Fig. 4: Example modular exponentiation block circuits for factoring N = 35 with base a = 4.
From: Realizing Shor’s algorithm with topological acoustic phase bits
Here \({U}_{4,35}^{k}\) denotes modular multiplication by 4k modulo 35 acting on the computational register. Shown for the three controlled stages used by the period finding routine: (a) \({U}_{4,35}^{4}\) (red; decomposed into substeps 2.1--2.2), (b) \({U}_{4,35}^{2}\) (green; substeps 5.1--5.5), and (c) \({U}_{4,35}^{1}\) (blue; substeps 8.1--8.7). Horizontal wires labeled \(\left|{{{\mathcal{P}}}}_{j}\right\rangle\) denote the computational register basis states (here j = 0, …, 5); 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.
