Learning states and noise is a crucial part of quantum computation, whether it is to verify state preparation, benchmark gates, or learn the outcome of a quantum circuit. My work falls under this broad theme and is divided into the following two parts.

The first part focuses on quantum state tomography of continuous-variable (CV) states and obtaining precision bounds using shadow tomography. Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called 'classical shadows', with powerful methods to bound the estimators used. We recast existing experimental protocols for CV quantum state tomography in the classical shadow framework, obtaining rigorous bounds on the number of independent measurements needed to estimate density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon-number-resolving, and photon-parity protocols using our method, and benchmark these results against numerical simulations as well as experimental data from optical homodyne experiments.

The second part focuses on open quantum systems with non-Markovian, or time-correlated, interactions. General studies in quantum characterization assume the absence of any time correlations, which does not hold up with increasing system sizes. We studied the effects of non-Markovianity through the output of a popular benchmarking technique called randomized benchmarking (RB). We introduced two models that exemplify Markovian and non-Markovian interactions, created an efficient algorithm to study the latter nontrivial model, and examined the outputs. We observed a non-exponential decay of average sequence fidelity in the non-Markovian model - a marked distinction from a Markovian model.

Finally, we designed a compressive tomography approach for learning non-Markovian quantum dynamics from RB data. Our method employs machine learning to reconstruct system-environment interaction unitaries from experimentally accessible information. We demonstrate successful characterization of complex quantum dynamics involving both Markovian and non-Markovian environments simultaneously. The approach leverages gradient descent with parametrization of SU(n) matrices to learn the complete interaction unitary, guided by RB fidelities. We applied this framework to realistic exchange-only qubit models to successfully quantify leakage from computational states, achieving high reconstruction fidelities. The learned models demonstrate robust extrapolation beyond training regimes, accurately predicting system behavior for sequence lengths up to five times longer than training data.

These results inform efforts to incorporate non-Markovian quantum noise in the characterization and benchmarking of quantum devices.

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