Quantum Information Processing
QIP 3 – more on quantum information;
quantum computational models; Grover’s
quantum search.
Dan C. Marinescu
Computer Science Division, EECS Department
University of Central Florida
Email: dcm@cs.ucf.edu
Contents

Last time









Classical gates
Quantum gates; Unitary transformations
One-qubit gates, two-qubit gates, CNOT
Three-qubit gates Fredkin and Toffoli gates
Universal quantum gates
Reversibility of quantum circuits
Decoherere; DiVincenzo’s criteria for realization of quantum circuits
A circuit for solving the balanced function problem posed by David Deutsch
Today

More about quantum information
 Quantum computational models
 Grover’s quantum search algorithms
The lectures are available at:
http://www.cs.ucf.edu/~dcm/Chile2012/ChileIndex.html
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Information
Landauer’s Principle
(Thermodinamics)
The erasure of one bit produces
at least kB T log 2 Joules of heat
and increases the thermodynamic
entropy by at least kB log 2
Laws of
Quantum
Mechanics
Energy
Matter
E = m c2
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Classical versus Quantum Information
Classical information is information written in stone…
Quantum information is more like the information in a dream.
Recalling a dream inevitably changes your memory of it. Eventually you
remember only your own description, not the original dream.
Charles Bennett at QIPP workshop, 2002
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Measurements, preparation, and extraction of
classical information from quantum information

Classical information is often carried by the same types of particles as quantum
information, e.g., electrons, or photons; why should we expect quantum information
to be different from classical information?

Is it possible to freely convert one type of information to another and then recover
the original information?
 We can convert classical to quantum information and then convert back the
quantum information to the original classical information. This process consists
of two stages: preparation, when the quantum information is generated from
the classical one and measurements, when classical information is obtained
from the quantum information.
 To convert quantum to classical information and then convert the classical
information to quantum information indistinguishable from the original one we
should first perform a measurement to extract classical information and then
use it to prepare quantum information. The only possibility to compare
quantum mechanical systems is in terms of statistical experiments and this is
not possible, since a measurement is an irreversible process, it alters the state
of a quantum system.
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



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Classical and quantum information and
conversion from one to another.
Classical information is represented by
thin arrows and quantum information as
thick arrows.
(a) Classical information can be
regarded as a particular form of
quantum information.
(b) Classical information can be
recovered from the quantum information
when the preparation phase is followed
by a measurement. The conversion
path is:
classical quantum classical.
(c) Quantum information cannot be
recovered when the preparation follows
the measurement; the measurement is
an irreversible process and alters the
state of quantum systems. The
conversion path is:
quantum classical quantum.
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Quantum teleportation

Pair of entangled qubits
particle
1
particle
2
particle
3

Carol
Bob
Alice
particle
1
particle
2
particle
3
Quantum
Channel
CNOT
particle 1 - target qubit
particle 3 - control qubit
The measurement on
the pair (1&3) changes
the state of particle 2 to
one of four states: S1,
S2, S3, S4
iY
Receive from Alice
results of measurements
00 01
10
11
Measurement
particle 3 - measured
particle 1 - unchanged
I
Send to Bob results of
measurement
00 01
10
11
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
The process of
transferring the state of a
quantum particle to
possibly distant one.
Based on entanglement.
No cloning - the original
state is destroyed in the
quantum teleportation
process.
X
Z
Z
Classical
Channel
Particle 2 is in the same
state as particle 3
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A teleportation experiment

Reflecting mirror
Reflecting mirror

A
D
P
o
l
a
r
I
z
e
r
Alice
B
h
v
Source
h
Bob
v
C

Reflecting mirror
Reflecting mirror
Carol
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Francesco De
Martini, University of
Rome, 1997.
A UV laser beam
interacts with a nonlinear medium, a
crystal of dihidrogen
phosphate to
generate two
photons for an
incoming one –
parametric
downconversion.
The polarization
entanglement of the
two photons is
converted into a
path entanglement.
9
Communication with entangled particles


Even when separated, two entangled particles continue to
interact with one another.
Basic idea. Consider three particles
 Two particles (particle 1 and particle 2)  in an anticorrelated state (spin up and spin down).
 We measure particle 1 and particle 3 and set them in an
anti-correlated state.
 Then particle 2 ends up in the same state particle 3 was
initially in.
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Quantum key distribution

Classical methods for key distribution are in principle insecure  physical
difficulty to detect the presence of an intruder when communicating through
a classical communication channel. All classical methods of key distribution
can be broken if enough computer power is available.

