Efficient Quantum Circuits for the Hilbert Transform

• URL: http://arxiv.org/abs/2601.10876v1
• Date: Thu, 15 Jan 2026 22:02:32 GMT
• Title: Efficient Quantum Circuits for the Hilbert Transform
• Authors: Henry Zhang, Joseph Li,
• Abstract summary: This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length $N$.<n>We generalize this algorithm to create any $d$-dimensional Hilbert transform in depth $O(dlog N)$.<n> Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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