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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
3
Classical Computation
History
4
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
8
What is computation?
Computation & Complexity Theory
9
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
10
Computation & Complexity Theory
11
Cbits, Logic Gates, & the Circuit Model
12
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
13
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
14
Moore’s Law Revisited
15
Moore’s Law
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Quantum Computation
17
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.
18
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
12
i iFields: N, Q, R, C, H
19
Mathematical Formalisms
Pauli Matrices Hadamard Gate
Pauli-X
Pauli-Y
Pauli-Z
Hadamard
Quantum Logic Gates = Linear Transformations
20
Mathematical Formalisms
Quantum Weirdness
Superposition
Entanglement
Teleportation
21
Mathematical Formalisms
Quantum Weirdness ISuperposition & Interference
22
Mathematical Formalisms
Quantum Weirdness ISuperposition & Interference
23
Mathematical Formalisms
Quantum Weirdness II
Entanglement – EPR Paradox
“Spookiness at a distance” - Einstein
24
Mathematical Formalisms
Quantum Weirdness IIITeleportation
BQP & the Power of Q.C.
17
If we have n qubits, how much information do we have?
26
Quantum Implementations
Ion Trap
NMR
Topological Q.C.
Superconducting Qubits
27
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.
28
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!
29
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)
31
Other Computational Paradigms
Relativity Computer
DONE
32
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
33
Other Computational Paradigms
DNA Computing
34
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.
35
Questions & Comments
Questions?
Comments?