Symmetries and Scale Invariance in Global Maps of Quantum Circuits

We present a combinatorial analysis of any maximal set of gate compositions parameterized over a finite alphabet which revealed an underlying fractal structure. Results are presented as global maps in the form of large 2D arrays where each row stands for an unfolded list of matrix elements for any unitary matrix resulting from a succession of Kronecker products while columns stand for all possible words representing such sequences. These can be given either in lexicographic order or via a non-binary (n-ary) Gray code in which case any such map can be factorized.