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，压缩感知在数据具有低复杂度且样本稀少时尤为有效，这与量子相位估计任务高度契合。本文提出了一种基于压缩感知的早期量子计算机海森堡极限量子相位估计算法。具体而言，给定适当初始态的多个副本以及对某些酉算子的查询，该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$T_{\max}\epsilon \ll \pi$，与当前最优算法相当，且该算法对采样过程中的一定噪声具有鲁棒性。我们还考虑了更广义的量子特征值估计问题，并数值证明了离网格压缩感知是解决该问题的有力候选方案。