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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?