Quantum ComputationThe Mathematics of Information

J. Caleb WherryAustin Peay State University

Departments of Computer Science, Mathematics, & Physics

Outline

I. Classical Computationi. History

a. Babbage, ENIAC, Vacuum Tubes, & the Transistorb. Moore’s Law

ii. Computation & Complexity Theoryiii. Cbits, Logic Gates, & the Circuit Modeliv. Moore’s Law Revisited

II. Quantum Computationi. Mathematical Formalisms (Linear Algebra & Quantum Mechanics)

a. Qubits, Quantum Gates, & the Quantum Circuit Model

ii. BQP & the Power of Q.C.iii. Quantum Q.C. Implementations

a. NMR, Iron Trap, Superconducting Qubits, & Topological Q.C.

iv. Quantum Algorithmsa. Grover’s Search & Shor’s Factoring Algorithms

III.Other Computational Paradigmsi. Zeno’s Computerii. Relativity Computeriii. Closed Timelike Curve Computationiv. DNA Computing 2

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Classical Computation

History

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Pascaline - 1623 Difference Engine - 1823

Step Reckoner - 1673

History

5

ENIAC - 1946

Vacuum Tubes

History

6

Texas Instruments 1954 Transistor

History

7

Moore’s Law

Computation & Complexity Theory

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What is computation?

Computation & Complexity Theory

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Strong Church-Turing Thesis A probabilistic Turing machine (e.g. a classical computer that can make fair

coin flips) can efficiently simulate any realistic model of computing.

ComputationA process following a well-defined model that is understood and can be

expressed in an algorithm, protocol, network topology, etc.

Computational ComplexityThe measure of the resources (e.g. time, space, basic operations, energy) used

by a computation. Measured as a function of the input size.

Turing MachineA very simplistic computer in which computations can be executed on.

1) Tape – Infinitely Long. Finite Alphabet.2) Head – Reads/Writes, Moves Tape 1 Cell

L/R.3) Table – Finite Set of Instructions.4) State Register – Current Finite State of TM.

Computation & Complexity Theory

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Computation & Complexity Theory

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Cbits, Logic Gates, & the Circuit Model

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Classical Bits• 2-state system (Boolean Algebra)• Possible states: 0 or 1 (Off or On)o 0 -> No voltageo 1 -> 0.5 voltage

If we have n classical bits, how much information do we have?

Cbits, Logic Gates, & the Circuit Model

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Basic Classical Logic Gates

•{One,Two}-ary Operations on our Boolean Algebra• Universal set of gates: (AND, NOT, & FANOUT)• What does universal mean?• Are they reversible?o What does reversible mean?

Logic Gates

Cbits, Logic Gates, & the Circuit Model

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Moore’s Law Revisited

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Moore’s Law

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Quantum Computation

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Mathematical Formalisms

|0 + |1|0 |1

Orthonormal Basis Set Superposition of 0 & 1

|0 |1 |

Bloch Sphere

Qubit – Quantum Bit

0 1

2

| E.g.

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

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Neat Mathematics

Where do qubits live?| lives in a Hilbert Space H .

H is a complete Vector Space with a defined inner product.

What does complete mean?Formal definition: a space is complete if every Cauchy Sequence converges to a point within the set.

But what does that mean?

?1

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i iFields: N, Q, R, C, H

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Mathematical Formalisms

Pauli Matrices Hadamard Gate

Pauli-X

Pauli-Y

Pauli-Z

Hadamard

Quantum Logic Gates = Linear Transformations

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Mathematical Formalisms

Quantum Weirdness

Superposition

Entanglement

Teleportation

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Mathematical Formalisms

Quantum Weirdness ISuperposition & Interference

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Mathematical Formalisms

Quantum Weirdness ISuperposition & Interference

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Mathematical Formalisms

Quantum Weirdness II

Entanglement – EPR Paradox

“Spookiness at a distance” - Einstein

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Mathematical Formalisms

Quantum Weirdness IIITeleportation

BQP & the Power of Q.C.

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If we have n qubits, how much information do we have?

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Quantum Implementations

Ion Trap

NMR

Topological Q.C.

Superconducting Qubits

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Quantum Algorithms

Grover’s Search

Normal amount of time a database search takes?

N items takes O(n) searches.

Grover’s Search takes O( SQRT(N) ) searches for N items.

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Quantum Algorithms

Shor’s Factoring

Fastest Classical Factoring Algorithm:

General Number Field SieveO(e^((log N)^1/3 (log log N)^2/3))

Shor’s Algorithm Factors in: O(log(N)^3)

Exponential Speedup!

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Other Computational Paradigms

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Other Computational Paradigms

Zeno’s Computer

STEP 1

STEP 2

STEP 3STEP 4

STEP 5Tim

e (s

econ

ds)

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Other Computational Paradigms

Relativity Computer

DONE

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Other Computational Paradigms

Closed Timelike Curve Computation

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

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Other Computational Paradigms

DNA Computing

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References

[1] Arora, S., Barak, B., “Computational Complexity: A Modern Approach.”

[2] Bernstein, E., Vazirani, U., “Quantum Complexity Theory.”

[3] Chuang, I., “Quantum Algorithms and their Implementations: QuISU – An Introduction for Undergraduates.”

[4] Lloyd, S., “Quantum Information Science.”

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

[6] Images Courtesy of Wikipedia.

[7] Thanks to Scott Aaronson & Michele Mosca for Slide Inspirations & Figures.

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Questions & Comments

Questions?

Comments?