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Quantum Detection of Sequency-Band Structure

Alok Shukla, Prakash Vedula

TL;DR

The paper addresses detecting structured components in quantum-encoded signals by probing energy in sequency bands. It introduces a fully quantum primitive based on the sequency-ordered $\text{QWHT}$, a comparator-based band oracle, and Quantum Amplitude Estimation to quantify band mass $P_{[a,a+M)}$. It shows that the sequency-ordered transform on amplitude-encoded data can be implemented in circuit depth $O(\log_2 N)$, offering an exponential advantage over classical $O(N \log_2 N)$ Walsh-Hadamard approaches when embedded as a modular component in larger quantum computations. The method supports coherent operation and provides a structure-based anomaly indicator for high-sequency content, with numerical validation on small systems demonstrating correctness of sequencing, band-energy estimation, and zero-crossing interpretation.

Abstract

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.

Quantum Detection of Sequency-Band Structure

TL;DR

The paper addresses detecting structured components in quantum-encoded signals by probing energy in sequency bands. It introduces a fully quantum primitive based on the sequency-ordered , a comparator-based band oracle, and Quantum Amplitude Estimation to quantify band mass . It shows that the sequency-ordered transform on amplitude-encoded data can be implemented in circuit depth , offering an exponential advantage over classical Walsh-Hadamard approaches when embedded as a modular component in larger quantum computations. The method supports coherent operation and provides a structure-based anomaly indicator for high-sequency content, with numerical validation on small systems demonstrating correctness of sequencing, band-energy estimation, and zero-crossing interpretation.

Abstract

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 (equivalently for ) when acting on an amplitude-encoded quantum state, whereas computing the full Walsh-Hadamard spectrum of an explicit length- classical signal requires 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.
Paper Structure (15 sections, 2 theorems, 39 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 15 sections, 2 theorems, 39 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Zero-crossings counting problem
  4. A technical lemma
  5. Sequency ordered Walsh-Hadamard transforms
  6. Walsh basis functions in sequency and natural ordering
  7. Quantum circuit for performing the sequency ordered Walsh-Hadamard transforms
  8. Quantum Algorithm for High-Sequency Noise Detection
  9. Problem Statement
  10. Overall Algorithm Approach
  11. Detailed Oracle Design ($U_S$)
  12. Amplitude Estimation for Probability Mass
  13. Computational Complexity
  14. Numerical Simulations
  15. Conclusion

Key Result

Lemma 2.1.1

For an integer $m$ with $1 < m \leq n$, and for $x = x_{n-1}\,x_{n-2} \,\ldots \, x_1\, x_0$ (or equivalently, $x$ with a decimal representation $x = \sum_{j=0}^{m-1} \, x_j 2^j$), with $x_j \in \{0,1\}$, define $Z_m(x)$ as Then, we have Here $s(m) = s_{m-1}\,s_{m-2}\,\ldots \, s_1\, s_0$ and $s_0 \, \oplus \, s_1 \, \oplus \, s_2 \, \oplus \, \ldots \, \oplus \, s_{m-1}= (s_0 + s_1 + s_2 + \ld

Figures (6)

  • Figure 1: Walsh basis functions in the sequency ordering for $N=8$.
  • Figure 2: Walsh basis functions in the natural ordering for $N=8$.
  • Figure 3: Quantum circuit for computing the Walsh-Hadamard transform in sequency ordering.
  • Figure 4: Numerical results for the constant (DC) input signal.
  • Figure 5: Numerical results for the piecewise-constant (edge-like) input signal.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Example 2.0.1
  • Lemma 2.1.1
  • proof
  • Corollary 2.1.2
  • proof
  • Example 2.1.3
  • Remark 4.4.1