Recent advances in probing and controlling quantum systems have brought us closer to Feynman's vision of simulating nature using quantum mechanics. Quantum processors promise efficient simulation of strongly correlated electrons, which suffer from the exponential scaling of classical simulation resources. Additionally, quantum processors provide access to quantum information properties that are not easily accessible in conventional condensed matter experiments. In this dissertation, I explore three distinct approaches to studying the dynamics of quantum information in the context of many-body physics simulations. First, I present an information-theoretic study of the dynamics of monitored variational circuits that prepare the ground states of strongly correlated Hamiltonians (Chapter 3). Viewing the optimization as a communication problem reveals "coding barren plateaus'', where a finite communication rate can be achieved between the sender and the receiver despite vanishing gradients. In a parallel study (Chapter 4), I present an adaptive variational algorithm to prepare the eigenstates of the Fermi-Hubbard model by building entanglement one gate at a time. This approach results in shallower circuits and circumvents barren plateaus, potentially enabling more efficient ground state preparation on near-term devices. Second, I focus on quantum information dynamics in strongly correlated electron systems, both clean and disordered. I present a real-device implementation for efficient disorder averaging, one of the central challenges in the computational study of many-body localization due to rare events, by leveraging quantum parallelism (Chapter 5). I examine the signature of localization in the dynamics of the second Renyi entropy, which can be extended to measure mutual information and detect rare events to determine the stability of the many-body localization phase. I also explore learning thermodynamics from the simulation of quantum dynamics by employing diffusion maps—an unsupervised machine learning method—to uncover the quantum phase diagram of the transverse field Ising model and map entropy across the phase diagram with only 500 shots per data point (Chapter 6). I do so by developing time average classical shadows (TACS), an efficient classical representation of the microcanonical ensemble. Additionally, I discuss a novel shadow tomographic technique called the valence bond shadows that offers intuitive and efficient method to estimate observables of quantum spin liquids prepared on quantum processors (Chapter 7). Third, simulating long-time dynamics of strongly correlated electrons on quantum processors will likely require fault-tolerant quantum computing using error-correcting codes. By studying the channel capacity of realistic quantum device models, I find emergent coding phases tailored to specific noise characteristics (Chapter 8). Carrying out this approach on current noisy devices could provide a systematic way to construct quantum codes for robust computation and communication. Although access to quantum information properties is challenging using conventional analytical and numerical methods, the same challenge provides a promising path to quantum advantage in near-term devices. My studies highlight the potential of near-term quantum simulations to deepen our understanding of strongly correlated electrons by enabling the direct probing of quantum information-theoretic quantities in their dynamics.