We present a quantum algorithm for estimating the amplitude content of user-specified sequency bands in quantum-encoded signals. The method employs a sequency-ordered Quantum Walsh-Hadamard Transform (QWHT), a comparator-based oracle that coherently marks basis states within an arbitrary sequency range, and Quantum Amplitude Estimation (QAE) to estimate the total probability mass in the selected band. This enables the detection of structured signal components, including both high- and low-sequency features, as well as the identification of rapid sign-change behavior associated with noise or anomalies. The proposed method can be embedded as a module within a larger quantum algorithm; in this setting, both the input and output remain fully quantum, enabling seamless integration with upstream and downstream quantum operations. We show that the sequency-ordered QWHT can be implemented with circuit depth $O(\log_2 N)$ (equivalently $O(n)$ for $N=2^n$ ) when acting on an amplitude-encoded quantum state, whereas computing the full Walsh-Hadamard spectrum of an explicit length- $N$ classical signal requires $O(N\log_2 N)$ operations via the fast Walsh-Hadamard transform. This results in an exponential quantum advantage when the QWHT is used as a modular block within a larger quantum algorithm, relative to classical fast Walsh-Hadamard transform-based approaches operating on explicit data. From an application perspective, the proposed sequency band-energy estimation may be interpreted as a structure-based anomaly indicator, enabling the detection of unexpected high-sequency components relative to a nominal low-sequency signal class. The algorithm is applicable to quantum-enhanced signal processing tasks such as zero-crossing analysis, band-limited noise estimation, and feature extraction in the Walsh basis.We present a quantum algorithm for estimating the amplitude content of user-specified sequency bands in quantum-encoded signals. The method employs a sequency-ordered Quantum Walsh-Hadamard Transform (QWHT), a comparator-based oracle that coherently marks basis states within an arbitrary sequency range, and Quantum Amplitude Estimation (QAE) to estimate the total probability mass in the selected band. This enables the detection of structured signal components, including both high- and low-sequency features, as well as the identification of rapid sign-change behavior associated with noise or anomalies. The proposed method can be embedded as a module within a larger quantum algorithm; in this setting, both the input and output remain fully quantum, enabling seamless integration with upstream and downstream quantum operations. We show that the sequency-ordered QWHT can be implemented with circuit depth $O(\log_2 N)$ (equivalently $O(n)$ for $N=2^n$) when acting on an amplitude-encoded quantum state, whereas computing the full Walsh-Hadamard spectrum of an explicit length-$N$ classical signal requires $O(N\log_2 N)$ operations via the fast Walsh-Hadamard transform. This results in an exponential quantum advantage when the QWHT is used as a modular block within a larger quantum algorithm, relative to classical fast Walsh-Hadamard transform-based approaches operating on explicit data. From an application perspective, the proposed sequency band-energy estimation may be interpreted as a structure-based anomaly indicator, enabling the detection of unexpected high-sequency components relative to a nominal low-sequency signal class. The algorithm is applicable to quantum-enhanced signal processing tasks such as zero-crossing analysis, band-limited noise estimation, and feature extraction in the Walsh basis.

quant-pharXiv:2602.08393Feb 10, 20260 citations

Algorithm of Walsh-Hadamard Transform with Optical Approach

Yunfu Shen

In this paper, the fast algorithm to Walsh-Hadamard transform (WHT) is studied from the point of view of the optical computation. Firstly, MSD addition and butterfly calculation in discrete WHT are presented in detail. Then the design and implementation scheme of fast algorithm of WHT is proposed and the software simulations to fast WHT are carried out optically. Meanwhile, the computational complexity of the algorithms in time and space are analyzed. The analysis shows that the problem of WHT with large-scale data can be solved in a few of clock cycles with optical computational method. The scheme also provides a good way for the design of fast optical Walsh-Hadamard transformer and an application in ternary optical computer.In this paper, the fast algorithm to Walsh-Hadamard transform (WHT) is studied from the point of view of the optical computation. Firstly, MSD addition and butterfly calculation in discrete WHT are presented in detail. Then the design and implementation scheme of fast algorithm of WHT is proposed and the software simulations to fast WHT are carried out optically. Meanwhile, the computational complexity of the algorithms in time and space are analyzed. The analysis shows that the problem of WHT with large-scale data can be solved in a few of clock cycles with optical computational method. The scheme also provides a good way for the design of fast optical Walsh-Hadamard transformer and an application in ternary optical computer.

