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arXiv:1505.04584 · Quantum Information Processing

Analogue algorithm for parallel factorization of an exponential number of large integers: II—optical implementation

Vincenzo Tamma

Published Nov 26, 2015• 11 citations
arXivHTMLPDF
DOI
10.1007/s11128-015-1189-4

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In a recent experiment, we have shown how optical interference can be exploited in order to perform computations and in particular factoring large integers. This has lead us to the developing of a novel analogue algorithm which allows the factorization of several large integers in a single run. This work represents a step stone towards the developing of novel physical networks able to implement quantum digital algorithms other than Shor’s algorithm. We describe the main ideas leading to this new line of research referring to future publications for a detailed description.
Quantum integer factorization is a groundbreaking field that explores the use of quantum algorithms, notably Shor's algorithm, to efficiently factorize large integers into their prime components. Its potential implications on modern encryption techniques, such as the Rivest-Shamir-Adleman (RSA) algorithm, are of considerable interest. This paper's central aim is to contrast the computational time and implementation of the Integer Factorization Problem (IFP) between classical and quantum computers. The inner workings of Quantum Shor's algorithm are broken down step by step, and the estimated processing time is contrasted with the classical General Number Field Sieve (GNFS) approach. Given the exponential speedup provided by quantum computing, this comparison helps assess Shor's algorithm's possible advantage over GNFS in factoring large numbers. Direct comparisons with GNFS are currently difficult due to Shor's algorithm's reliance on large-scale quantum computer development for practical execution speeds. This paper focuses on solving integer factorization up to2n2^{n}2nbits.
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.