A quantum algorithm for simulating non-sparse Hamiltonians

We present a quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse. Our algorithm is based on the assumption that the entries of the Hamiltonian are stored in a data structure that allows for the efficient preparation of states that encode the rows of the Hamiltonian. We use a linear combination of quantum walks to achieve a poly-logarithmic dependence on the precision. The time complexity measured in terms of circuit depth of our algorithm is $O(t\sqrt{N}\lVert H \rVert \text{polylog}(N, t\lVert H \rVert, 1/\epsilon))$, where $t$ is the evolution time, $N$ is the dimension of the system, and $\epsilon$ is the error in the final state, which we call precision. Our algorithm can directly be applied as a subroutine for unitary Hamiltonians and solving linear systems, achieving a $\widetilde{O}(\sqrt{N})$ dependence for both applications.