Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

Lin Lin1,2 and Yu Tong1

1Department of Mathematics, University of California, Berkeley, CA 94720, USA
2Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

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We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal $\mathcal{\widetilde{O}}(d\kappa\log(1/\epsilon))$ query complexity for a $d$-sparse matrix, where $\kappa$ is the condition number, and $\epsilon$ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.

We present a quantum eigenstate filtering algorithm that allows us to efficiently prepare a target eigenstate of a given Hamiltonian to high precision under reasonable assumptions. We apply this algorithm to the quantum linear system problem, and present two algorithms based on quantum adiabatic computing and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal query complexity. Neither algorithm uses phase estimation or amplitude amplification.

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