"As mentioned [earlier](#shorsalgorithm), many of the experimental demonstrations of Shor's algorithm rely on significant optimisations based on apriori knowledge of the expected results. We will follow the formulation in [this](http://science.sciencemag.org/content/351/6277/1068) paper, which demonstrates a reasonably scalable realisation of Shor's algorithm using $N = 15$. Below is the first figure from the paper, showing various quantum circuits, with the following caption: _Diagrams of Shor’s algorithm for factoring $N = 15$, using a generic textbook approach (**A**) compared with Kitaev’s approach (**B**) for a generic base $a$. (**C**) The actual implementation for factoring $15$ to base $11$, optimized for the corresponding single-input state. Here $q_i$ corresponds to the respective qubit in the computational register. (**D**) Kitaev’s approach to Shor’s algorithm for the bases ${2, 7, 8, 13}$. Here, the optimized map of the first multiplier is identical in all four cases, and the last multiplier is implemented with full modular multipliers, as depicted in (**E**). In all cases, the single QFT qubit is used three times, which, together with the four qubits in the computation register, totals seven effective qubits. (**E**) Circuit diagrams of the modular multipliers of the form $a \\bmod N$ for bases $a = {2, 7, 8, 11, 13}$._\n",
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