Random circuits have emerged as an invaluable tool in the quantum computing toolkit. On the one hand, the task of sampling outputs from a random circuit has established itself as a promising approach to experimentally demonstrate the superiority of quantum computers using near-term, noisy platforms. On the other hand, random circuits have also been used to deduce far-reaching conclusions about the theoretical foundations of quantum information and communication.

One intriguing aspect of random circuits is exemplified by the entanglement phase transition that occurs in monitored circuits, where unitary gates compete with projective measurements to determine the entanglement structure of the resulting quantum state. When the measurements are sparse, the circuit is unaffected and entanglement grows ballistically; when the measurements are too frequent, the unitary dynamics is arrested or frozen. The two phases are separated by a sharp-phase transition. In this work, we discuss an experiment probing such phases using a trapped-ion quantum computer.

While entanglement is an important resource in quantum communication, it does not fully capture the non-classicality necessary to achieve universal quantum computation. A family of measures, termed "magic", is used to quantify the extent to which a quantum state can enable universal quantum computation. In this dissertation, we also discuss a newly uncovered phase transition in magic using quantum circuits that implement a random stabilizer code. This phase transition is intimately related to the error correction threshold. In this work, we present numerical and analytic characterizations of the magic transition.

Finally, we use a statistical mechanical mapping from random circuits acting on qubits to Ising models to suggest thresholds in error mitigation whenever the underlying noise of a quantum device is imperfectly characterized. We demonstrate the existence of an error-mitigation threshold in dimensions D>=2.