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Quartic quantum speedups for community detection

Alexander Schmidhuber, Alexander Zlokapa

TL;DR

This work develops a quantum algorithm that achieves a quartic speedup over the best classical Kikuchi-based method for hypergraph community detection in p-marginal HSBMs with even p ≥ 4, while delivering a superpolynomial space advantage. By extending the Kikuchi hierarchy to generalized SBMs and constructing an efficiently preparable guiding state, the authors quantify a spectral-detection approach that bridges information-theoretic limits and computational hardness. They prove LCDF-based lower bounds matching the classical Kikuchi performance up to log factors, and show the quantum algorithm attains nearly quartic speedup across a broad parameter regime, with the speedup interpolating as parameters vary. The results suggest the robustness of Kikuchi-based quantum speedups beyond Tensor PCA and pXORSAT, and point toward a marginal-order criterion as a key predictor of quantum advantage in planted inference problems.

Abstract

Community detection is a foundational problem in data science. Its natural extension to hypergraphs captures higher-order correlations beyond pairwise interactions. In this work, we develop a quantum algorithm for hypergraph community detection that achieves a quartic quantum speedup over the best known classical algorithm, along with superpolynomial savings in space. Our algorithm is based on the Kikuchi method, which we extend beyond previously considered problems such as Tensor PCA and $p$XORSAT to a broad family of generalized stochastic block models. To demonstrate (near) optimality of this method, we prove matching lower bounds (up to logarithmic factors) in the low-degree framework, showing that the algorithm saturates a smooth statistical-computational tradeoff. The quantum speedup arises from a quantized version of the Kikuchi method and is based on the efficient preparation of a guiding state correlated with the underlying community structure. Our work suggests that prior quantum speedups using the Kikuchi method are sufficiently robust to encompass a broader set of problems than previously believed; we conjecture that a quantity known as marginal order characterizes the existence of these quantum speedups.

Quartic quantum speedups for community detection

TL;DR

This work develops a quantum algorithm that achieves a quartic speedup over the best classical Kikuchi-based method for hypergraph community detection in p-marginal HSBMs with even p ≥ 4, while delivering a superpolynomial space advantage. By extending the Kikuchi hierarchy to generalized SBMs and constructing an efficiently preparable guiding state, the authors quantify a spectral-detection approach that bridges information-theoretic limits and computational hardness. They prove LCDF-based lower bounds matching the classical Kikuchi performance up to log factors, and show the quantum algorithm attains nearly quartic speedup across a broad parameter regime, with the speedup interpolating as parameters vary. The results suggest the robustness of Kikuchi-based quantum speedups beyond Tensor PCA and pXORSAT, and point toward a marginal-order criterion as a key predictor of quantum advantage in planted inference problems.

Abstract

Community detection is a foundational problem in data science. Its natural extension to hypergraphs captures higher-order correlations beyond pairwise interactions. In this work, we develop a quantum algorithm for hypergraph community detection that achieves a quartic quantum speedup over the best known classical algorithm, along with superpolynomial savings in space. Our algorithm is based on the Kikuchi method, which we extend beyond previously considered problems such as Tensor PCA and XORSAT to a broad family of generalized stochastic block models. To demonstrate (near) optimality of this method, we prove matching lower bounds (up to logarithmic factors) in the low-degree framework, showing that the algorithm saturates a smooth statistical-computational tradeoff. The quantum speedup arises from a quantized version of the Kikuchi method and is based on the efficient preparation of a guiding state correlated with the underlying community structure. Our work suggests that prior quantum speedups using the Kikuchi method are sufficiently robust to encompass a broader set of problems than previously believed; we conjecture that a quantity known as marginal order characterizes the existence of these quantum speedups.

Paper Structure

This paper contains 28 sections, 28 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Technical summary
  3. Organization of the paper
  4. The Kikuchi method for -marginal HSBM
  5. Kikuchi method
  6. Lower bounds against low coordinate degree functions
  7. Quantum algorithm for -marginal HSBM
  8. Quantum algorithm
  9. Quantum space advantage
  10. Conclusion
  11. Correctness and efficiency of the classical Kikuchi algorithm
  12. Kikuchi hierarchy
  13. Analysis in the random case
  14. Analysis in the planted case
  15. Proof of the detection threshold
  16. ...and 13 more sections

Key Result

Theorem 4

No function with coordinate degree $\ell$ can solve hypergraph community detection on a $p$-marginal HSBM with $p > 2$ if

Theorems & Definitions (79)

  • Remark 1
  • Definition 2: $p$-marginal HSBM with $k$ communities
  • Theorem 4: Lower bound on hypergraph community detection, informal
  • Theorem 5: Classical upper bound on hypergraph community detection, informal
  • Theorem 6: Quantum upper bound on hypergraph community detection, informal
  • Remark 7: Generality of quantum algorithm
  • Definition 8
  • Definition 9
  • Definition 10: Bosonic Kikuchi matrix
  • Theorem 11
  • ...and 69 more