Efficient Quantum Circuits for the Hilbert Transform

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.