The Learning Stabilizers with Noise problem

TL;DR

This paper introduces the Learning Stabilizers with Noise (LSN) problem, a quantum analog of the classical Learning Parity with Noise (LPN) problem, providing algorithms, complexity analysis, and applications in quantum cryptography and learning theory.

Contribution

The paper defines the LSN problem, offers quantum algorithms for various noise regimes, and establishes its hardness and relevance to quantum cryptography and learning.

Findings

Quantum algorithms solve LSN in low to high noise regimes.

LSN includes LPN as a special case, indicating classical hardness.

Proves a worst-case to average-case reduction for LSN.

Abstract

Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory. Is there a natural quantum analog of the LPN problem? In this work, we introduce the Learning Stabilizers with Noise (LSN) problem, the task of decoding a random stabilizer code in the presence of local depolarizing noise. We give both polynomial-time and exponential-time quantum algorithms for solving LSN in various depolarizing noise regimes, ranging from extremely low noise, to low constant noise rates, and even higher noise rates up to a threshold. Next, we provide concrete evidence that LSN is hard. First, we show that LSN includes LPN as a special case, which suggests that it is at least as hard as its classical counterpart. Second, we prove a worst-case to average-case reduction for variants of LSN. We then ask: what is the computational complexity of solving LSN? Because the task features quantum inputs, its complexity cannot be characterized by traditional complexity classes. Instead, we show that the LSN problem lies in a recently introduced (distributional and oracle) unitary synthesis class. Finally, we identify several applications of our LSN assumption, ranging from the construction of quantum bit commitment schemes to the computational limitations of learning from quantum data.