Characterising the universal phenomena occurring in quantum many-body systems out-of-equilibrium — such as thermalisation or growth of entanglement — and the emergent laws governing them is one of the central themes of modern physics. A particularly interesting question concerns the role played in these processes by quantum mechanics, i.e., are the universal phenomena occurring in quantum many-body systems fundamentally different from those observed in classical many-body systems? If so, which of their features are genuinely quantum? I will discuss this question considering quantum many-body systems in discrete space-time, i.e. quantum circuits. I will introduce “permutation circuits”, a class of local quantum circuits that act classically — do not generate superpositions — in a special basis. Considering random (or Floquet random) permutation circuits I will show that these systems have dynamical and spectral properties that are remarkably similar to those of generic quantum circuits while I will point out and explain the key differences.

References:
BB, Klobas, Kos, Malz, Phys. Rev. X 15, 011015 (2025)
Szász-Schagrin, Mazzoni, BB, Klobas, Piroli, Phys. Rev. Res. 8, L012061 (2026)
BB, Klobas, Kos, Malz, Phys. Rev. B 113, L100302 (2026)
BB, Horvath, Klobas, Orlov, Prosen, in preparation.

Finding a construction and security proof for pseudorandom unitaries has been a major open problem in quantum cryptography. Recently, a number of works provided the first constructions and security proofs of pseudorandom unitaries (and their inverse-secure and transpose-secure variants). Now that we know how to construct pseudorandom unitaries, what do we do with them? In my talk, I will talk about the current state of applications of pseudorandom unitaries, why I think it’s been difficult to find uses that make use of the full power of pseudorandom unitaries, and some directions I think we should be looking towards to find interesting problems and applications.

In this talk we discuss the theory of random symplectic lattices and moment formulas and compare them to the notion of unitary designs. To this end, I will give a brief introduction into GKP codes as well as symplectic Hecke operators and prove the symplectic mean value theorem.

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.

I will present an exact compressed oracle for Haar-random group actions arising from arbitrary finite-dimensional unitary representations of compact groups. Based on the Peter–Weyl theorem, the oracle stores its memory in a truncated Fourier basis and updates it using Clebsch–Gordan transforms, naturally supporting forward, inverse, conjugate, and transpose queries; for the defining representation of the unitary group, we also give an efficient implementation using high-dimensional Clebsch–Gordan transforms. I will then explain how this Fourier-space construction is related, via Schur transforms, to the path-recording oracle developed by Ma and Huang, and briefly comment on concurrent and independent work by Foxman, Lombardi, Ma, Nehoran, Wright. The main goal of the talk is to stimulate discussion about potential applications of these new constructions in quantum algorithms, cryptography, and lower-bound arguments.

For generic quantum many-body systems, the circuit complexity of a time-evolved pure state is believed to grow linearly with time for an exponentially long time. This linear complexity growth has been rigorously proven in certain models, such as random quantum circuits. In this talk, we’ll consider the circuit complexity of subsystems of time-evolved states. We'll prove a sharp transition that occurs in the subsystem complexity for random quantum circuits and discuss a number of conjectures for subsystem complexity growth inspired by holography. We’ll also discuss both the dynamics of subsystem entanglement as well as the distinguishability of subsystems.

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.

Random circuits yield shallow, approximate unitary k-designs even in constrained connectivity, though these circuits are spatially and temporally structured depending on k. It has remained open whether this structure is essential, or if time-homogeneous and more spatially symmetric circuit ensembles also yield shallow k-designs. Here I address this question using the tools of quantum relative entropy decay. Canonical examples of unstructured random circuits such as 1-dimensional brickwork and gates placed randomly via graph edges yield fast relative entropy decay, thereby converging quickly in diamond norm distance. This talk will emphasize the aspects of quantum relative entropy that enable these results and seem ripe for broader applications.

In this talk, I will introduce the Magic Hierarchy, a new circuit model that alternates between arbitrary Clifford circuits and constant-depth layers of two-qubit gates. This model unifies several previously studied models, including QAC0f and circuits with adaptive intermediate measurements. This talk I first present lower bounds at the first level of the hierarchy. I will then pose various open problems about this circuit model and show that resolving these questions has interesting implications in various areas: such as classical and quantum circuit complexity, many-body physics, communication complexity, and Hamiltonian complexity.
This talk is based on my paper.

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 ∉ QAC⁰.