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Quantum Computation and Quantum Information

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Quantum Computation and Quantum Information Lecture 2 Part 1 of CS406 Research Directions in Computing Dr. Rajagopal Nagarajan Assistant: Nick Papanikolaou – PowerPoint PPT presentation

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Title: Quantum Computation and Quantum Information


1
Quantum Computation and Quantum Information
Lecture 2
  • Part 1 of CS406 Research Directions in Computing

Dr. Rajagopal Nagarajan Assistant Nick
Papanikolaou
2
Lecture 2 Topics
  • Physical systems on the atomic scale
  • State vectors and basis states Qubits
  • Systems of many qubits
  • Quantum Measurement
  • Entanglement
  • Quantum gates
  • Quantum coin-flipping and teleportation

3
Quantum physics and Nature
  • There exists a vast array of minute objects on
    the atomic scale electrons, protons, neutrons,
    photons, quarks, neutrinos,
  • Quantum mechanics is a system of laws that
    describes the behaviour of such objects
  • With computer chips getting smaller and smaller,
    by 2020 we will store 1 bit of data on objects of
    that size!

4
Quantum physics and Nature (2)
  • Atom-sized objects behave in unusual ways their
    state is generally unknown at any given time,
    and changes if you try to observe it!
  • Several properties of these systems can be
    manipulated and measured.

5
(No Transcript)
6
Qubits
  • A qubit is any quantum system with exactly two
    degrees of freedom we use them to represent
    binary 0 and 1
  • Hydrogen atom
  • Spin-1/2 electron

Ground state
Excited state
Spin-down (-h/2) state
Spin-up (h/2) state
7
Qubits (2)
  • In general, the state of a qubit is a
    combination, or superposition, of two basis
    states
  • The rest state and the excited state are the
    basis states of the hydrogen atom
  • The spin-up and spin-down states are basis states
    for the spin-1/2 particle

8
The State Vector
  • The state of a quantum system is described by a
    state vector, written yñ
  • If the basis states for a qubit are written 0ñ
    and 1ñ, then the state vector for the qubit is
  • yñ a 0ñ b 1ñ
  • where a and b are complex numbers with
  • a2 b2 1

9
Basis States
  • Instead of 0ñ and 1ñ we can use any other basis
    states, as long as we can distinguish clearly
    between the two.
  • Mathematically, basis states must be given by
    orthogonal vectors.

The inner product of the two vectors must be
0 á0 1ñ 0
10
Basis states (2)
  • For example, we could use the basis ñ, -ñ
    to describe the state of a qubit


Now yñ g ñ d -ñ orthogonality á
-ñ 0

ñ

11
Systems of many qubits
  • If we know the individual states of the electrons
    in the system below

y1ñ 0 0ñ 1 1ñ 1ñ y2ñ 1 0ñ 0 1ñ
0ñ y3ñ 0 0ñ 1 1ñ 1ñ
  • ... then what is the overall state of the
    three-particle system?

12
Systems of many qubits (2)
  • The state of a composite quantum system, when all
    the component states are known, is their tensor
    product
  • yñ y1ñ Ä y2ñ Ä y3ñ
  • This is the outer product of vectors
  • Note that this is different from the inner
    product áf½cñ

13
Systems of many qubits (3)
  • We have
  • yñ y1ñ Ä y2ñ Ä y3ñ
  • (0 0ñ 1 1ñ) Ä (1 0ñ 0 1ñ) Ä (0 0ñ
    1 1ñ)
  • 1ñ Ä 0ñ Ä 1ñ
  • By convention, we write 1ñ Ä 0ñ Ä 1ñ as
    101ñ

14
Quantum Measurement
  • To extract any information out of a quantum
    system, you have to perform a physical
    measurement
  • By measuring a quantum system
  • you automatically change its state, the very
    state youre trying to measure
  • you obtain, in general, a random result, which
    may be different from the original state

15
Quantum Measurement (2)
  • When you try to measure a qubit
  • yñ a 0ñ b 1ñ
  • ... you will never be able to obtain the values
    of a and b.
  • A measurement has to be made with respect to a
    particular basis.

16
Quantum Measurement (3)
  • If you measure with respect to the 0ñ, 1ñ
    basis
  • if yñ 0ñ the answer will be 0ñ with
    probability 100
  • if yñ 1ñ the answer will be 1ñ with
    probability 100
  • in all other cases (e.g. ab0.5), the result
    will be probabilistic.
  • After measurement, the value of yñ will change
    permanently to the result obtained.

17
Quantum Measurement (4)
  • If you measure with respect to a different basis,
    things are worse!
  • Measuring yñ a 0ñ b 1ñ with respect to
    ñ, -ñ will give one of the results ñ and
    -ñ with particular probabilities.
  • Also, the value of yñ will change permanently to
    the result obtained.

