Transcript Quantum Computation and Quantum Information – Lecture 2

Quantum Computation and
Quantum Information – Lecture 2
Part 1 of CS406 – Research
Directions in Computing
Dr. Rajagopal Nagarajan
Assistant: Nick Papanikolaou
Lecture 2 Topics
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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
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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)
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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.
Qubits
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A qubit is any quantum system with exactly
two degrees of freedom; we use them to
represent binary ‘0’ and ‘1’
Hydrogen atom:
Ground state:
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Excited state:
Spin-1/2 electron:
Spin-down (-ħ/2) state:
Spin-up (+ħ/2) state:
Qubits (2)
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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
The State Vector
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The state of a quantum system is described
by a state vector, written 
If the basis states for a qubit are written 
and , then the state vector for the qubit is

where  and  are complex numbers with
2 2 
Basis States
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Instead of  and  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:


Basis states (2)
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For example, we could use the basis
{-} to describe the state of a qubit:

-
Now: -


orthogonality: -
Systems of many qubits
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If we know the individual states of the
electrons in the system below:

2
1
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2
3
3
... then what is the overall state of the threeparticle system?
Systems of many qubits (2)
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The state of a composite quantum system,
when all the component states are known, is
their tensor product:
 2 3
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This is the “outer product” of vectors
Note that this is different from the inner
product 
Systems of many qubits (3)
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We have
 2 3
() () ()
 
By convention, we write as 
Quantum Measurement
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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
you’re trying to measure
you obtain, in general, a random result, which
may be different from the original state
Quantum Measurement (2)
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When you try to measure a qubit

... you will never be able to obtain the values
of  and .
A measurement has to be made with respect
to a particular basis.
Quantum Measurement (3)
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If you measure with respect to the {}
basis:
–
–
–
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if  the answer will be  with probability 100%
if  the answer will be  with probability 100%
in all other cases (e.g. .5), the result will be
probabilistic.
After measurement, the value of  will change
permanently to the result obtained.
Quantum Measurement (4)
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If you measure with respect to a different
basis, things are worse!
Measuring  with respect to
{-} will give one of the results  and -
with particular probabilities.
Also, the value of  will change permanently
to the result obtained.
Quantum Measurement, Formally
Formally, when you measure

with respect to {} you will get:
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–
–
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result  with probability |2
result  with probability |2
If you use a different measurement basis, the
result will be one of the basis states, with
different probabilities
Measuring many qubits
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We want to know the possible outcomes of
measuring the two qubit state:
()()

prob. 2 2
prob. 2 2
the first measurement will reduce  to
one of these smaller states
Measuring many qubits (2)
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The second measurement will reduce to
one of the four states .

|  |2
|  |2  |  |2

|  |2
|  |2  |  |2


|  |2
|  |2  |  |2

|  |2
|  |2  |  |2

Measuring many qubits (3)
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By multiplying the branches in the overall
tree, we can obtain the probability of each
result. So for the state

two consecutive measurements will give
–
–
–
–
result  with probability |2
result  with probability |2
result  with probability |2
result  with probability |2
Entanglement
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There exist states of many-qubit systems that
cannot be broken down into a tensor
product
E.g.: there do not exist  for which
()()
These are termed entangled states.
The Bell states
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For a two-qubit system, the
four possible entangled
states are named Bell
states:



- 



- 
1
( 00
2
1
( 00
2
1
( 01
2
1
( 01
2
 11
)
- 11
)
 10
)
- 10
)
Measuring Entangled States
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After measuring an entangled pair for the first
time, the outcome of the second measurement is
known 100%



1

( 00  11
2
)
1

0.5
0.5

1

Review
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Thus far we have seen:
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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
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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, σx, σy, σz)
the Controlled Not (CNot)
The Hadamard gate
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The Hadamard gate acts on one qubit, and
places it in a superposition of  and  :
2
(0  1 )
H0 
2
2
(0 - 1 )
H1 
2
The Pauli gates
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The Pauli gates act on one qubit, as follows:
–
phase shift, σz:
σz()- 
–
bit flip, σx:
σx()
–
phase shift and bit flip, σy:
σy()- 
–
identity, I, does not change the input
The Controlled Not Gate
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The CNot gate acts on two qubits:
CNot( ) = 
CNot( ) = 
CNot( ) = 
CNot( ) = 
Quantum Coin Flipping
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Quantum coin flipping is based on the
following game:
–
–
–
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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)
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Assume that Alice can only perform a
flipping operation, i.e. gate σx
Remember: σx()
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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
σx
1
(0 1
2
1
(1  0
2
Bob
H

)
)
The No-cloning principle
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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
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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
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Alice wants to teleport particle 1 to Bob
 1   0 1  1 1

Two particles, 2 and 3, are prepared in an
entangled state


23
1

0
(
2
2
0 3  1 2 1 3)
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:
CNot (  1 , 


2
);
H( 
1
)
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
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Depending on the outcome of Alice’s
measurement, Bob applies a Pauli operator
to particle 3, “reincarnating” the original qubit
If outcome=00, Bob uses operator I
If outcome=01, Bob uses operator σx
If outcome=11, Bob uses operator σy
If outcome=10, Bob uses operator σz
Bob’s measurement produces the original
state of particle 1.
Quantum Teleportation (Summary)
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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]
Review
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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!