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的应用。实践中，量子系统的许多模拟任务聚焦于低能子空间而非整个希尔伯特空间中的量子态。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，即使存在热化导致的不完美态制备，该特性仍允许改进先前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同样存在。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的改进。此外，我们为低能子空间中的一般数字量子模拟提供了查询复杂度下界。