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Society for Industrial and Applied Mathematics eBooks

A Classical Quadratic Speedup for Planted kkk xor

Meghal Gupta, William He, Ryan O’Donnell, Noah Singer

Published Jan 1, 2026• 0 citations
Publisher

Abstract

A recent work of Schmidhuber et al. (QIP, SODA, & Phys. Rev. X 2025) exhibited a quantum algorithm for the noisy planted \(k\)xor problem running quartically faster than all known classical algorithms. In this work, we design a new classical algorithm that is quadratically faster than the best previous one, in the case of large constant \(k\). Thus for such \(k\), the quantum speedup of Schmidhuber et al. becomes only quadratic (though it retains a space advantage). Our algorithm, which also works in the semirandom case, combines tools from sublinear-time algorithms (essentially, the birthday paradox) and polynomial anticoncentration.

DOI
10.1137/1.9781611978971.88

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A recent work of Schmidhuber et al. (QIP, SODA, & Phys. Rev. X 2025) exhibited a quantum algorithm for the noisy planted (k)xor problem running quartically faster than all known classical algorithms. In this work, we design a new classical algorithm that is quadratically faster than the best previous one, in the case of large constant (k). Thus for such (k), the quantum speedup of Schmidhuber et al. becomes only quadratic (though it retains a space advantage). Our algorithm, which also works in the semirandom case, combines tools from sublinear-time algorithms (essentially, the birthday paradox) and polynomial anticoncentration.
We describe a quantum algorithm for the Planted Noisy k XOR Problem (also known as Sparse Learning Parity with Noise) that achieves a nearly (fourth-power) speedup over the best known classical algorithm while using exponentially less space. 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 superquadratic quantum attacks.
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum algorithms could then lead to an exponential space saving as well as a quartic speedup over classical. Here we show how to accelerate both classical and quantum algorithms quadratically, while maintaining the same quartic separation between them. That is, our classical algorithm here is quadratically faster than the original classical algorithm, while the quantum algorithm is eigth-power faster than the original classical algorithm. We then give a further modification of the quantum algorithm, increasing its speedup over the modified classical algorithm to the sixth power. We only prove these speedups for detection, rather than recovery, but we give a strong plausibility argument that our algorithm achieves recovery also. Note added: After this paper was prepared, A. Schmidhuber pointed out to me Ref. 3. This improves the best existing bounds on the spectral norm of a certain random operator. Because the norm of this operator enters into the runtime, with this improvement on the norm, we no longer have a provable polynomial speedup. Our results are phrased in terms of certain properties of the spectrum of this operator (not merely the largest eigenvalue but also the density of states). So, if these properties still hold, the speedup still holds. Rather than modify the paper, I have left it unchanged but added a section at the end discussing the needed property of density of states and considering for which problems (there are several problems for which this kind of quartic quantum speedup has been used and the techniques here will likely be applicable to several of them) the property is likely to hold.
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 [Has20], 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.
Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods that obtain quantum speedups with respect to the dimension nnn and number of constraints mmm. While their dependence on other parameters suggests no overall speedup over classical methodologies, some quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to our advantage, and present an iterative refinement scheme for the Hamiltonian Updates algorithm of Brandão et al. (Quantum 6, 625 (2022)) to exponentially improve the dependence of their algorithm on precision. As a result, we obtain a classical algorithm to solve the semidefinite relaxation of Quadratic Unconstrained Binary Optimization problems (QUBOs) in matrix multiplication time. Provided access to a quantum read/classical write random access memory (QRAM), a quantum implementation of our algorithm exhibits a worst case running time of O(ns+n1.5⋅polylog(n,∥C∥F,1ε))\mathcal{O} \left(ns + n^{1.5} \cdot \text{polylog} \left(n, \| C \|_F, \frac{1}ε \right) \right)O(ns+n1.5⋅polylog(n,∥C∥F​,ε1​)).
This paper's aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover\cite{grover}. Grover's algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this paper is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover's algorithm or similar techniques, which can be used as subroutines in many quantum algorithms.}{The tools we provide are carefully constructed: they are easy to use, and in many cases they are asymptotically faster than the best tools previously available. The tools we build on include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer.}{After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known deterministic classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible deterministic classical algorithm. These algorithms solve problems from the fields of graph theory and computational geometry, and some employ dynamic programming techniques. We discuss a breadth-first search that is faster than Θ(edges)\Theta(\text{edges})Θ(edges) (the classical limit) in a dense graph, maximum-points-on-a-line in O(N3/2lg⁡N)O(N^{3/2}\lg N)O(N3/2lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.
Search is one of the most commonly used primitives in quantum algorithm design. It is known that quadratic speedups provided by Grover’s algorithm are optimal, and no faster quantum algorithms for Search exist. While it is known that at least some quantum computation is required to achieve these speedups, the existing bounds do not rule out the possibility of an equally fast hybrid quantum-classical algorithm where most of the computation is classical. In this work, we study such hybrid algorithms, and we show that classical computation, unless it by itself can solve the Search problem, cannot assist quantum computation. In addition, we generalize this result to algorithms with subconstant success probabilities.