A quantum algorithm for simulating non-sparse Hamiltonians (pp597-615)
         Chunhao Wang and Leonard Wossnig
          doi: https://doi.org/10.26421/QIC20.7-8-5
Abstracts: We present a quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse. Our algorithm is based on the input model where the entries of the Hamiltonian are stored in a data structure in a quantum random access memory (qRAM) which allows for the efficient preparation of states that encode the rows of the Hamiltonian. We use a linear combination of quantum walks to achieve poly-logarithmic dependence on precision. The time complexity of our algorithm, measured in terms of the circuit depth, is O(t\sqrt{N}\norm{H}\,\polylog(N, t\norm{H}, 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 be directly applied as a subroutine for unitary implementation and quantum linear systems solvers, achieving \widetilde{O}(\sqrt{N}) dependence for both applications.
Key words: Hamiltonian simulation, quantum algorithm, linear combination of unitaries, quantum walk, quantum machine learning, qRAM, linear systems solver