18
Quantum Measurement, Formally
  • Formally, when you measure
  • yñ a 0ñ b 1ñ
  • with respect to 0ñ, 1ñ you will get
  • result 0ñ with probability a2
  • result 1ñ with probability b2
  • If you use a different measurement basis, the
    result will be one of the basis states, with
    different probabilities

19
Measuring many qubits
  • We want to know the possible outcomes of
    measuring the two qubit state
  • yñ (a 0ñ b 1ñ) Ä (g 0ñ d 1ñ)
  • ag 00ñ ad 01ñ bg 10ñ bd 11ñ

prob. ag2 ad2
prob. bg2 bd2
the first measurement will reduce yñ to one of
these smaller states
20
Measuring many qubits (2)
  • The second measurement will reduce yñ to one of
    the four states 00ñ, 01ñ, 10ñ, 11ñ.

ag 00ñ ad 01ñ
bg 10ñ bd 11ñ
00ñ
01ñ
10ñ
11ñ
21
Measuring many qubits (3)
  • By multiplying the branches in the overall tree,
    we can obtain the probability of each result. So
    for the state
  • yñ ag 00ñ ad 01ñ bg 10ñ bd 11ñ
  • two consecutive measurements will give
  • result 00ñ with probability ag2
  • result 01ñ with probability ad2
  • result 10ñ with probability bg2
  • result 11ñ with probability bd2

22
Entanglement
  • There exist states of many-qubit systems that
    cannot be broken down into a tensor product
  • E.g. there do not exist a, b, g, d for which
  • m 00ñ n 11ñ (a 0ñ b 1ñ) Ä (g 0ñ d
    1ñ)
  • These are termed entangled states.

23
The Bell states
  • For a two-qubit system, the four possible
    entangled states are named Bell states

24
Measuring Entangled States
  • After measuring an entangled pair for the first
    time, the outcome of the second measurement is
    known 100

1


0.5
1


0.5
25
Review
  • Thus far we have seen
  • how qubits are represented
  • how many qubits can be combined together
  • what happens when you measure one or more qubits
  • where entangled pairs come from, and what happens
    when you measure them
  • Now we will take a look at quantum gates

26
Quantum gates
  • As in classical computing, a gate is an operation
    on a unit of data, here a qubit
  • A quantum gate is represented by a matrix that
    may be applied to a state vector
  • We will talk about this in more detail next time
    for now we will look at some examples of commonly
    used quantum gates
  • the Hadamard gate (H)
  • the Pauli gates (I, sx, sy, sz)
  • the Controlled Not (CNot)

27
The Hadamard gate
  • The Hadamard gate acts on one qubit, and places
    it in a superposition of 0ñ and 1ñ

28
The Pauli gates
  • The Pauli gates act on one qubit, as follows
  • phase shift, sz
  • sz(a 0ñ b 1ñ) a 0ñ - b 1ñ
  • bit flip, sx
  • sx(a 0ñ b 1ñ) a 1ñ b 0ñ
  • phase shift and bit flip, sy
  • sy(a 0ñ b 1ñ) a 1ñ - b 0ñ
  • identity, I, does not change the input

29
The Controlled Not Gate
  • The CNot gate acts on two qubits
  • CNot( 00ñ ) 00ñ
  • CNot( 01ñ ) 01ñ
  • CNot( 10ñ ) 11ñ
  • CNot( 11ñ ) 10ñ

30
Quantum Coin Flipping
  • Quantum coin flipping is based on the following
    game
  • Alice places a coin, head upwards in a box.
  • Alice and Bob then take turns to optionally turn
    the coin over (without looking at it).
  • At the end of the game, the box is opened and and
    Bob wins if the coin is head upwards.
  • In the quantum version of the game, the coin is a
    quantum state

31
Quantum Coin Flipping (2)
  • Assume that Alice can only perform a flipping
    operation, i.e. gate sx
  • Remember sx(a 0ñ b 1ñ) a 1ñ b 0ñ
  • There is a strategy that allows Bob to win
    always he must perform Hadamard operations.
  • Thus Bob places the state of the coin in a
    superposition of heads and tails!

32
Quantum Coin Flipping (3)
Person Action performed State

Bob H
Alice sx
Bob H 0ñ
33
The No-cloning principle
  • It has been proved by Wootters and Zurek that it
    is impossible to clone, or duplicate, an unknown
    quantum state.
  • However, it is possible to recreate a quantum
    state in a different physical location through
    the process of quantum teleportation.

34
Quantum Teleportation The Basics
  • If Alice and Bob each have a single particle from
    an entangled pair, then
  • It is possible for Alice to teleport a qubit to
    Bob, using only a classical channel
  • The state of the original qubit will be destroyed
  • How?
  • Using the properties of entangled particles

35
Quantum Teleportation
  • Alice wants to teleport particle 1 to Bob
  • Two particles, 2 and 3, are prepared in an
    entangled state
  • Particle 2 is given to Alice, particle 3 is given
    to Bob

36
Quantum Teleportation (2)
  • In order to teleport particle 1, Alice now
    entangles it with her particle using the CNot and
    Hadamard gates
  • Thus, particle 1 is disassembled and combined
    with the entangled pair
  • Alice measures particles 1 and 2, producing a
    classical outcome 00, 01, 10 or 11.

37
Quantum Teleportation
  • Depending on the outcome of Alices measurement,
    Bob applies a Pauli operator to particle 3,
    reincarnating the original qubit
  • If outcome00, Bob uses operator I
  • If outcome01, Bob uses operator sx
  • If outcome11, Bob uses operator sy
  • If outcome10, Bob uses operator sz
  • Bobs measurement produces the original state of
    particle 1.

38
Quantum Teleportation (Summary)
  • The basic idea is that Alice and Bob can perform
    a sequence of operations on their qubits to
    move the quantum state of a particle from one
    location to another
  • The actual operations are more involved than we
    have presented here see the standard texts on
    quantum computing for details
  • Recommended S. Lomonaco, A Rosetta Stone for
    Quantum Computation see www

39
Review
  • Quantum gates allow us to manipulate quantum
    states without measuring them
  • Quantum states cannot be cloned
  • Teleportation allows a quantum state to be
    recreated by exchanging only 2 bits of classical
    information
  • Quantum coin flipping is more fun than classical
    coin flipping!
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