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A Classical Quadratic Speedup for Planted kXOR
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A Classical Quadratic Speedup for Planted kXOR
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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.
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The Kikuchi Hierarchy is Sharp for $k$XOR
Normalized Kikuchi matrices achieve the sharp m ~ rho^{-2} n^{k/2} / ell^{k/2-1} trade-off with no logarithmic loss for detection, recovery, and two-sided refutation in kXOR, with matching low-degree lower bounds.
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