Quantum phase estimation is one of the most powerful quantum primitives. This work proposes a new approach for the problem of multiple eigenvalue estimation: Quantum Multiple Eigenvalue Gaussian filtered Search (QMEGS). QMEGS leverages the Hadamard test circuit structure and only requires simple classical postprocessing. QMEGS is the first algorithm to simultaneously satisfy the following two properties: (1) It can achieve the Heisenberg-limited scaling without relying on any spectral gap assumption. (2) With a positive energy gap and additional assumptions on the initial state, QMEGS can estimate all dominant eigenvalues to $\epsilon$ accuracy utilizing a significantly reduced circuit depth compared to the standard quantum phase estimation algorithm. In the most favorable scenario, the maximal runtime can be reduced to as low as $\log(1/\epsilon)$. This implies that QMEGS serves as an efficient and versatile approach, achieving the best-known results for both gapped and gapless systems. Numerical results validate the efficiency of our proposed algorithm in various regimes.

中文翻译：

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量子多重特征值高斯滤波搜索：一种高效且通用的量子相位估计方法


量子相位估计是最强大的量子基元之一。这项工作为多特征值估计问题提出了一种新方法：量子多重特征值高斯过滤搜索 （QMEGS）。QMEGS 利用 Hadamard 测试电路结构，只需要简单的经典后处理。QMEGS 是第一个同时满足以下两个特性的算法：（1） 它可以在不依赖任何频谱间隙假设的情况下实现海森堡极限缩放。（2） 在正能量间隙和对初始状态的额外假设下，QMEGS 可以利用与标准量子相位估计算法相比显著减少的电路深度，以 $\epsilon$ 的精度估计所有主要特征值。在最有利的情况下，最大运行时间可以减少到 $\log（1/\epsilon）$。这意味着 QMEGS 是一种高效且通用的方法，可在有间隙和无间隙系统中实现最知名的结果。数值结果验证了我们提出的算法在各种情况下的效率。