# Quartic quantum speedups for planted inference

arXiv: https://arxiv.org/abs/2406.19378 · HTML: https://mlsanity.com/paper/2406.19378/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2024.

## Built on by (most cited first, 1 of 1)

- [Practical advantage beyond the quadratic speedup limit with fully-quantum walks](https://mlsanity.com/paper/2607.22818/) (2026)

## Abstract (arXiv)

We describe a quantum algorithm for the Planted Noisy kXOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic (4th 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 kXOR 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.