A MATHEMATICAL INTRODUCTION TO SHOR’S QUANTUM FACTORING ALGORITHM

J. Caleb WherryDepartments of Computer Science & MathematicsAustin Peay State University

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OUTLINE Introduction to Quantum Computing

Classical vs. Quantum Computing Qubits & Quantum Logic Gates

Classical Factoring Classical Methods for Factoring NP vs. BQP Complexity Classes

Quantum Factoring Mathematical Method Quantum Fourier Transform

Conclusion Future of Classical Cryptography Future of Quantum Computing

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INTRODUCTION TO QUANTUM COMPUTING

Bits On/Off Voltage

0 or 1

n bits = n bits of info

Classical Logic Gates Universal And, Or, & Not Copy – non-universal

{One,Two}-ary Operations on the Boolean Algebra

Quantum Bits (Qubits) Elementary Particle Spin

Photon, Electron, Ion, etc. 0, 1, or Superposition of

0 &1 n qubits = 2n bits of info

Quantum Logic Gates Universal And, Or, & Not Copy – non-universal

No-copy Theorem Linear Transformations

Classical Computing Quantum Computing

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INTRODUCTION TO QUANTUM COMPUTING

0

10 1

2

|0 + |1|0 |1

Orthonormal Basis Set

Superposition of 0 & 1

|0 |1 |

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INTRODUCTION TO QUANTUM COMPUTING

Pauli Matrices

Hadamard Gate

Pauli-X

Pauli-Y

Pauli-Z

Hadamard

Pauli-X : Not gatePauli-Y: Not gate with i multiplePauli-Z: Flips sign of second entangled state

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INTRODUCTION TO QUANTUM COMPUTING Experimental Problems

Decoherence Quantum Noise Collapse of Quantum States

Uncertainty Principle NMR Measurement No-cloning Theorem

Quantum Weirdness Superposition Entanglement Teleportation Superdense Coding

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CLASSICAL FACTORING Exponentially Hard

Thought to be in NP Complexity Class

Best Known Classical Method: General Number Field Sieve

~O(e(log N)1/3 (log log N)2/3)

Other Methods GCD Brute Force

RSA

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CLASSICAL & QUANTUM COMPLEXITY

Image Source: http://en.wikipedia.org/wiki/Quantum_computer

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QUANTUM FACTORING Exponentially Faster

BQP ~O(log(N)3)

2 Main Parts 1) Classical

Reduction to Order-Finding Problem Done on a Classical Computer

2) Quantum Solve the Order-Finding Problem Done on a Quantum Computer

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QUANTUM FACTORING Classical Part

Start with an odd composite number N Pick a random q < N

Note that this is random which will come up later… GCD(q,N) == 1 ? continue : halt

Halt because there is a non-trivial factor. Cannot use for period finding.

We now use our quantum computer to compute r, the period of: f(x) = qx mod N

We restrict r to be even (if odd, repick q). r satisfies:

ar 1 mod Nar – 1 0 mod N

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QUANTUM FACTORING Classical Part cont…

ar -1 = (ar/2 -1)(ar/2 + 1) 0 mod N Thus, N divides (ar/2 -1)(ar/2 + 1) If both terms are prime (which is what we want), then

these are the only two solutions. If not, then return and repick q.

This seems pretty straight forward except for the quantum part. Finding r is not an easy problem.

We will employ the use of the quantum Fourier transform and make use of quantum superposition to solve this problem…

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QUANTUM FACTORING Quantum Part

N is still our composite number We start off by finding a Q such that: Q = 2q , N2 ≤ Q ≤ 2N2

When we find such a Q, we can see that:Q/r > N

We then initialize our input and output registers to hold Q qubits and apply the Hadamard gate to entangle all the states.

We now have an entangled state Apply the Quantum Fourier Transform to this state

Destructive interference will occur and cancel out certain states (Double-slit experiment)

|

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QUANTUM FACTORING Quantum Part cont…

Perform a measurement and retrieve an approximation of r

We then check with a classical computer if r is correct. If not then we redo the calculation

Probabilistic Algorithm If answer is incorrect, then the q we picked at the

beginning is not correct so we start over This causes this algorithm to be in BQP

Exponentially Faster

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CONCLUSION Future of Classical Cryptography

A Look Into The Future: Moore’s Law 10-20 Years Until Quantum Scale is Reached

Other Classical Cryptography Methods Immune to Quantum Parallel Attacks Lattice-based Cryptography

Future of Quantum Computing Extreme Conditions

4 - 20 Kelvin Applications Are Minimal Currently No Connection To Solving NP(-complete) Problems

Although This is Commonly Thought True!

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REFERENCES Bernstein, E., Vazirani, U., “Quantum Complexity

Theory.”

Chuang, I., “Quantum Algorithms and Their Implementations: QuISU – An Introduction for Undergraduates.”

Lloyd, S., “Quantum Information Science.”

Nielson, M., Chuang, I., “Quantum Computation and Quantum Information.”

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QUESTIONS?

COMMENTS?