Random unitaries from Hamiltonian dynamics

2025-10-09

AI Summary

Can a time-independent Hamiltonian generate quantum dynamics indistinguishable from Haar-random unitaries? While local random circuits approximate the Haar measure, they do not correspond to physically realizable stationary Hamiltonians. Method: We combine Hamiltonian dynamics analysis, unitary *k*-design theory, and efficient simulation techniques grounded in standard cryptographic assumptions. Results: We prove that constant-range local Hamiltonians cannot produce pseudorandom unitaries. In contrast, one-dimensional polylog-local random Hamiltonians achieve unitary *k*-designs and cryptographic pseudorandomness within constant evolution time. Our construction yields the first physically realizable, efficiently simulatable short-time pseudorandom evolution model—rigorously establishing Haar-indistinguishable dynamical randomness under a stationary Hamiltonian framework for the first time.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a $2$-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary $k$-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.

Problem

• Studying indistinguishability of Hamiltonian dynamics from random unitaries
• Establishing limitations of constant-local Hamiltonians for randomness
• Proving polylog-local Hamiltonians can achieve true quantum randomness

Innovation

• Polylog-local Hamiltonians simulate random unitaries
• Hamiltonian dynamics form unitary k-designs efficiently
• One-dimensional systems achieve Haar-random indistinguishability