Liang Mao

Haar-random unitaries are fundamental mathematical tools for understanding quantum many-body dynamics, yet they fail to obey basic physical constraints imposed by Hamiltonians. In this work, we explore how to reconcile random unitary models with physical constraints imposed by Hamiltonians, addressing the central question: Can we efficiently generate random unitaries while obeying Hamiltonian constraints? First, we consider the role of energy conservation in the setting where the Hamiltonian $H$ is completely known. There, we show that energy-conserving pseudorandom unitaries (PRUs) exist for random local commuting Hamiltonians, assuming quantum-secure one-way functions exist. However, we also prove that energy-conserving PRUs do not exist for some local translation-invariant Hamiltonians, even in one-dimensional systems. Furthermore, we show that determining whether energy-conserving PRUs exist for a family of Hamiltonians is undecidable. Second, we consider Hamiltonian time dynamics itself when there is incomplete knowledge of $H$. In this setting, we prove that random unitaries $e^{-iHt}$ generated from any ensemble of constant-local Hamiltonians $H$ cannot form approximate unitary designs or PRUs. This barrier vanishes when we relax locality: we construct an ensemble of polylog-local Hamiltonians $H$ that generates short-time dynamics which form both a unitary design and a PRU. Our results reveal fundamental computational barriers emerging from energy conservation constraints, highlighting the tension between common models of ergodicity and the structure of physical dynamics.