Qubit noise spectroscopy is an important tool for the experimental investigation of open quantum systems. However, conventional techniques for noise spectroscopy are time-consuming, because they require measurements of the noise spectral density at many different frequencies. In this dissertation, I describe an alternative approach to noise spectroscopy, which requires fewer resources, and relies on direct measurement of arbitrary linear functionals of the noise spectral density. This method uses random pulse sequences with carefully-controlled correlations, which are chosen using algorithms for phase retrieval. These measurements allow us to  reconstruct sparse noise spectra via compressed sensing. Our simulations of the performance of the random pulse sequences on a realistic physical system, self-assembled quantum dots, reveal a speedup of an order of magnitude in extracting the noise spectrum, compared to conventional dynamical decoupling approaches.

I also advance this method in two complementary directions: expanding its applicability to a larger class of noise spectra, and simplifying its implementation. First, we extend the random pulse sequences method to reconstruct piecewise-linear noise spectra, which more realistically model many physical systems. We show through numerical simulations that the new method resolves finer spectral features while maintaining an order-of-magnitude speedup over conventional approaches to noise spectroscopy. Second, we introduce a simplified variant using Rademacher measurements, optimized for sparse spectra, that greatly reduces experimental complexity without compromising reconstruction accuracy. Together, these developments broaden the reach of random pulse sequences noise spectroscopy and enhance its practicality for high-resolution, resource-efficient noise characterization in realistic quantum systems.

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