Permutation-invariant quantum circuits

Permutation invariant quantum circuits are SWAP invariant circuits. They form a closed subgroup of SU(n) with a corresponding subalgebra. We show how all possible circuit elements can be constructure, the structure of the group and characterize the algebra with a symmetrized construction. The structure hints at a more general mathematical structure of two-qubit-operation invariant circuits. Permutation invariant quantum circuits have a completely different scaling compared to arbitrary circuits with only a polynomial growth in the number of parameters. The reduction in parameters connects to results of the simulatability of this class of circuits and their lack of barren plateaus. We provide some applications of permutation-invariant circuits and indicate how the construction is an important stepping stone to the implementation of any discrete symmetry onto a quantum circuit.