Abstract Current quantum computers have a limited number of resources and are heavily affected by internal and external noise. Therefore, small, noise-tolerant circuits are of great interest. With regard to circuit size, it is especially important to reduce the number of required qubits. Concerning to fault-tolerance, circuits entirely built with Clifford+T gates allow the use of error correction codes. However, the T-gate has an excessive cost, so circuits with a high number of T-gates should be avoided. This work focuses on optimising in such terms an operation that is widely used in larger circuits and algorithms: the calculation of the absolute-value of two’s complement encoded integers. The proposed circuit halves the number of required T gates with respect to the best circuit currently available in the literature. Moreover, our circuit requires at least 2 qubits less than the other circuits for such an operation.

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The structure of multivariate serial codes over a finite chain ring R is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some nontrivial codes over R. The structure of the dual codes in the serial abelian case is also derived, and some conditions for the existence of self‐dual codes over R are studied.