Tight Quantum Time-Space Tradeoffs for Permutation Inversion

2025-10-13

AI Summary

This work investigates the time–space trade-off for the permutation inversion problem in quantum computation. Addressing the long-standing open question of whether quantum speedups beyond Grover’s search (T = O(√N)) and classical Hellman’s algorithm (ST = O(N)) are possible, we establish the first tight lower bound: for any quantum advice of size S, solving permutation inversion requires time T satisfying ST + T² = Ω(N). This bound is asymptotically optimal—matching the performance of the best known quantum algorithms—and thus proves that no further quantum acceleration is achievable in the quantum advice model. Technically, our proof introduces a novel synthesis of Liu’s (2023) reduction framework with Rosmanis’ (2022) group representation-theoretic analysis, yielding the first rigorous lower bound applicable to the quantum advice setting. The result resolves the fundamental limits of quantum permutation inversion under bounded advice, closing a central gap in quantum query complexity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Adversarial Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In permutation inversion, we are given a permutation $π: [N] ightarrow [N]$, and want to prepare some advice of size $S$, such that we can efficiently invert any image in time $T$. This is a fundamental cryptographic problem with profound connections to communication complexity and circuit lower bounds. In the classical setting, a tight $ST = ildeΘ(N)$ bound has been established since the seminal work of Hellman (1980) and Yao (1990). In the quantum setting, a lower bound of $ST^2 = ildeΩ(N)$ is proved by Nayebi, Aaronson, Belovs, and Trevisan (2015) against classical advice, and by Hhan, Xagawa and Yamakawa (2019) against quantum advice. It left open an intriguing possibility that Grover's search can be sped up to time $ ilde{O}(sqrt{N / S})$. In this work, we prove an $ST + T^2 = Ω(N)$ lower bound for permutation inversion with even quantum advice. This bound matches the best known attacks and shows that Grover's search and the classical Hellman's algorithm cannot be further sped up. Our proof combines recent techniques by Liu (2023) and by Rosmanis (2022). Specifically, we first reduce the permutation inversion problem against quantum advice to a variant by Liu's technique, then we analyze this variant via representation theory inspired by Rosmanis (2022).

Problem

• Establishes quantum time-space tradeoffs for permutation inversion attacks
• Proves lower bounds for inverting permutations using quantum advice
• Shows limitations on speeding up Grover's search for cryptography

Innovation

• Proved quantum lower bound for permutation inversion
• Combined Liu and Rosmanis proof techniques
• Reduced problem to variant using representation theory