The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.

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Rupert C. D. Young, Philip M. Birch, and Chris R. Chatwin, "A simplification of the Shor quantum factorization algorithm employing a quantum Hadamard transform," Proc. SPIE 10649, Pattern Recognition and Tracking XXIX, 1064903 (Presented at SPIE Defense + Security: April 18, 2018; Published: 27 April 2018); https://doi.org/10.1117/12.2309468.