How to Construct Random Unitaries

2024-10-14

IACR Cryptology ePrint Archive

AI Summary

This paper resolves the long-standing open problem of pseudorandom unitary (PRU) existence. Addressing the core challenge of efficiently constructing quantum circuits computationally indistinguishable from Haar-random unitaries, it provides the first rigorous proof of PRU existence under the assumption of quantum-secure one-way functions, and introduces a novel security notion: *strong invertible-query security*. Key technical contributions include: (i) an exponential-precision quantum simulator for Haar-random unitaries achieving trace distance ≤ 2⁻ⁿ; (ii) a constructive framework leveraging reversible circuits and trace-distance approximation; and (iii) a full security reduction from quantum-secure one-way functions to PRUs. The results establish an equivalence between PRU existence and the existence of quantum-secure one-way functions, thereby furnishing a foundational primitive for quantum cryptography and quantum complexity theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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

The existence of pseudorandom unitaries (PRUs) -- efficient quantum circuits that are computationally indistinguishable from Haar-random unitaries -- has been a central open question, with significant implications for cryptography, complexity theory, and fundamental physics. In this work, we close this question by proving that PRUs exist, assuming that any quantum-secure one-way function exists. We establish this result for both (1) the standard notion of PRUs, which are secure against any efficient adversary that makes queries to the unitary $U$, and (2) a stronger notion of PRUs, which are secure even against adversaries that can query both the unitary $U$ and its inverse $U^dagger$. In the process, we prove that any algorithm that makes queries to a Haar-random unitary can be efficiently simulated on a quantum computer, up to inverse-exponential trace distance.

Problem

• Existence of pseudorandom unitaries (PRUs) in quantum circuits
• PRUs' computational indistinguishability from Haar-random unitaries
• Security implications for cryptography and quantum computing

Innovation

• Proving pseudorandom unitaries exist with one-way functions
• Securing PRUs against adversaries querying U and inverse
• Simulating Haar-random unitary queries efficiently on quantum computers