Quantum Computing Introduction Part II

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Quantum Computing Introduction Part II
Presented by Chensheng Qiu Supervised by Dplm.
Ing. Gherman

• Advanced topic seminar SS02
• Innovative Computer architecture and concepts
• Examiner Prof. Wunderlich

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QC -- Interdisciplinary Subject
Quantum Computation Quantum Information
Study of information processing tasks that can be
accomplished using quantum mechanical systems
Cryptography
Quantum Mechanics
Information Theory
Computer Science
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Contents

• Classical and Quantum Experiments
• Basic concepts in Quantum mechanics
• Qubits and Quantum Registers
• Quantum Gates and Networks
• Summary and Concluding remarks

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Classical vs. Quantum Exper.

• Classical Experiments
• Experiment with bullets
• Experiment with waves
• Quantum Experiments
• Two slits Experiment with electrons
• Stern-Gerlach Experiment

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Experiment with bullets
(b)
Figure 1 Experiment with bullets
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Experiment with bullets
(b)
Figure 1 Experiment with bullets
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Experiment with bullets
(b)
Figure 1 Experiment with bullets
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Experiment with bullets
(b)
Figure 1 Experiment with bullets
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Experiment with Waves
wave source
H1
H2
wall
(a)
Figure 2 Experiments with waves
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Experiment with Waves
wave source
H1
H2
wall
(b)
(a)
Figure 2 Experiments with waves
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Experiment with Waves
wave source
H1
H2
I2(x)
wall
(b)
(a)
Figure 2 Experiments with waves
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Experiment with Waves
wave source
H1
H2
wall
(a)
Figure 2 Experiments with waves
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Two Slit Experiment
Figure 3 Two slit experiment
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Two Slit Experiment
Figure 3 Two slit experiment
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Two Slit Exp. With Observ.
H1
H2
? Decoherence
wall
(a)
Figure 4 Two slit experiment with observation
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Stern-Gerlach Experiment
Will be discussed in more detail later
Figure 5 Stern-Gerlach experiment with spin-1/2
particles
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Conclusions From the Exps

• Limitations of classical mechanics
• Particles evince wavelike behavior
• Effect of observations cannot be ignored
• Evolution and measurement must be distinguished

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Conclusions From the Exps

• Quantum Phenomena
• ?
• Quantum Mechanics

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Contents

• Classical and Quantum Experiments
• Basic concepts in Quantum mechanics
• Qubits and Quantum Registers
• Quantum Gates and Networks
• Summary and Concluding remarks

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Requirs On Math. Apparatus

• Physical states ? Mathematic entities
• Interference phenomena
• Nondeterministic predictions
• Model the effects of measurement
• Distinction between evolution and measurement

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Whats Quantum Mechanics

• A mathematical framework
• Description of the world known
• Rather simple rules
• but counterintuitive
• applications

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Introduction to Linear Algebra

• Quantum mechanics
• The basis for quantum computing and quantum
  information
• Why Linear Algebra?
• Prerequist
• What is Linear Algebra concerning?
• Vector spaces
• Linear operations

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Basic concs in LA concerning QM

• Complex numbers
• Vector space
• Linear operators
• Inner products
• Unitary operators
• Tensor products

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Dirac-notation

• For the sake of simplification
• ket stands for a vector in Hilbert
• bra stands for the adjoint of
• Named after the word bracket

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Inner Products

• A function combining two vectors
• Yields a complex number
• Obeys the following rules

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Hilbert Space

• Inner product space linear space equipped with
  inner product
• Hilbert Space (finite dimensional) can be
  considered as inner product space of a quantum
  system
• Orthogonality
• Norm
• Unit vector parallel to

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Hilbert Space (Contd)

• Orthonormal basis
• a basis set where
• Can be found from an arbitrary basis set by
  Gram-Schmidt Orthorgonalizatioin

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Unitary Operator

• An operator U is unitary, if
• Preserves Inner product

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Tensor Product

• Larger vector space formed from two smaller ones
• Combining elements from each in all possible ways
• Preserves both linearity and scalar
  multiplication

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Postulates in QM

• Why are postulates important?
• they provide the connections between the
  physical, real, world and the quantum mechanics
  mathematics used to model these systems
• - Isaal L.
  Chuang

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Physical Systems - Quantum Mechanics Connections
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Contents

• Classical and Quantum Experiments
• Basic concepts in Quantum mechanics
• Qubits and Quantum Registers
• Quantum Gates and Networks
• Summary and Concluding remarks

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Whats a qubit (1)

• A qubit has two possible states
• Unlike bits, a quibit can be in a state other
  than
• We can form linear combinations of states
• A quibit state is a unit vector in a two
  dimensional complex vector space

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Whats a qubit (2)

• and are known as computational basis
  states
• orthonormal basis, we can assume
• we cannot examine a qubit to determine its
  quantum state
• A measurement yields

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Qubits Cont'd

• We may rewrite as
• From a single measurement one obtains only a
  single bit of information about the state of the
  qubit
• There is "hidden" quantum information and this
  info grows exponentially

We can ignore eia as it has no observable effect
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Bloch Sphere
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How can a qubit be realized?

