Strong random unitaries and fast scrambling
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| Format: | Preprint |
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2025
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| _version_ | 1866916980272922624 |
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| author | Schuster, Thomas Ma, Fermi Lombardi, Alex Brandao, Fernando Huang, Hsin-Yuan |
| author_facet | Schuster, Thomas Ma, Fermi Lombardi, Alex Brandao, Fernando Huang, Hsin-Yuan |
| contents | 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 $O(\log n)$ for systems of $n$ qubits. We further show that strong unitary designs can form in circuit depth $O(\log^2 n)$ in circuits composed of independent two-qubit Haar-random gates, and that strong PRUs can form in circuit depth $\text{poly}(\log n)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_26310 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong random unitaries and fast scrambling Schuster, Thomas Ma, Fermi Lombardi, Alex Brandao, Fernando Huang, Hsin-Yuan Quantum Physics Strongly Correlated Electrons Computational Complexity Cryptography and Security High Energy Physics - Theory 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 $O(\log n)$ for systems of $n$ qubits. We further show that strong unitary designs can form in circuit depth $O(\log^2 n)$ in circuits composed of independent two-qubit Haar-random gates, and that strong PRUs can form in circuit depth $\text{poly}(\log n)$ 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. |
| title | Strong random unitaries and fast scrambling |
| topic | Quantum Physics Strongly Correlated Electrons Computational Complexity Cryptography and Security High Energy Physics - Theory |
| url | https://arxiv.org/abs/2509.26310 |