Learning with errors may remain hard against quantum holographic attacks
Yunfei Wang, Xin Jin, Junyu Liu
TL;DR
This work identifies a fourth resolution to the tension between LWE hardness and holographic entropy: estimating bulk surface areas to the precision required for entropy is itself computationally hard. By constructing two holographic quantum algorithms, it shows that leading-order RT-length measurements demand exponentially many samples in the boundary size $N$, while subleading FLM corrections require exponential-time reconstruction of the bulk covariance, aligning holographic entropy with quantum complexity bounds. A comparative analysis with BKZ-based LWE attacks places holographic and lattice-based approaches in the same exponential regime, albeit with different resource types (measurements vs. computation). The results preserve the quantum-extended Church–Turing thesis and illuminate the nuanced complexity of bulk observables in AdS/CFT, with implications for quantum cryptanalysis and the operational limits of holographic duality.
Abstract
The Learning with Errors (LWE) problem underlies modern lattice-based cryptography and is assumed to be quantum hard. Recent results show that estimating entanglement entropy is as hard as LWE, creating tension with quantum gravity and AdS/CFT, where entropies are computed by extremal surface areas. This suggests a paradoxical route to solving LWE by building holographic duals and measuring extremal surfaces, seemingly an easy task. Three possible resolutions arise: that AdS/CFT duality is intractable, that the quantum-extended Church-Turing thesis (QECTT) fails, or that LWE is easier than believed. We develop and analyze a fourth resolution: that estimating surface areas with the precision required for entropy is itself computationally hard. We construct two holographic quantum algorithms to formalize this. For entropy differences of order N, we show that measuring Ryu-Takayanagi geodesic lengths via heavy-field two-point functions needs exponentially many measurements in N, even when the boundary state is efficiently preparable. For order one corrections, we show that reconstructing the bulk covariance matrix and extracting entropy requires exponential time in N. Although these tasks are computationally intractable, we compare their efficiency with the Block Korkine-Zolotarev lattice reduction algorithm for LWE. Our results reconcile the tension with QECTT, showing that holographic entropy remains consistent with quantum computational limits without requiring an intractable holographic dictionary, and provide new insights into the quantum cryptanalysis of lattice-based cryptography.