A Complexity Analysis of Shor’s Quantum Factoring Algorithm

J. Caleb WherryAustin Peay State University

Department of Computer Science

Outline

I. Introduction

II. Classical Factoring

III. Quantum Computing

IV. Shor’s Quantum Factoring Algorithm

V. Conclusion

2

Introduction

3

DNA Computing

3

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.

R CTC R CR

C

0 0 0

Answer

“Causality-Respecting Register”

“Closed Timelike

Curve Register”

Polynomial Size Circuit

Closed Timelike Curve Computation

Relativity Computer Quantum Computing

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

RSANormal RSA public key (above N) is around 1024-2048 bits.

Quantum Computing

5

|0 + |1|0 |1

Orthonormal Basis Set Superposition of 0 & 1

|0 |1 |

Bloch Sphere

0 12|

E.g.

=Qubits: Photons, Electrons, Ions, etc.*Spin of above particles.

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:

Nxaf a mod)(

Where x < N and coprime to N.

Using rules of modular arithmetic to yield:

Nxx

Nx

Nx

Nx

rr

r

r

r

mod0)1)(1(

mod01)(

mod1)(

mod1

22

22

22

Quantum Register:

22 2NqN

1

0

0,|1 q

a

aq

Set up quantum side:

Shor’s Quantum Factoring Algorithm

7

Quantum Register:

1

0

0,|1 q

a

aq

Nx a mod|

1

0

mod,|1 q

a

a Nxaq

State After Transformation:

Measure Register 1:

'

,'||'|

1avec

kaa

Quantum Fourier Transform

New State:

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.

rq

Shor’s Quantum Factoring Algorithm

8Fourier Transform

http://www.academictutorials.com/graphics/graphics-fourier-transform.asp

Shor’s Quantum Factoring Algorithm

),1(

),1(

mod0)1)(1(

2

2

22

NxGCD

NxGCD

Nxx

r

r

rr

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.

Conclusion

10

We have seen how hard factoring is for even a small amount of bits.

Quantum computing is a computational paradigm that can potentially give exponential

speed up over classical computing.

Drawbacks Extremely hard to control quantum systems. Extreme engineering environments. 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.

Shafi Goldwasser. IEEE Computer Society Press, 1994. 124-136. Print.

Questions &| Comments

12

Questions &| Comments?