Strong random unitaries and fast scrambling

2025-09-30

AI Summary

Simulating Haar-random unitaries on physical systems is central to quantum gravity and many-body physics—especially for phenomena involving information scrambling, the butterfly effect, and the Hayden–Preskill protocol, which require simultaneous oracle access to a unitary $U$ and its inverse $U^dagger$, complex conjugate $U^*$, and transpose $U^T$. Conventional unitary designs and pseudorandom unitaries (PRUs) fail to guarantee security against such multi-query adversaries. We introduce the notions of **strong unitary designs** and **strong pseudorandom unitaries**, which formally capture joint indistinguishability under arbitrary combinations of these four queries. Leveraging shallow random circuits composed of independent two-qubit Haar gates—without ancillary qubits—we construct strong designs of depth $O(log^2 n)$ and strong PRUs of $mathrm{poly}(log n)$ depth, via rigorous combinatorial analysis and low-degree polynomial approximations. Our results yield the first constructive proof of fast scrambling and close a foundational gap in approximate unitary modeling for physical experiments.

Technology Category

• Machine Learning: Quantum Machine Learning
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

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 $ ext{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.

Problem

• Capturing random unitary behavior under inverse, conjugate, and transpose queries
• Achieving robust unitary designs with optimal logarithmic circuit depth
• Providing operational proof for fast scrambling conjecture in black holes

Innovation

• Strong unitary designs robust under all queries
• Optimal circuit depth logarithmic in qubit count
• Fast scrambling conjecture proven with operational evidence