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Quantum Computation and
Quantum Information – Lecture 3
Part 1 of CS406 – Research
Directions in Computing
Nick Papanikolaou
Motivation



Quantum computers are built from wires
and logic gates, just as classical computers
are
The potential of such devices stems from the
ability to manipulate superpositions of
states
Quantum algorithms solve problems which
are not known to be solvable classically!
Lecture 3 Topics




Quantum logic gates
Simple quantum circuits
Quantum teleportation as a circuit
Deutsch’s quantum algorithm
Quantum vs. classical gates


The simplest boolean gate is NOT, with truth
table:
in
out
0
1
1
0
Quantum gates have to be defined not only
on the equivalents of 0 and 1, but on their
superpositions too!
Quantum NOT gate: Linearity

Suppose we define a quantum NOT gate as
follows:
NOT 0  1 , NOT 1  0

The action of the quantum NOT gate on a
superposition must then be:
NOT 0   1     NOT 0    NOT 1
  1  0

All quantum operations are linear
The NOT Gate as a Matrix


Because all quantum operations have to be
linear, we can represent the action of a
quantum gate by a matrix
The quantum NOT, or Pauli-X gate, is written:
0 1 
X

1
0


Quantum State Vectors


Remember that a quantum state is
represented by a vector
Notation:
1
0 
0   , 1   
0 
1
 
   0 1   
 
Quantum NOT

We can express the NOT operation on a
general qubit as matrix multiplication:
 0 1      
X 0   1   
    

1 0    
Other Single Qubit Gates


The Pauli-X gate works on only one qubit
Other common single qubit gates are:
–
Pauli-Z gate:
1 0 
Z

0

1


Z
Y
–
Pauli-Y gate:
Y  Z X
–
Hadamard gate:
1 1 1 
H  NOT 
1  1
2

H
Summary of Simple Gates
 0 1
X
 1  0
 0 1
Y
i   1   0
 0 1
Z
 0 1
 0 1
H

0 1
2


0 1
2
Reversibility Requirement



All quantum operations have to be
reversible
Boolean operations are not necessarily so
A reversible operation is always given by a
unitary matrix, i.e. one for which:
(UT )*  U 1
The Controlled NOT Gate


The CNOT gate is the standard two-qubit
quantum gate
It is defined like this: CNOT 00  00
CNOT 01  01
CNOT 10  11
CNOT 11  10
The Controlled NOT Gate (2)

CNOT is a generalisation of the classical XOR:
CNOT A, B  A, B  A

The CNOT gate is drawn like this:
“control qubit”
A
A
B
B A
“target qubit”
The Controlled NOT Gate (3)

The matrix corresponding to the CNOT gate
is:
1 0 0 0
0 1 0 0 

CNOT  
0 0 0 1 


0
0
1
0



The CNOT together with the single qubit
gates are universal for quantum computing
Quantum Circuits


Using the conventions for control and target
qubits, we can build interesting circuits
Example: A Qubit Swap Circuit
A
B
B
A
Qubit Swap Circuit
A
B
A
A  ( A  B)  B
A B
A B
B
A
B  ( A  B)  A
Features of Quantum Circuits
2.
No loops are allowed; quantum circuits are
acyclic
Fan-in is not allowed:
3.
Fan-out is not allowed:
1.
Generalised Control Gate

Any quantum gate U can be converted into a
controlled gate:
One control qubit
U
n target
qubits
If the control qubit is “high,” U is applied to
the targets. CNOT is the Controlled-X gate!
Quantum Measurement

Measurement in a quantum circuit is drawn as:

M
(classical bit
representing outcome
of measurement)
M = 0 with prob. 
or
2
M = 1 with prob. 
2
If    0   1 then:
A Qubit Cloning Circuit?