Journal of Information and Computational ScienceSep 20, 20133 citations

A hardware accelerator approach for quantum computer modeling

Vyacheslav F. Guzik, Sergey M. Gushanskiy, Кубраков Евгений Сергеевич

WIT transactions on information and communication technologiesApr 1, 20140 citations

A quantum algorithm for counting zero-crossings

Shukla, Alok

We present a zero-crossings counting problem that is a generalization of the Bernstein-Vazirani problem. The goal of this problem is to count the number of zero-crossings (or sign changes) in a special type of sequence S, whose definition depends upon a secret string. A quantum algorithm is presented to solve this problem. The proposed quantum algorithm requires only one oracle query to solve the problem, whereas a classical algorithm would need at least n oracle queries, where $2^n$ is the size of the sequence S. In addition to solving the zero-crossings counting problem, we also give a quantum circuit for performing the Walsh-Hadamard transforms in sequency ordering. The Walsh-Hadamard transform in sequency ordering is used in a wide range of scientific and engineering applications, including in digital signal and image processing. Therefore, the proposed quantum circuit for computing the Walsh-Hadamard transforms in sequency ordering may be helpful in quantum computing algorithms for applications for which the computation of the Walsh-Hadamard transform in sequency ordering is required.We present a zero-crossings counting problem that is a generalization of the Bernstein-Vazirani problem. The goal of this problem is to count the number of zero-crossings (or sign changes) in a special type of sequence S, whose definition depends upon a secret string. A quantum algorithm is presented to solve this problem. The proposed quantum algorithm requires only one oracle query to solve the problem, whereas a classical algorithm would need at least n oracle queries, where $2^n$ is the size of the sequence S. In addition to solving the zero-crossings counting problem, we also give a quantum circuit for performing the Walsh-Hadamard transforms in sequency ordering. The Walsh-Hadamard transform in sequency ordering is used in a wide range of scientific and engineering applications, including in digital signal and image processing. Therefore, the proposed quantum circuit for computing the Walsh-Hadamard transforms in sequency ordering may be helpful in quantum computing algorithms for applications for which the computation of the Walsh-Hadamard transform in sequency ordering is required.

arXiv:2212.11814Dec 21, 20220 citations

High-Speed Emulation of the Quantum Computing Based on Logic Operations

Kosuke Saito, Yasufumi Suzuki, Minoru Fujishima +1 more

International Conference on Solid State Devices and Materials (SSDM)Sep 18, 20022 citations

O(\log_2 N)

(equivalently

O(n)

for

N=2^n

) when acting on an amplitude-encoded quantum state, whereas computing the full Walsh-Hadamard spectrum of an explicit length-

N

classical signal requires

O(N\log_2 N)

operations via the fast Walsh-Hadamard transform. This results in an exponential quantum advantage when the QWHT is used as a modular block within a larger quantum algorithm, relative to classical fast Walsh-Hadamard transform-based approaches operating on explicit data. From an application perspective, the proposed sequency band-energy estimation may be interpreted as a structure-based anomaly indicator, enabling the detection of unexpected high-sequency components relative to a nominal low-sequency signal class. The algorithm is applicable to quantum-enhanced signal processing tasks such as zero-crossing analysis, band-limited noise estimation, and feature extraction in the Walsh basis.

2^n

is the size of the sequence S. In addition to solving the zero-crossings counting problem, we also give a quantum circuit for performing the Walsh-Hadamard transforms in sequency ordering. The Walsh-Hadamard transform in sequency ordering is used in a wide range of scientific and engineering applications, including in digital signal and image processing. Therefore, the proposed quantum circuit for computing the Walsh-Hadamard transforms in sequency ordering may be helpful in quantum computing algorithms for applications for which the computation of the Walsh-Hadamard transform in sequency ordering is required.