Efficient Quantum Circuits for the Hilbert Transform

TL;DR

This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length N, with exponentially fewer operations than classical algorithms for the same mapping.

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.

Figures (5)

• Figure 1: The circuit diagram of the $d$-dimensional quantum Hilbert transform. Each register is $n$ qubits, with an ancilla as the most significant qubit.
• Figure 2: The quantum Hilbert transform agrees exactly with classical results calculated using the fast Fourier transform for $f(x) = \sin(x)/(1+x^4)$.
• Figure 3: The quantum Hilbert transform precisely captures fast multi-fault events in HVDC grids.
• Figure 4: The absolute amplitudes of the quantum Hilbert transform detect the corners of a chessboard.
• Figure S1: A simple implementation of the $d$-dimensional quantum Hilbert transform when dynamic circuits are not supported by hardware; $d$ ancillae are used to avoid increased depth.

Theorems & Definitions (10)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof