Abstract
A quantum time-dependent spectrum analysis, or simply, quantum spectral analysis (QSA) is presented in this work, and it’s based on Schrödinger’s equation. In the classical world, it is named frequency in time (FIT), which is used here as a complement of the traditional frequency-dependent spectral analysis based on Fourier theory. Besides, FIT is a metric which assesses the impact of the flanks of a signal on its frequency spectrum, not taken into account by Fourier theory and lets alone in real time. Even more, and unlike all derived tools from Fourier Theory (i.e., continuous, discrete, fast, short-time, fractional and quantum Fourier Transform, as well as, Gabor) FIT has the following advantages, among others: 1) compact support with excellent energy output treatment, 2) low computational cost, O(N) for signals and O(N2) for images, 3) it does not have phase uncertainties (i.e., indeterminate phase for a magnitude = 0) as in the case of Discrete and Fast Fourier Transform (DFT, FFT, respectively). Finally, we can apply QSA to a quantum signal, that is, to a qubit stream in order to analyze it spectrally.
Highlights
The main concepts related to Quantum Information Processing (QIP) may be grouped in the next topics: quantum bit, Bloch’s Sphere
A quantum circuit is a model for quantum computation in which a computation is a sequence of quantum gates; which are reversible transformations on a quantum mechanical analog of an n-bit register
This can be compared to the classical discrete Fourier transform which takes O(2n2) gates, which is exponentially more than O(n2)
Summary
The main concepts related to Quantum Information Processing (QIP) may be grouped in the next topics: quantum bit (qubit, which is the elemental quantum information unit), Bloch’s Sphere In developing the QSP framework, we are at liberty to impose quantum mechanical constraints that we find useful and to avoid those that are not This framework provides a unifying conceptual structure for a variety of traditional processing techniques and a precise mathematical setting for developing generalizations and extensions of algorithms; leading to a potentially useful paradigm for signal processing, with applications in areas including frame theory, quantization and sampling methods, detection, parameter estimation, covariance shaping, and multiuser wireless communication systems. The discrete Fourier transform can be implemented as a quantum circuit consisting of only O(n2) Hadamard gates and controlled phase shift gates, where n is the number of qubits [1] This can be compared to the classical discrete Fourier transform which takes O(2n2) gates (where n is the number of bits), which is exponentially more than O(n2).
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