  We propose a novel quantum approach to signal processing, including a quantum
algorithm for low-pass and high-pass filtering, based on the sequency-ordered
Walsh-Hadamard transform. We present quantum circuits for performing the
sequency-ordered Walsh-Hadamard transform, as well as quantum circuits for
low-pass, high-pass, and band-pass filtering. Additionally, we provide a proof
of correctness for the quantum circuit designed to perform the sequency-ordered
Walsh-Hadamard transform. The performance and accuracy of the proposed approach
for signal filtering were illustrated using computational examples, along with
corresponding quantum circuits, for DC, low-pass, high-pass, and band-pass
filtering. Our proposed algorithm for signal filtering has a reduced gate
complexity and circuit depth of $O (\log_2 N)$, compared to at least $O
((\log_2 N )^2)$ associated with Quantum Fourier Transform (QFT) based
filtering (excluding state preparation and measurement costs). In contrast,
classical Fast Fourier Transform (FFT) based filtering approaches have a
complexity of $O (N \log_2 N )$. This shows that our proposed approach offers a
significant improvement over QFT-based filtering methods and classical
FFT-based filtering methods. Such enhanced efficiency of our proposed approach
holds substantial promise across several signal processing applications by
ensuring faster computations and efficient use of resources via reduced circuit
depth and lower gate complexity.