Quantum key distribution ensures that an eavesdropper can succeed only
with a very low probability. No amount of computing power will allow
breaking of a quantum key distribution protocol.

Information encoding for quantum key distribution; the sender uses photons
polarized in two bases, Vertical/Horizontal and diagonal (45/135 deg).
 A photon with Vertical/Horizontal (VH) polarization
 1  a photon with vertical polarization
 0  a photon with a horizontal polarization.
 A photon with Diagonal (DG) polarization
 1  a photon with vertical polarization
 0  a photon with a horizontal polarization.
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Vertical
Horizontal
45 deg
Vertical/Horizontal (VH)
Using a classical channel,
after the exchange, Bob will
tell Alice what basis he used
for each photon and Alice will
tell him when he guessed
right. Bob will guess right and
measure in the basis the
photons were sent about half
of the time if there is no
eavesdropper. Eve will change
the ratio of photons measured
in the correct basis.
135 deg
Diagonal (DG)
(a)
(b)
Quantum communication channel
Source of
polarized
photons
Quantum wiretap
Photon
separation
system
Eve
Classical wiretap
Alice
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Classical communication channel
Bob
(c)
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Quantum parallelism

In quantum systems the amount of parallelism increases exponentially
with the size of the system, thus with the number of qubits. For example,
a 21-qubit quantum computer is twice as powerful as as a 20-qubit one.

An exponential increase in the power of a quantum computer requires
linear increase in the amount of matter and space needed to build the
larger quantum computing engine.

A quantum computer will enable us to solve problems with a very large
state space.
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Deutsch’s problem

Consider a black box characterized by a transfer function that maps
a single input bit x into an output, f(x). It takes the same amount of
time, T, to carry out each of the four possible mappings performed
by the transfer function f(x) of the black box:
f(0) = 0
f(0) = 1
f(1) = 0
f(1) = 1

The problem posed is to distinguish if
f (0)  f (1)
f (0)  f (1)
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0
f(0)
1
0
f(0)
1
f(1)
f(1)
2T
(a)
T
(b)
|x>
|x>
Uf
|y>
| y > O+ f(x) >
T
(c)
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A quantum circuit to solve Deutsch’s problem
|0>
H
|x>
|x>
H
Uf
|1>
H
0
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|y>
| y > +O f(x)
1
2
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16
0
 
1
0
    1
 0  0 1         
 0 1  0
0
 
1

1  1 1  1  1 1  1 1

  
  
G1  H  H 
1

1
1

1
2
2

 2 1
1

1

1 1
1  G1 0  
2 1

1

1 1 1

 1 1  1
1  1  1

 1  1 1 
1 1 1  0 
1
 
 
 1 1  1 1  1   1
  



1 1 1 0 2 1
 
 



  1
 1  1 1  0 
 
 0  1  0  1 
1
1  ( 00  01  10  11 )  


2
 2  2 
0 1
0 1
x 
y 
2
2
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y  f ( x) 

0 1
2
 f ( x)
0  f ( x)  1  f ( x)
y  f ( x)  (1)
2
f ( x)
2
0 1
 0 1

2
y  f ( x)  
 0  1

2
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
f ( x)  1  f ( x)
2
if
f ( x)  0
if
f ( x)  1
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x 
0 1


0

y 
 
2


 2  x  ( y  f ( x))  

 0
 
 


2
0 1


 0
 


 2  x  ( y  f ( x))  

 0
 
 


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 1  0  1

2  2
 1  0  1

2  2
 1  0  1

2  2
 1  0  1

2  2
1
 
 1   1
    if f (0)  f (1)  0
 2 1 
  1
 
1
 

1   1
     if f (0)  f (1)  1
2 1

 
  1
 
1
 
 1   1
    if
 2   1
1
 
1
 

1   1


if




1
2

 
1
 
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f (0)  0, f (1)  1
f (0)  1, f (1)  0
19
 1
  
