Quantum Phase Estimation by Compressed Sensing

TL;DR

This paper introduces a new quantum phase estimation algorithm leveraging compressed sensing, achieving Heisenberg-limited precision, robustness to noise, and efficient runtime suitable for early quantum computers.

Contribution

It presents a novel Heisenberg-limited QPE algorithm based on compressed sensing, with improved runtime and noise robustness, applicable to quantum eigenvalue estimation.

Findings

Achieves $ ext{O}(rac{1}{ extepsilon} ext{poly} extlog(rac{1}{ extepsilon}))$ runtime

Maintains $T_{ extmax} extepsilon extless extless extpi$, comparable to existing algorithms

Demonstrates robustness against sampling noise and effectiveness for quantum eigenvalue estimation

Abstract

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.