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# A Complexity Analysis of Shor's Quantum Factoring Algorithm

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• We can be clever and only look at SQRT(N_bits).-Classical factoring algorithm is of order O(e^log(N_bits)^3) and called General Number Field Sieve.-Number of particles in universe: 10^80 ~ 2^256
• -Coprime meaning no common factor other than 1. GCD(x,N)=1.-Exponential number of a’s to check since we don’t know it. a is from 0-q-1-r has to be even or above won’t work.-GCD of last factors and N will yield non-trivial factor of N.
• -take O(M^3) steps, where M is the number of bits N is.
• ### A Complexity Analysis of Shor's Quantum Factoring Algorithm

1. 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. 2. Outline<br />Introduction<br />Classical Factoring<br />Quantum Computing<br />Shor’s Quantum Factoring Algorithm <br />Conclusion<br />2<br />
3. 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. 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. 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. 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. 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. 8. Shor’s Quantum Factoring Algorithm<br />8<br />Fourier Transform<br />http://www.academictutorials.com/graphics/graphics-fourier-transform.asp<br />
9. 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. 10. Conclusion<br />10<br /><ul><li> We have seen how hard factoring is for even a small amount of bits.
11. 11. Quantum computing is a computational paradigm that can potentially give exponential speed up over classical computing.
12. 12. Drawbacks
13. 13. Extremely hard to control quantum systems.
14. 14. Extreme engineering environments.
15. 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 />