• Two polarizations of a photon
• Alignment of a nuclear spin in a uniform magnetic
  field
• Two energy states of an electron

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Qubit in Stern-Gerlach Exp.
Oven
Figure 6 Abstract schematic of the Stern-Gerlach
experiment.
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Qubit in Stern-Gerlach Exp.
Oven
Figure 7 Three stage cascade Stern-Gerlach
measurements
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Qubit in Stern-Gerlach Exp.
Figure 8 Assignment of the qubit states
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Qubit in Stern-Gerlach Exp.
Figure 8 Assignment of the qubit states
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Quantum Registers

• A collection of qubits
• Size is the number of qubits it contains
• State is decided by the tensor product of all the
  single qubits
• Example the number 6 can be represented by a
  register in state
• where the stands for tensor product

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Contents

• Classical and Quantum Experiments
• Basic concepts in Quantum mechanics
• Qubits and Quantum Registers
• Quantum Gates and Networks
• Summary and Concluding remarks

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Quantum Gates and Networks

• Operation on qubit
• Unitarity constraint
• Quantum network
• Size of quantum network

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Quantum Gates Why Unitary?

• Length preserving
• We require the normalization condition
• for
• and the result after the gate
  has acted

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Quantum Gates Why Unitary?

• Information preserving
• A unitary operator fulfills
• ? Reversible!

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Quantum Not-Gate

• NOT gate representation
• for any
• we get
• to summarize
  

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Hadamard Gate

• Most common quantum gate
• Like a square-root of NOT
• but

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Hadamard Gate

• Bloch Sphere Representation

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Single Qubit Gate Class. vs. Quan.
Figure 8 Operation on single bit in classical
gate (top) and operation on single qubit in
quatum gates
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Phase Shift Gate
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Separable vs. Entangled States
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Multiple Qubit Gates

• Controlled-NOT (CNOT) Gate
• two input qubits control and target
• In General

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CNOT quantum gate

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Controlled-U Gate

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Universal set of gates

• Classical Universal Gates (example)
• - The NAND gate is a classical Universal
• Gate. Why?

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Universal set of gates

• Universal Quantum Gates
• Infinite set e.g. Hadamard gate, all phase
  gates, and C-NOT
• Finite set e.g. Hadamard and the controlled-V
  can be used to APPROXIMATE any unitary
  transformation on qubits

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Contents

• Classical and Quantum Experiments
• Basic concepts in Quantum mechanics
• Qubits and Quantum Registers
• Quantum Gates and Networks
• Summary and Concluding remarks

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Summary and Concluding Remarks

• Experimental motivations for QM
• Hilbert space
• Unitary transformation
• Qubits as mathematical objects
• Quantum gates
• Gedanken experiments

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7- Qubit Q-Computer by IBM
Quantum computing researchers (l-r) Isaac Chuang
and Costantino Yannoni

• Could be Most advanced model of QC
• Finding the factors of the number 15 with Shors
  algorithm
• Nuclei of five fluorine and two carbon atoms
  interacting with each other
• Programmed by RF pulses
• Detected by NMR technique

Diagram of the 7-qubit molecule
From IBM research news
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Thanks !
Questions and Discussion
You can click mouse to view the following slide,
I. E. definition of tensor product of matrix and
the part of Quantum Arithmetic, which werent
shown during my presentation
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Tensor Product

• In QC, we usually use the tensor product of
  matrix

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Contents

• Classical and Quantum Experiments
• Basic concepts in Quantum mechanics
• Qubits and Quantum Registers
• Quantum Gates and Networks
• Quantum Arithmetic
• Summary and Concluding remarks

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Quantum Arithmetic

• Hadamard and C-V gates as building block

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Quantum Arithmetic

• Three subsequent
• Toffoli-Gate

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Quantum Arithmetic

• Auxiliary bit
• Reversible

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Quantum Arithmetic

• Function evaluation
• E.g Toffoli Gate
• Requires at least quantum registers
• Possible need of working-bit

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Quantum Arithmetic

• Modular arithmetic
• Commutative
• Associative
• distributive
• A trick for mod
• For l-bit modulus n
• intermediate result lt 2l-bit

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Quantum Arithmetic

• Function evaluation for supperposition
• f(x) for all x in a single run
• Problem of measurement