Quartic quantum speedups for planted inference (2406.19378v2)

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.

• The paper presents a quantum algorithm that achieves nearly quartic speedup on the Planted Noisy kXOR problem, significantly outperforming classical methods.
• The approach maps the problem to a Guided Sparse Hamiltonian framework and prepares guiding states with quadratically larger overlap for efficient spectral analysis.
• The findings have important cryptographic implications, suggesting that security parameters may need adjustment to counter emerging quantum vulnerabilities.

Quartic Quantum Speedups for Planted Inference

In the domain of quantum computation, the quest for identifying problems that can exploit quantum advantages continues to be a prominent area of exploration. The paper "Quartic quantum speedups for planted inference," authored by Alexander Schmidhuber, Ryan O'Donnell, Robin Kothari, and Ryan Babbush, asserts significant advancements in this line of research. The authors introduce a quantum algorithm demonstrating a nearly quartic speedup for the Planted Noisy $k$XOR problem, a task within the domain of planted inference and learning parity with noise (LPN).

Overview and Classical Context

The Planted Noisy $k$XOR problem involves distinguishing between random instances of $k$-sparse linear equations in $n$ variables (over the field $\F_2$) and those instances containing a hidden (or "planted") solution, corrupted by noise. This problem is essential in both theoretical computer science and cryptographic constructions. The prevalent classical approach to this problem involves the Kikuchi method, where the problem is transformed into a degree-2 polynomial optimization problem, facilitating spectral analysis of a corresponding Kikuchi matrix.

Quantum Speedup

The authors build upon classical methods and propose a quantum algorithm that leverages these transformations but significantly improves on the time complexity. The key contributions can be summarized as follows:

1. Quantum Algorithm for Planted Noisy kXOR: The authors describe an efficient quantum algorithm that achieves a quartic speedup over the best-known classical algorithms. The quantum algorithm executes in a time complexity of $n^{\ell/4} \cdot \poly(n)$. Here, $\ell$ is a parameter that determines the trade-off between classical and quantum complexity.
2. Guided Sparse Hamiltonian Problem: The Planted Noisy $k$XOR problem is mapped to the Guided Sparse Hamiltonian problem. This involves defining a Hamiltonian whose eigenvalues correspond to the constraints and verifying whether its ground state has notably large eigenvalues. Their quantum algorithm makes efficient use of guiding states, which have substantial overlap with the Hamiltonian's leading eigenvector, to achieve the speedup.
3. Efficient Guiding State Preparation: A significant contribution is the method for preparing these guiding states. Leveraging properties of the problem, the authors construct these states such that they have a quadratically larger overlap with the leading eigenspace compared to random states. This is achieved using a tensorized structure that relates back to the planted vector.
4. Implications for Cryptographic Applications: The implications of these results are profound for cryptographic constructions that rely on the hardness of the Planted Noisy $k$XOR problem. By showing that certain cryptographic assumptions may be vulnerable to quantum attacks, the authors indicate that security parameters may need to be adjusted in light of quantum capabilities.

Addressing Technical Challenges

The authors face several complexities in achieving a nearly quartic speedup:

• Simulation of Kikuchi Hamiltonian: Efficient simulation of the Kikuchi Hamiltonian on a quantum computer, given its sparse nature.
• Overlap Analysis: Proving that the guiding state prepared has significant overlap with the top few eigenstates of the Hamiltonian.
• Second-moment Method: Applying rigorous probabilistic methods to establish the desired overlaps and distributions with high confidence.

The authors successfully navigate these challenges using a blend of probabilistic arguments, tensor analysis, and quantum algorithmic techniques, specifically adapted to the planted inference problems.

Future Directions and Theoretical Implications

The explicit transformation of planted inference problems into instances of the Guided Sparse Hamiltonian problem opens new directions for applying quantum algorithms beyond the presented cases. The potential for extending these speedups to other planted problems and further refining the theoretical frameworks suggests a fertile ground for future studies. The techniques developed could be instrumental in demonstrating quantum advantages for a broader class of average-case problems, potentially impacting fields like combinatorial optimization and machine learning.

Furthermore, the general framework presented may also encourage the development of new cryptographic protocols resistant to quantum attacks, ensuring the advancement of quantum-secure cryptography.

Conclusion

"Quartic quantum speedups for planted inference" marks a significant milestone in quantum algorithms for planted problems, providing a nearly quartic speedup for tackling the Planted Noisy $k$XOR problem. By mapping these problems to the Guided Sparse Hamiltonian framework and utilizing efficient guiding state preparation, the authors not only improve on classical complexities but also lay the groundwork for future quantum advancements in planted inference and cryptographic security.