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.

中文翻译：

---

基于最优多项式的量子本征滤波在求解量子线性系统中的应用

我们提出了一种基于量子信号处理 (QSP) 和极大极小多项式的量子本征态滤波算法。该算法允许我们有效地准备给定哈密顿量的目标本征态，如果我们可以访问与目标本征态具有非平凡重叠的初始状态并且具有合理的光谱间隙下限。我们将该算法应用于量子线性系统问题（QLSP），并分别提出了基于量子绝热计算（AQC）和量子芝诺效应的两种算法。两种算法都将最终解准备为纯状态，并实现了接近最优的 $\mathcal{\widetilde{O}}(d\kappa\log(1/\epsilon))$ 对 $d$-sparse 矩阵的查询复杂度，其中 $\kappa$ 是条件数，$\epsilon$ 是所需的精度。