Applications of Harmonic Analysis in Quantum Computing

Abstract

The quantum Fourier transformation has been demonstrated to be a useful tool in dealing with quantum-mechanical problems. This paper discusses how quantum Fourier transformation is applied in finding the period of functions within Shor’s algorithm, allowing the algorithm to be much more efficient in factorizing large integers than classic algorithms. This paper also discusses how Grover’s algorithm uses the concept of harmonic process, where amplitude amplification is likened to harmonic oscillation for improving the efficiency of searching unsorted data base. The paper further explores the application of harmonic analysis in quantum error correction, particularly in stabilizer codes. The paper also introduces wavelet analysis as an alternative approach for detecting and correcting localized quantum errors. Finally, the paper extends the discussion into the potential future directions of harmonic analysis in quantum computing, such as extending Fourier transformation to non-abelian groups and using spherical harmonics in quantum algorithms which may optimize current quantum algorithms and solve currently unsolved quantum questions.