Hamiltonian simulation in the low energy subspace

We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a Hamiltonian $H$. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm of $H$. We find some improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain our main results, we prove exponentially-decaying upper bounds on the leakage or transitions to high-energy subspaces due to the terms in the product formula that may be of independent interest.