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Quantum Computation and Quantum Information


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

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

Dr. Rajagopal Nagarajan Assistant Nick
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

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!

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.

(No Transcript)
  • 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
Qubits (2)
  • In general, the state of a qubit is a
    combination, or superposition, of two basis
  • 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

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

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
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


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?

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

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

Quantum Measurement
  • To extract any information out of a quantum
    system, you have to perform a physical
  • 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

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.

Quantum Measurement (3)
  • If you measure with respect to the 0ñ, 1ñ
  • 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.

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.

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

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
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
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

  • 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
  • These are termed entangled states.

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

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



  • 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

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)

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

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

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

Quantum Coin Flipping
  • Quantum coin flipping is based on the following
  • 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

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!

Quantum Coin Flipping (3)
Person Action performed State

Bob H
Alice sx
Bob H 0ñ
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.

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

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

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.

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.

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

  • 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
  • Quantum coin flipping is more fun than classical
    coin flipping!
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