 1   1 if
 2 1 
  
   1
 2  x  ( y  f ( x))  
  1 
 1   1
   if
 2   1
  1 

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f (0)  f (1)
f (0)  f (1)
20
1

1 1 1   1 0  1  0

  
 
G3  H  I 
2 1  1  0 1 
2 1

0







 3  G3 2  






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1

1 0
2 1

0

1

1 0
2 1

0

0
1
0
1
0
1
0
1
0 1
  
0 1  1   1




1 0 2 1
  
0  1   1
1 0 1
  
0 1  1   1




1 0 2 1
  
0  1  1 
1
0 1 0

1 0 1
0 1 0 

1 0  1
1
 
0 1
1   1
0


2 0
2
 
0
 
0
 
0 1
1 0


1
2 1 
2
 
  1
 
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if
f (0)  f (1)
if
f (0)  f (1)
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Evrika!!

By measuring the first output qubit we are able to determine
performing a single evaluation.
3   f (0)  f (1)
f (0)  f (1)
0 1
2
0 if f (0)  f (1)

f (0)  f (1)  
1 if f (0)  f (1)
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Circuit model of computation




A circuit model of computation  relates a practical implementation of an
algorithm with the circuits to compute Boolean functions; in terms of power
this model is equivalent to the Turing Machine model. The model should
distinguish between computable and non-computable functions.
A circuit model deals with practical implementation thus, with finite
systems, while the Turing Machine model assumes unbounded resources,
e.g., an infinite tape.
To construct the circuit for a particular algorithm we have to design a
protocol telling us what types of classical gates are needed, how to
interconnect them, and how to locate the results of the computations.
The need for the circuit design protocol leads to a deeper connection
between the circuit computational model and the Turing Machine
computational model. We should prohibit the ability to “hide”' the
complexity of an algorithm in the protocol for building the circuit, or even to
consider embedding a non-computable function e.g., a function for the
“halting problem,” in the protocol.
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Uniform circuit family





A circuit, Cn has n input bits and any number of auxiliary and output bits;
when the input is a string x of n bits the output of the circuit is denoted as
Cn(x).
A circuit is consistent if when the input is a string of length m<n then
Cm(x)=Cn(x).
A circuit family is uniform and then it is denoted as {Cn(x)} if an algorithm
for a Turing Machine to generate a description of the protocol for the
design of the circuit, given n, the size of the input, exists.
If such an algorithm for a Turing Machine does not exist then the circuit
family is called non-uniform.
The equivalence of the circuit and the Turing computational models is
restricted to uniform circuit families.
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Finite resources and computational models


To capture the finiteness of resources required by a physical realization of
a computing device David Deutsch restated the Church-Turing hypothesis
as: every realizable physical system can be perfectly simulated by a
universal model computing machine operating by finite means.
The much stronger formulation of the Church-Turing hypothesis is not
satisfied by a Turing Machine T operating in the realm of classical physics;
indeed, the set of states of a classical physical system form a continuum
due to the continuity of classical dynamics. The classical Turing Machine
T cannot simulate every classical dynamics system because there are
only countable ways of preparing the input for T.
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Quantum computational models


A quantum computational model specifies the resources needed for a
quantum computer, as well as the means to specify and control a quantum
computation.
To process quantum information, a computational model requires several
steps from the set:

free-time quantum evolution,
 controlled-time evolution,
 preparation, and
 Measurement

Several quantum computational models exist:






quantum Turing Machine (QTM),
quantum circuit,
topological quantum computer,
adiabatic,
one-way quantum computer, and
the measurements models
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QTM



The model is a generalization of the classical Turing Machine model; it
was the first quantum computational model introduced by Benioff in 1980
and further developed by Bernstein and Vazirani in 1997.
Though based on quantum kinematics and dynamics, Benioff's model was
classical in the sense that it required specification of the state as a set of
numbers measurable at any instant of time.
In 1982, Richard Feynman introduced a ``universal quantum simulator''
consisting of a lattice of spin systems with near-neighbor interactions;
Feynman's model could simulate any physical system with a finitedimensional state space but did not include a mechanism to select
arbitrary dynamic laws.
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The quantum circuit model



