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- 1. A Complexity Analysis of Shor’s Quantum Factoring Algorithm<br />J. Caleb Wherry<br />Austin Peay State University<br />Department of Computer Science<br />
- 2. Outline<br />Introduction<br />Classical Factoring<br />Quantum Computing<br />Shor’s Quantum Factoring Algorithm <br />Conclusion<br />2<br />
- 3. Introduction<br />3<br />Relativity Computer<br />QuantumComputing<br />Closed Timelike Curve Computation<br />DNA Computing<br />S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669. <br />3<br />
- 4. Classical Factoring<br />4<br />Composite Number<br />For RSA, p & q are prime.<br />Why is Factoring Hard?<br />Example 1: N being 16 bits long<br />Example 2: N being 32 bits long<br />216 = 65,536 possible values<br />232 = 4,294,967,296 possible values<br />RSA<br />Normal RSA public key (above N) is around 1024-2048 bits.<br />
- 5. Quantum Computing<br />5<br />Orthonormal Basis Set<br />Superposition of 0 & 1<br />|0 + |1<br />|0<br />|1<br />|0<br />|<br /><br />|1<br />E.g.<br /><br />=<br />Qubits: Photons, Electrons, Ions, etc.<br />*Spin of above particles.<br />|<br />Bloch Sphere<br />
- 6. Shor’s Quantum Factoring Algorithm<br />6<br />We have an integer N that we want to factor.<br />To factor, we have to find the period of this function:<br />Where x < N and coprime to N.<br />Using rules of modular arithmetic to yield:<br />Set up quantum side:<br />Quantum Register:<br />
- 7. Shor’s Quantum Factoring Algorithm<br />7<br />Quantum Register:<br />State After Transformation:<br />Measure Register 1:<br />New State:<br />Quantum Fourier <br />Transform<br />What does this do?<br />“Peaks” values in Register 1 around multiples of:<br />Measure this register to get one of these values, then compute classically r with continued fractions. <br />
- 8. Shor’s Quantum Factoring Algorithm<br />8<br />Fourier Transform<br />http://www.academictutorials.com/graphics/graphics-fourier-transform.asp<br />
- 9. Shor’s Quantum Factoring Algorithm<br />Once we have r, computing above will yield a non-trivial factor of N. QFT takes no more than O(M3), where M is the number of bits N is.<br />*Caveat: Shor’s Algorithm resides in BQP, so the answer could be wrong. Re-run to get another answer or to verify first answer. Note though, running polynomial algorithm multiple times is much better than running 1 exponential algorithm.<br />
- 10. Conclusion<br />10<br /><ul><li> We have seen how hard factoring is for even a small amount of bits.
- 11. Quantum computing is a computational paradigm that can potentially give exponential speed up over classical computing.
- 12. Drawbacks
- 13. Extremely hard to control quantum systems.
- 14. Extreme engineering environments.
- 15. Algorithms are counterintuitive.</li></li></ul><li>References<br />11<br />Arora, Sanjeev;Barak, Boaz. “Computational Complexity: A Modern Approach.” New York: Cambridge University Press. 2009. Print.<br />Hayward, Matthew. “Quantum Computing and Shor’s Algorithm.” Sydney: Macquarie University Mathematics Department. 2008. Print.<br />Nielsen, Michael A.; Chuang, Isaac L. “Quantum Computation and Quantum Information.” New York: Cambridge University Press. 2000. Print.<br />Shor, Peter W. “Algorithms for Quantum Computation: Discrete Logarithms and Factoring.” Proc. 35th Annual Symposium on Foundations of Computer Science. Ed. ShafiGoldwasser. IEEE Computer Society Press, 1994. 124-136. Print.<br />
- 16. Questions &| Comments<br />12<br />Questions &| Comments?<br />

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