Quartic quantum speedups for planted inference

TL;DR

This paper introduces a quantum algorithm that significantly accelerates solving the Planted Noisy k-XOR problem, achieving nearly quartic speedup over classical methods with minimal qubit requirements, impacting cryptographic security.

Contribution

It presents a novel quantum algorithm for the Planted Noisy k-XOR problem, extending prior work and demonstrating a nearly quartic speedup with logarithmic qubits.

Findings

Achieves nearly quartic quantum speedup over classical algorithms.

Uses logarithmically many qubits, making it resource-efficient.

Suggests cryptographic schemes based on this problem may be vulnerable to quantum attacks.

Abstract

We describe a quantum algorithm for the Planted Noisy $k$XOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic ($4$th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings, by building on his quantum algorithm for the Tensor Principal Component Analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi Method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the Guided Sparse Hamiltonian problem. Since the Planted Noisy $k$XOR problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to super-quadratic quantum attacks.