The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this talk, we discuss challenges in connecting these two pictures, as well as our recent work which addresses the gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random. On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary k-design and a PRU. Finally, we discuss opportunities for future work. Based on arXiv:2510.08434.

This talk will give an overview of some recent developments in the theory of deep thermalization, the emergence of universal wavefunction ensembles on a local subsystem as a result of measuring the rest of the system. Standard thermodynamic intuition says that, at late times, these distributions should become "as random as possible" within any macroscopic constraints on the dynamics, such as symmetries and conservation laws. While this intuition is typically right, I will present a class of surprising exceptions based on the concept of quantum resource theory (QRT): random circuits of "free" (resource-nonincreasing) unitaries can undergo sharp transitions between a resourceful, deep-thermalizing phase and a resourceless, "deep-ergodicity breaking" phase upon tuning the amount of resource present in the system. These transitions are described, across a wide variety of QRTs, by a unified framework called "block sharpening". This leads to universal predictions for the location and critical exponents of these transitions for disparate physical settings, from coherence in random permutation circuits to magic in random stabilizer codes.

Shallow-depth quantum circuit complexity has received growing interest in recent years due to a number of exciting results leading to developments in quantum circuit lower-bounds, state synthesis, learning theory, and pseudorandomness. In this talk, I will give a high-level overview of the shallow-depth quantum circuit landscape and recent works in the area. The talk will aim to highlight open questions and promising directions for future research, with a focus towards resolving the central long-standing conjecture Parity ∉ QAC^0.