quantum computation: the physics of information
TRANSCRIPT
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Quantum ComputationThe Physics of Information
J. Caleb WherryAustin Peay State University
Departments of Computer Science, Mathematics, & Physics
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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
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History
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Pascaline - 1623 Difference Engine - 1823
Step Reckoner - 1673
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History
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ENIAC - 1946
Vacuum Tubes
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History
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Texas Instruments 1954 Transistor
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History
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Moore’s Law
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Computation & Complexity Theory
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What is computation?
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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.
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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?
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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
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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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Mathematical Aside
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
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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 items1
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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?