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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, arXiv:2505.06158
BB, Klobas, Kos, Malz, arXiv:2505.06158
BB, Horvath, Klobas, Orlov, Prosen, in preparation.