Tight Quantum Time-Space Tradeoffs for Function Inversion

TL;DR

This paper establishes tight quantum time-space lower bounds for function inversion, showing Grover's search is optimal even with quantum advice, and introduces a general framework for quantum time-space lower bounds applicable to various problems.

Contribution

The paper proves a new quantum lower bound of ST + T^2 = Ω(N) for function inversion, extending prior bounds and developing a versatile framework for quantum time-space complexity analysis.

Findings

Quantum advice does not significantly improve inversion bounds.

Grover's search remains optimal with quantum advice for S = O(√N).

Framework applies to Yao's box problem and cryptography.

Abstract

In function inversion, we are given a function $f: [N] \mapsto [N]$, and want to prepare some advice of size $S$, such that we can efficiently invert any image in time $T$. This is a well studied problem with profound connections to cryptography, data structures, communication complexity, and circuit lower bounds. Investigation of this problem in the quantum setting was initiated by Nayebi, Aaronson, Belovs, and Trevisan (2015), who proved a lower bound of $ST^2 = \tilde\Omega(N)$ for random permutations against classical advice, leaving open an intriguing possibility that Grover's search can be sped up to time $\tilde O(\sqrt{N/S})$. Recent works by Hhan, Xagawa, and Yamakawa (2019), and Chung, Liao, and Qian (2019) extended the argument for random functions and quantum advice, but the lower bound remains $ST^2 = \tilde\Omega(N)$. In this work, we prove that even with quantum advice, $ST + T^2 = \tilde\Omega(N)$ is required for an algorithm to invert random functions. This demonstrates that Grover's search is optimal for $S = \tilde O(\sqrt{N})$, ruling out any substantial speed-up for Grover's search even with quantum advice. Further improvements to our bounds would imply new classical circuit lower bounds, as shown by Corrigan-Gibbs and Kogan (2019). To prove this result, we develop a general framework for establishing quantum time-space lower bounds. We further demonstrate the power of our framework by proving quantum time-space lower bounds for Yao's box problem and salted cryptography.