Quantum computing promises to revolutionize several scientific and technological domains through fundamentally new ways of processing information. Among its most compelling applications is digital quantum simulation, where quantum computers are used to replicate the behavior of other quantum systems. This could enable the study of problems that are otherwise intractable on classical computers, transforming fields such as quantum chemistry, condensed matter physics, and materials science.

Despite this potential, realizations of practical quantum advantage for relevant problems are hindered by imperfections of current devices. This also affects quantum hardware based on superconducting circuits which is among the most advanced and scalable platforms.The envisaged long-term solution of fault-tolerant quantum computers that correct their own errors remains out of reach mainly due to the associated qubit number overhead. As a result, the field has developed strategies that combine quantum and classical resources, exploit hardware-native operations, and employ error mitigation techniques to extract meaningful results from noisy data. This thesis contributes to this broader effort by exploring methods for advancing quantum simulation across the full computational stack, including hardware-level innovations, refined techniques for noise modeling and error mitigation, and algorithmic improvements enabled by efficient measurement processing.

On the hardware side, we develop a framework for full qudit-based quantum computation using superconducting transmon circuits. This makes optimal use of the available quantum resources by exploiting higher energy levels of transmons beyond the usually employed two-dimensional qubit subspace. We demonstrate how to implement a universal qudit gate set, enabling more efficient circuit synthesis and implementations of informationally-complete (IC) measurements. On the algorithmic and software side, we present techniques for the efficient classical processing of quantum data using IC measurements. We introduce a method that reduces the variance of observable estimators in post-processing, thereby improving the efficiency of quantum algorithms without additional quantum resources. On the error mitigation side, we employ IC measurements to perform accurate quantum computations that mitigate noise in classical post-processing. We demonstrate two such error mitigation techniques on IBM Quantum hardware at scales that lie beyond the brute-force simulation capabilities of classical computers. Firstly, we use IC measurements to parallelize the subspace expansion algorithm for ground state estimation. Secondly, we leverage a tensor-network pipeline that systematically compensates for the effects of noise to simulate many-body quantum dynamics. The latter method relies on accurate models of the device's noise channels, which were in the past challenged by experimental ambiguities that led to biased noise models. We demonstrate that a novel noise learning scheme can remove these inconsistencies, improving both the bias and variance of noise-model-based error mitigation.

The results of this thesis showcase how digital quantum simulation can be advanced despite the presence of noise, pushing the frontiers of quantum computing with superconducting hardware on several fronts. This both extends the reach of error mitigation and informs future strategies towards fault-tolerant quantum computing.