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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. 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.

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
for now we will look at some examples of commonly
used quantum gates
• the Pauli gates (I, sx, sy, sz)
• the Controlled Not (CNot)

27
• 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

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
• 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!