Thomas Schuster

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.

We prove that recognizing the phase of matter of an unknown quantum state is quantum computationally hard. More specifically, we show that the worst-case runtime of any phase recognition algorithm must grow exponentially in the correlation length $\xi$ of the state. This exponential growth renders the problem practically infeasible even for moderate constant values of the correlation length $\xi$, and leads to super-polynomial quantum computational time in the system size $n$ whenever $\xi = \omega(\log n)$. Our results apply to a substantial portion of all known phases of matter, including symmetry-breaking phases and symmetry-protected topological phases for any discrete on-site symmetry group in any spatial dimension. To establish this hardness, we extend the study of pseudorandom unitaries to quantum systems with symmetries. We prove that symmetric pseudorandom unitaries exist under standard cryptographic conjectures, and can be constructed in extremely low circuit depths for any discrete on-site group. We also provide extensions of our results to systems with translation invariance and purely classical phases of matter. A key technical limitation is that the locality of the parent Hamiltonian of the states we consider is linear in $\xi$; removing this constraint remains an important open question.

Understanding how fast physical systems can resemble Haar-random unitaries is a fundamental question in physics. Many experiments of interest in quantum gravity and many-body physics, including the butterfly effect in quantum information scrambling and the Hayden-Preskill thought experiment, involve queries to a random unitary~$U$ alongside its inverse~$U^\dagger$, conjugate~$U^*$, and transpose~$U^T$. However, conventional notions of approximate unitary designs and pseudorandom unitaries (PRUs) fail to capture these experiments. In this work, we introduce and construct strong unitary designs and strong PRUs that remain robust under all such queries. Our constructions achieve the optimal circuit depth of $\mathcal{O}(\log n)$ for systems of $n$ qubits. We further show that strong unitary designs can form in circuit depth $\mathcal{O}(\log^2 n)$ in circuits composed of independent two-qubit Haar-random gates, and that strong PRUs can form in circuit depth $\poly(\log n)$ in circuits with no ancilla qubits. Our results provide an operational proof of the fast scrambling conjecture from black hole physics: every observable feature of the fastest scrambling quantum systems reproduces Haar-random behavior at logarithmic times.