A quantum algorithm for simulating non-sparse Hamiltonians
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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√(N) H polylog(N, t H , 1/ϵ)), where t is the evolution time, N is the dimension of the system, and ϵ 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 O(√(N)) dependence for both applications.
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