Quantum Hilbert Transform in Logarithmic Time

Overview

The Hilbert transform is a critical tool in signal processing, used to extract instantaneous amplitude, frequency, and phase from signals. Its wide -ranging applications include ECG signal analysis to detect heart arrhythmias, power grid control to prevent blackouts during natural disasters, and early warning systems for earthquakes and tsunamis. However, classical Hilbert transform algorithms are bottlenecked by the 𝑂(𝑛 𝑙𝑜𝑔 𝑛) time complexity of the fast Fourier transform, especially as datasets grow to contain quadrillions to quintillions of data points. This project presents a quantum algorithm for the Hilbert transform that uses quantum parallelism to calculate the Hilbert t ransform in just 𝑂(𝑙𝑜𝑔 𝑛) depth. Although general state multiplication is impossible on quantum computers, this project creates a novel quantum DC filtering algorithm which uses postselective measurement to bypass the unitary constraints of quantum mechanics. Additionally, single-qubit rotation gates are used to compute phase shifts needed for the Hilbert transform in constant time. Numerical results from IBM Qiskit on power systems and ECG data verify that the algorithm’s quantum results match exactly with classical results while requiring exponentially fewer operations, for scalable and efficient information processing on quantum computers.

Competition history

• JSHS 2025 Category not listed

Resources

• National JSHS abstract book ↗

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