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.

Transcript

  • 1. A Complexity Analysis of Shor’s Quantum Factoring Algorithm
    J. Caleb Wherry
    Austin Peay State University
    Department of Computer Science
  • 2. Outline
    Introduction
    Classical Factoring
    Quantum Computing
    Shor’s Quantum Factoring Algorithm
    Conclusion
    2
  • 3. Introduction
    3
    Relativity Computer
    QuantumComputing
    Closed Timelike Curve Computation
    DNA Computing
    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.
    3
  • 4. Classical Factoring
    4
    Composite Number
    For RSA, p & q are prime.
    Why is Factoring Hard?
    Example 1: N being 16 bits long
    Example 2: N being 32 bits long
    216 = 65,536 possible values
    232 = 4,294,967,296 possible values
    RSA
    Normal RSA public key (above N) is around 1024-2048 bits.
  • 5. Quantum Computing
    5
    Orthonormal Basis Set
    Superposition of 0 & 1
    |0 + |1
    |0
    |1
    |0
    |

    |1
    E.g.

    =
    Qubits: Photons, Electrons, Ions, etc.
    *Spin of above particles.
    |
    Bloch Sphere
  • 6. Shor’s Quantum Factoring Algorithm
    6
    We have an integer N that we want to factor.
    To factor, we have to find the period of this function:
    Where x < N and coprime to N.
    Using rules of modular arithmetic to yield:
    Set up quantum side:
    Quantum Register:
  • 7. Shor’s Quantum Factoring Algorithm
    7
    Quantum Register:
    State After Transformation:
    Measure Register 1:
    New State:
    Quantum Fourier
    Transform
    What does this do?
    “Peaks” values in Register 1 around multiples of:
    Measure this register to get one of these values, then compute classically r with continued fractions.
  • 8. Shor’s Quantum Factoring Algorithm
    8
    Fourier Transform
    http://www.academictutorials.com/graphics/graphics-fourier-transform.asp
  • 9. Shor’s Quantum Factoring Algorithm
    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.
    *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.
  • 10. Conclusion
    10
    • 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.
  • References
    11
    Arora, Sanjeev;Barak, Boaz. “Computational Complexity: A Modern Approach.” New York: Cambridge University Press. 2009. Print.
    Hayward, Matthew. “Quantum Computing and Shor’s Algorithm.” Sydney: Macquarie University Mathematics Department. 2008. Print.
    Nielsen, Michael A.; Chuang, Isaac L. “Quantum Computation and Quantum Information.” New York: Cambridge University Press. 2000. Print.
    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.
  • 16. Questions &| Comments
    12
    Questions &| Comments?