Hardness of Quantum Distribution Learning and Quantum Cryptography

2025-07-01

AI Summary

This work investigates the fundamental relationship between the existence of quantum one-way puzzles (OWPuzzs) and the hardness of quantum distribution learning. Method: We formulate an average-case quantum distribution learning model, extending the PAC learning framework, and employ black-box reductions alongside complexity-theoretic analysis involving classes such as PP, BQP, and SampBQP. Contribution/Results: We establish, for the first time, a necessary and sufficient condition for the existence of OWPuzzs: they exist if and only if accurate quantum distribution learning—measured by KL divergence—is average-case hard. Furthermore, we prove that PP ≠ BQP is equivalent to the hardness of adversarial quantum learning; however, deriving OWPuzzs existence from this requires overcoming the infinite polynomial hierarchy barrier. Our results forge a critical bridge between computational complexity theory and quantum learning theory, yielding several new complexity class separations, including conditional separations among PP, BQP, and SampBQP under standard complexity assumptions.

Technology Category

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

Application Category

• Security and Privacy: Blockchains and distributed ledgers
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The existence of one-way functions (OWFs) forms the minimal assumption in classical cryptography. However, this is not necessarily the case in quantum cryptography. One-way puzzles (OWPuzzs), introduced by Khurana and Tomer, provide a natural quantum analogue of OWFs. The existence of OWPuzzs implies $PP eq BQP$, while the converse remains open. In classical cryptography, the analogous problem-whether OWFs can be constructed from $P eq NP$-has long been studied from the viewpoint of hardness of learning. Hardness of learning in various frameworks (including PAC learning) has been connected to OWFs or to $P eq NP$. In contrast, no such characterization previously existed for OWPuzzs. In this paper, we establish the first complete characterization of OWPuzzs based on the hardness of a well-studied learning model: distribution learning. Specifically, we prove that OWPuzzs exist if and only if proper quantum distribution learning is hard on average. A natural question that follows is whether the worst-case hardness of proper quantum distribution learning can be derived from $PP eq BQP$. If so, and a worst-case to average-case hardness reduction is achieved, it would imply OWPuzzs solely from $PP eq BQP$. However, we show that this would be extremely difficult: if worst-case hardness is PP-hard (in a black-box reduction), then $SampBQP eq SampBPP$ follows from the infiniteness of the polynomial hierarchy. Despite that, we show that $PP eq BQP$ is equivalent to another standard notion of hardness of learning: agnostic. We prove that $PP eq BQP$ if and only if agnostic quantum distribution learning with respect to KL divergence is hard. As a byproduct, we show that hardness of agnostic quantum distribution learning with respect to statistical distance against $PPT^{Σ_3^P}$ learners implies $SampBQP eq SampBPP$.

Problem

• Characterizing one-way puzzles via quantum distribution learning hardness
• Linking PP≠BQP to agnostic quantum distribution learning difficulty
• Exploring worst-case to average-case reductions for quantum learning

Innovation

• Characterizes OWPuzzs via quantum distribution learning
• Links PP≠BQP to agnostic learning hardness
• Shows worst-case hardness implies SampBQP≠SampBPP