CNOT-count optimized quantum circuit of the Shor's algorithm

We present improved quantum circuit for modular exponentiation of a constant, which is the most expensive operation in Shor's algorithm for integer factorization. While previous work mostly focuses on minimizing the number of qubits or the depth of circuit, we try to minimize the number of CNOT gate which primarily determines the running time on a ion trap quantum computer. First, we give the implementation of basic arithmetic with known lowest number of CNOT gate and the construction of improved modular exponentiation of a constant by accumulating intermediate date and windowing technique. Then, we precisely estimate the number of improved quantum circuit to perform Shor's algorithm for factoring a $n$-bit integer, which is $217\frac{n^3}{\log_2n}+4n^2+n$. According to the number of CNOT gates, we analyze the running time and feasibility of Shor's algorithm on a ion trap quantum computer. Finally, we discuss the lower bound of CNOT numbers needed to implement Shor's algorithm.