# A quantum Fourier transform (QFT) based note detection algorithm

**Authors:** Shlomo Kashani, Maryam Alqasemi, Jacob Hammond  
**arXiv:** [2204.11775](https://arxiv.org/abs/2204.11775)  
**Date:** 2022-04-25  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2204.11775v2/

## Key points

- The study focuses on the Quantum Fourier Transform (QFT) and its application in developing a quantum music note detection algorithm.
- QFT is a fundamental component of quantum information processing and has been used in various quantum algorithms, including Shor's factorization algorithm.
- Shor's algorithm efficiently finds prime factors of large numbers on quantum computers, with implications for cryptography and number theory.
- The researchers examine the structure and implementation of QFT for creating a music note detection algorithm on simulated and real quantum computers.
- The study uses a symbolic representation validated using the SymPy package to simplify verifying the algorithm.
- The algorithm is implemented as a quantum circuit using IBM's qiskit library.
- Period detection on an actual single musical tone is demonstrated using different numbers of qubits, showcasing the potential application of QFT in signal processing tasks.
- Detailed explanations are provided for Discrete Fourier Transform (DFT), including nth roots of unity and its matrix form.
- The transition from DFT to QFT is discussed, highlighting concepts such as relative phase shift and binary number representation.
- Numerical examples are provided to verify the QFT algorithm manually and symbolically using universal quantum gates.
- Scaling considerations for implementing QFT in larger circuits are also discussed.
- Other potential applications of QFT in signal processing tasks are explored, emphasizing how leveraging quantum computing techniques can offer superior solutions compared to classical computers.

## Summary

This study focuses on the Quantum Fourier Transform (QFT) and its application in the development of a quantum music note detection algorithm. The QFT is a fundamental component of quantum information processing and has been widely used in various quantum algorithms, including Shor's factorization algorithm. Shor's algorithm efficiently finds prime factors of large numbers on quantum computers, which has significant implications for cryptography and number theory. The researchers examine the structure and implementation of the QFT for creating a music note detection algorithm on both simulated and real quantum computers. While formal approaches exist for verifying quantum algorithms, this study adopts a simpler symbolic representation validated using the SymPy package, which symbolically replicates quantum computing processes. The algorithm is then implemented as a quantum circuit using IBM's qiskit library. The researchers demonstrate period detection on an actual single musical tone using different numbers of qubits, showcasing the potential application of the QFT in signal processing tasks. The paper provides detailed explanations of the Discrete Fourier Transform (DFT), including nth roots of unity and its matrix form. It also discusses the transition from DFT to QFT, highlighting concepts such as relative phase shift and binary number representation. To verify the QFT algorithm, numerical examples are provided for manually applying the QFT and symbolically applying it using universal quantum gates. The authors also discuss scaling considerations for implementing the QFT in larger circuits. In addition to music note detection, this study explores other potential applications of the QFT in signal processing tasks. It emphasizes how leveraging quantum computing techniques can offer superior solutions to previously unsolvable problems on classical computers. Overall, this research contributes to our understanding of the QFT's structure and implementation while demonstrating its potential use in music note detection and signal processing tasks.