A complexity efficient penta-diagonal quantum smoothing filter for bio-medical signal denoising: a study on ECG

Abstract

Extracting information-bearing signal from a noisy environment has been a practical challenge in both classical and quantum computing formalism, especially in critical signal processing applications. To filter out the effect of noise, we propose a quantum smoothing filter built upon quantum formalism-based circuits applied for electrocardiogram signal denoising. The proposed quantum filter is a conceptually novel framework with an advantage in computational complexity as compared to the existing classical filters, such as discrete wavelet transform and empirical mode decomposition, whereas it achieves similar performance metrics for the accuracy of the filter. Further, we exploit the penta-diagonal Toeplitz structure of the smoothing filter, which gives approximately 48%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$48\\%$$\\end{document} gate cost reduction for 10 qubit circuit compared to the standard Hamiltonian simulation without structure. The run-time complexity using the quantum matrix inversion technique for the structured matrix is given by O~κ2poly(logN)εP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ilde{{\\mathscr {O}}}\\left( \\frac{\\kappa ^2 \ ext {poly}(\\log {N})}{\\varepsilon _P}\\right)$$\\end{document} for condition number κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} of the N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\ imes N$$\\end{document} filter matrix within precision εP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon _P$$\\end{document}. Embedding fixed sparsity of the banded matrix, the quantum filter shows potentially better run-time complexity than classical filtering techniques. For the quantifiable research results of our work, we have shown several performance metrics, such as mean-square error and peak signal-to-noise ratio analysis, with a bound of error due to observation noise, simulation error and quantum measurement uncertainty.