  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.
