更少的Trotter步数。这种改进在对称性保护的数字量子模拟中依然存在。我们证明了在幂律量子相互作用动力学模拟中也存在类似改进。同时，我们给出了低能子空间中一般数字量子模拟的查询复杂度下界。