Quantum planted-clique detection in the conjectured classical-hard regime

Determine whether a uniform polynomial-time quantum algorithm can achieve constant-advantage planted-clique detection from one classical graph when k=floor(n^(1/2-epsilon)) for any fixed 0<epsilon<1/2.

Background

The paper studies planted-clique detection in the regime k=n1/2-epsilon, where statistical detection is possible but polynomial-time classical detection is conjectured to be hard. A quantum algorithm operating on the same single classical graph and achieving constant distinguishing advantage would establish a conditional quantum advantage under the classical planted-clique conjecture. The paper does not resolve whether such an algorithm exists.

References

A uniform polynomial-time quantum algorithm that achieves constant-advantage detection at $k=\lfloor n{1/2-\varepsilon}\rfloor$, for even one fixed $0<\varepsilon<1/2$, would establish an average-case quantum advantage under the classical conjecture. Here both algorithms receive the same input: one classical graph. Whether such a quantum algorithm exists remains open.

— Planted Cliques and Quantum Symmetry-Adapted Measurements  (2609.40310 - Havlicek et al., 30 Sep 2026) in Section 1, Introduction

An efficient measurement with constant advantage on these reduced states would give an efficient planted-clique distinguisher. Our contribution is to formulate this measurement problem explicitly and establish its statistical sufficiency; its computational complexity remains open.

— Planted Cliques and Quantum Symmetry-Adapted Measurements  (2609.40310 - Havlicek et al., 30 Sep 2026) in Section 1, paragraph “Schur measurements and retained information”

We determine how many copies of the state are necessary and sufficient to discriminate the signal. We prove a lower and upper bounds for joint measurements. We then identify the limiting experiment as observing the graph up to complementation and derive its Helstrom and pretty good decision rules. These results quantify the information in the encoding; efficient detection in the conjectured hard regime remains open.

— Planted Cliques and Quantum Symmetry-Adapted Measurements  (2609.40310 - Havlicek et al., 30 Sep 2026) in Section 3, Binary Phase States, opening paragraph

The computational question is whether a smaller, efficiently implementable action can expose a useful part of this separation through its labels.

— Planted Cliques and Quantum Symmetry-Adapted Measurements  (2609.40310 - Havlicek et al., 30 Sep 2026) in Section 4.5, “Which smaller actions retain an informative label?”