As a signal recovery algorithm, compressed sensing is particularly useful when the data has low-complexity and samples are rare, which matches perfectly with the task of quantum phase estimation (QPE). In this work we present a new Heisenberg-limited QPE algorithm for early quantum computers based on compressed sensing. More specifically, given many copies of a proper initial state and queries to some unitary operators, our algorithm is able to recover the frequency with a total runtime $\mathcal{O}(\epsilon^{-1}\text{poly}\log(\epsilon^{-1}))$, where $\epsilon$ is the accuracy. Moreover, the maximal runtime satisfies $T_{\max}\epsilon \ll \pi$, which is comparable to the state of art algorithms, and our algorithm is also robust against certain amount of noise from sampling. We also consider the more general quantum eigenvalue estimation problem (QEEP) and show numerically that the off-grid compressed sensing can be a strong candidate for solving the QEEP.

翻译：作为一种信号恢复算法，压缩感知特别适用于数据复杂度低且样本稀缺的场景，这与量子相位估计（QPE）任务高度契合。本文针对早期量子计算机，提出了一种基于压缩感知的海森堡极限QPE算法。具体而言，给定若干份合适的初始态副本以及对某些酉算子的查询，我们的算法能以总运行时间$\mathcal{O}(\epsilon^{-1}\text{poly}\log(\epsilon^{-1}))$恢复频率，其中$\epsilon$为精度。此外，最大运行时间满足$T_{\max}\epsilon \ll \pi$，与现有最优算法相当，且该算法对采样中的一定噪声具有鲁棒性。我们还进一步讨论了更一般的量子本征值估计问题（QEEP），数值实验表明，离网格压缩感知有望成为求解QEEP的有力工具。