Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 324 / 2 / b / i / Solution

The map $x\mapsto -x(\mathrm{mod}K)$ is a bijection and an involution, so its permutation matrix satisfiesIn particular $S^{†}S=I$, which proves that $S$ is a unitary operator. Equivalently, it permutes the computational basis and therefore preserves every inner product.