Fig. 3: Example modular exponentiation block circuits for factoring N = 15 with bases a ∈ {2, 7, 8, 11, 13}.
From: Realizing Shor’s algorithm with topological acoustic phase bits
Here \({U}_{a,15}^{k}\) denotes modular multiplication by ak modulo 15 acting on the computational register. Panels show the three controlled stages used in the period finding routine: (a) the \({U}_{a,15}^{4}\) block (red; executed at step 2), (b) the \({U}_{a,15}^{2}\) block (green; step 5; identical for the shown bases except the special case a = 11), and (c)-(g) the \({U}_{a,15}^{1}\) block (blue; decomposed into substeps 8.1--8.4 depending on a), shown explicitly for a = 2, 7, 8, 11, 13. Horizontal wires labeled \(\left|{{{\mathcal{P}}}}_{j}\right\rangle\) denote the computational register basis states (here j = 0, 1, 2, 3); each block is applied conditionally, controlled by the corresponding phibit from the period finding register. Dashed vertical markers label the internal substep sequencing within a block realization.
