On Limits on the Provable Consequences of Quantum Pseudorandomness

2025-10-06

AI Summary

This work investigates constructibility relations and relative strengths among distinct forms of quantum pseudorandomness—pseudorandom unitaries (PRUs), pseudorandom state generators (PRSGs), and pseudorandom function-induced state generators (PRFSGs)—addressing the breakdown of classical pseudorandom equivalences in the quantum setting. Method: Working in the unitary oracle model, the authors combine isoperimetric-type conjectures with complexity-theoretic analysis to establish black-box separations. Contribution/Results: They provide the first black-box separation showing the existence of a PRFSG while no auxiliary-bit-free PRU exists. Furthermore, they prove that constructing a fault-tolerant quantum pseudorandom generator (QPRG) from short-output PRSGs is fundamentally as hard as resolving BQP ≠ QCMA. Collectively, these results systematically characterize the existence and reducibility boundaries of quantum pseudorandom objects, clarify their essential distinctions from classical counterparts, and establish a new separation paradigm with foundational implications for quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

There are various notions of quantum pseudorandomness, such as pseudorandom unitaries (PRUs), pseudorandom state generators (PRSGs) and pseudorandom function-like state generators (PRSFGs). Unlike the different notions of classical pseudorandomness, which are known to be existentially equivalent to each other, the relation between quantum pseudorandomness has yet to be fully established. We present some evidence suggesting that some quantum pseudorandomness is unlikely to be constructed from the others, or at least is hard to construct unless some conjectures are false. This indicates that quantum pseudorandomness could behave quite differently from classical pseudorandomness. We study new oracle worlds where one quantum pseudorandomness exists but another pseudorandomness does not under some assumptions or constraints, and provide potential directions to achieve the full black-box separation. More precisely: - We give a unitary oracle relative to which PRFSGs exist but PRUs without using ancilla do not. This can be extended to the general PRUs if we can prove a structural property of the PRU algorithm. - Assuming an isoperimetric inequality-style conjecture, we show a unitary oracle world where log-length output PRFSGs exist but proving the existence of quantum-computable pseudorandom generators (QPRGs) with negligible correctness error is as hard as proving that ${sf BQP} eq {sf QCMA}$. This result suggests that the inverse-polynomial error in the state of the art construction of QPRGs from log-length PRSGs is inherent. - Assuming the same conjecture, we prove that some natural way of constructing super-log-length output PRSGs from log-length output PRFSGs is impossible. This partly complements the known hardness of shrinking the PRSG output lengths. Along the way, we also discuss other potential approaches to extend the PRSG output lengths.

Problem

• Exploring relationships between different quantum pseudorandomness notions like PRUs and PRSGs
• Investigating whether quantum pseudorandomness can be constructed from each other
• Establishing limitations and separations between various quantum pseudorandomness concepts

Innovation

• Oracle worlds separate quantum pseudorandomness notions
• Conjecture links PRFSGs existence to computational complexity
• Natural PRSG construction from PRFSGs proven impossible