Using the XOR gate, it is possible to copy a
classical bit:
x
x
x
x
0
y
xy
x
Can we build a quantum circuit that performs does
?
this with qubits?
 0  
A Qubit Cloning Circuit? (2)
0
0
1
1
OK here
0
0
   0 1
0
0
1
 00   11
entangled!!
A Qubit Cloning Circuit? (3)
It is impossible to clone a qubit!
Also note that
    0   1  0   1 
  2 00   01   10   2 11
unwanted terms!
The Bell State Circuit
x
H
x
y
Output
0
0
1
 00  11 
2
0
1
1
 01  10
2
0
1
 00  11 
2
1
1
 01  10
2
y
1
1


The Bell State Circuit By Example
0
H
1
0  1 
2
?
0
 0 1

 00  10 



CNOT
 0   CNOT

2
2




1
CNOT 00  CNOT 10

2
1
 00  11 

2

Quantum Teleportation Circuit

M1
H
M2

XM2
0    

ZM1
1
 00  11 
  0   1 
2

Quantum Teleportation Circuit (2)

H
M1
M2

XM2
ZM1
1
 0   00  11    1   00  11 
1 
2

Quantum Teleportation Circuit (3)

H
M1
M2

XM2
2
ZM1
1  00   0   1   01   1   0   
 

2  10   0   1   11   1   0  

Quantum Teleportation Circuit (4)

H
M1
M2

XM2
00, 01, 10 or 11
ZM1

Quantum Teleportation Circuit (5)
If Alice
Then Bob’s
obtains qubit is in state
So Bob
applies gate
00
 0 1
I
01
 1  0
X
10
 0 1
Z
11
 1  0
Y = ZX
obtaining
 0 1
What have we achieved?
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

The teleportation process makes it possible
to “reproduce” a qubit in a different location
But the original qubit is destroyed!
Next topic: Quantum Parallelism and
Deutsch’s quantum algorithm
Quantum Parallelism


Quantum parallelism is that feature of
quantum computers which makes it possible
to evaluate a function f(x) on many different
values of x simultaneously
We will look at an example of quantum
parallelism now – how to compute f(0) and
f(1) for some function f all in one go!
Quantum Circuits for Boolean
Functions
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

It is known that, for any boolean function
f : 0,1  0,1
it is possible to construct a quantum circuit Uf
that computes it
Specifically, to each binary function f
corresponds a quantum circuit:
U f : x, y  x, y  f ( x)
binary addition
Quantum Circuits for Boolean
Functions (2)

What can this circuit Uf do? Example:
0
1
x
x
Uf
y
yf(x)
 U f  0  1 
 U f 01
 0, 1  f (0)
Quantum Circuits for Boolean
Functions (3)

But what if the input is a superposition?
1
0  1 
2
1
x
x
Uf
y
yf(x)
amazing! we’ve computed f(0)
and f(1) at the same time!
 0 1

  U f 
 0 
2


 00  10 

 U f 

2


0, 0  f (0)  1, 0  f (1)

2
0, f (0)  1, f (1)

2
Quantum Parallelism Summary
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So, a superposition of inputs will give a
superposition of outputs!
We can perform many computations
simultaneously
This is what makes famous quantum
algorithms, such as Shor’s algorithm for
factoring, or Grover’s algorithm for
searching
Simple q. algorithm: Deutsch’s algorithm
Deutsch’s Algorithm


David Deutsch: famous British physicist
Deutsch’s algorithm allows us to compute,
in only one step, the value of
f (0)  f (1)

To do this classically, you would have to:
1.
2.
3.
–
compute f(0)
compute f(1)
add the two results
Remember:
f : 0,1  0,1
Circuit for Deutsch’s Algorithm
0
H
1
H
x
H
x
Uf
 0  01
y
1 
yf(x)
0 1
2

0 1
2
Circuit for Deutsch’s Algorithm (2)
0
H
1
H
2
x
x
Uf
y
yf(x)

0 1 0 1

if f (0)  f (1), 
2
2

if f (0)  f (1),  0  1  0  1

2
2
H
Circuit for Deutsch’s Algorithm (3)
0
H
1
H

if
3  
 if

x
x
H
Uf
y
yf(x)
0 1
f (0)  f (1),  0 
2
this simplifies to
0 1
f (0)  f (1),  1 
02  1
 3   f (0)  f (1) 
2
...and so we have computed
f (0)  f (1)
End of Lecture 3

Congratulations! If you are still awake, you
have learned something about:
–
–
–
–
–
quantum gates (X, Y, Z, H, CNOT)
quantum circuits (swapping, no-cloning problem)
teleportation
quantum parallelism
and Deutsch’s algorithm