Proposed by David Deutsch and further developed by Andrew Yao.
The model applies only to uniform families of quantum circuits.
Some of the challenges posed by the quantum circuit model are:

the no-cloning theorem prohibits fanout;
 the decoherence of quantum states makes even the implementation of
quantum wires non-trivial;
 we can only approximate the function of a one-qubit gate with gates from a
small set of universal quantum gates.


In the quantum circuit model quantum computation is carried out by
quantum circuits that transform information under the control of external
stimuli.
A quantum circuit operating on n qubits performs a unitary operation in the
Hilbert space H2n and consists of a finite collection of quantum gates; each
quantum gate implements a unitary transformation on a small number k of
qubits.
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The quantum search algorithm



The algorithm was proposed by Lov Grover in 2004.
Preskill called Grover's algorithm ``perhaps the most important new
development'' in quantum computing.
The quantum search illustrates

The quintessence of a quantum algorithm: take advantage of quantum
parallelism to create a superposition of all possible values of a function and
then amplify, i.e., increase the probability amplitude of the solution;
 The contrast between classical and quantum algorithm strategies: a classical
search algorithm continually reduces the amplitude of non-target states, while
a quantum search algorithm amplifies the amplitude of the target states. In this
context, to amplify means to increase the probability.
 The algorithm can be applied directly to a wide range of problems. Even
problems not generally regarded as searching problems, can be reformulated
to take advantage of quantum parallelism and entanglement, and lead to
algorithms which show a square root speedup over their classical counterparts
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The problem and the solution




Consider a search space Q = {qi } consisting of N=2n unsorted items; each
item qi,1≤i≤N, is uniquely identified by a binary n-tuple i, called the index of
the item. We assume that M≤N items satisfy the requirements of a query
and we wish to identify one of them.
The classic approach is to repeatedly select an item qi, then decide if the
item is a solution to the query, and if so, terminate the search. If there is a
single solution (M = 1) then a classical search algorithm requires O(N)
iterations; in the worst case, we need to examine all N elements; if we
repeat the experiment many times then, on average, we will end up
examining N/2 elements before finding the desired one.
Grover’s search requires O ( N ) iterations. No classical or quantum
algorithm can solve this problem in less than O ( N ) iterations.
The main idea of the quantum search algorithm is to rotate the state vector
in a two-dimensional Hilbert space defined by an initial and a final (target)
state vector. The algorithm is iterative and each iteration causes the same
amount of rotation.
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The intuition behind the quantum search
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33
Inversion about the mean;
the amplitude of vertical bars
is proportional to the
modulus of the numbers.
 (a) A set of eight integers
A={34,66,47,63,54,28,42,48}
with the average

1 8
1
m   qi  (34  66  47  ...  42  48)  49
8 i 1
8

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(b) Inversion about the mean
of the set A; the integer qi
becomes
q’i = m+(m- qi )= 2 m - q’i
34
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


Inversion abut the mean and phase
inversion
(c) A set of eight complex numbers
of equal amplitude qi  1 / 8
(d) Phase inversion of the 6-th item,
the one we search for.
(e) Inversion about the mean from
step (d) m  3 /( 4 8)
q1 '  q2 '  q3 '  q4 '  q5 '  q'7  q8 '  1 /( 2 8 )
q6 '  5 / 2 8

(f) A second phase inversion of the
item we search for

(g) A second inversion about the
new mean from step (e), :m'  1 /(8 8)
q1 ' '  q2 ' '  q3 ' '  q4 ' '  q5 ' '  q' '7  q8 ' '  1 /( 4 8 )
q6 ' '  11 / 4 8
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UTFSM - June 2012
38
Geometric interpretation of a Grover iteration: two successive reflections
coresspond to a rotation with an angle 2
7/26/2016
UTFSM - June 2012
39
7/26/2016
UTFSM - June 2012
40