Efficient Quantum Circuits for the Hilbert Transform

TL;DR

This paper introduces an efficient quantum algorithm for the discrete Hilbert transform, significantly reducing complexity compared to classical methods, with applications demonstrated in power systems and image processing.

Contribution

It provides the first quantum implementation of the Hilbert transform with polylogarithmic size and depth, enabling faster processing of non-stationary signals.

Findings

Quantum Hilbert transform achieved in polylogarithmic size and depth.

Simulation results match classical outcomes exactly.

Applicable to multi-dimensional signals with efficient scaling.

Abstract

The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length $N$, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any $d$-dimensional Hilbert transform in depth $O(d\log N)$. Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.