In this paper, we provide three methods for constructing quantum error-correcting (QEC) codes via the Hermitian or Euclidean sums and hulls of cyclic codes over R, where \(R=\mathbb {F}_l[\gamma ]/\langle \gamma ^2\rangle \) with l to be prime power. We define two Gray maps \(\Psi _1\) and \(\Psi _2\) , and study the Hermitian or Euclidean sums and hulls of cyclic codes over R. Under the Gray maps \(\Psi _1\) and \(\Psi _2\) , two Gray map images are obtained for Hermitian or Euclidean sums of cyclic codes, respectively. Then, three new classes of QEC codes are obtained via two Gray map images and the Calderbank–Shor–Steane construction or Quantum construction X. Moreover, the QEC codes constructed are new in the sense that their parameters are different from all the previously known ones.
