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We construct a new solution to the tetrahedron equation and thethree-dimensional (3D) reflection equation by extending the quantum clusteralgebra approach by Sun and Yagi concerning the former. We consider theFock-Goncharov quivers associated with the longest elements of the Weyl groupsof type $A$ and $C$, and investigate the cluster transformations correspondingto changing a reduced expression into a `most distant' one. By devising a newrealization of the quantum $y$-variables in terms of $q$-Weyl algebra, thesolutions are extracted as the operators whose adjoint actions yield thecluster transformations of the quantum $y$-variables. Explicit formulas oftheir matrix elements are also derived for some typical representations.

Quantum cluster algebras and 3D integrability: Tetrahedron and 3D reflection equations