Quantum Information Processing
QIP 1 –Basic Concepts
Dan C. Marinescu
Computer Science Division, EECS Department
University of Central Florida
Email: dcm@cs.ucf.edu
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Contents
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Laws of physics and the limits of solid-state technology
The mathematical model
A bit or a qubit of history
The lectures are available at:
http://www.cs.ucf.edu/~dcm/Chile2012/ChileIndex.html
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Moore’s Law
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Microprocessor
Year
4004
1971
2,250
8008
1972
2,500
8080
1974
5,000
8086
1978
29,000
286
1982
120,000
386
1985
275,000
486
1989
1,180,000
Pentium
1993
3,100,000
Pentium II
1997
7,500,000
Pentium IV
2000
42,000,000
Itanium
2002
220,000,000
Itanium II
2003
410,000,000
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# transistors
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Limits of solid-state technology
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To increase the clock rate we have to pack transistors as densely as
possible because the speed of light is finite.
The power dissipation increases with the cube of the clock rate. When
we double the speed of a device its power dissipation increases 8
(eight) fold.
The computer technology vintage year 2000 requires some 3 x 10-18
Joules/elementary operation.
An exponential growth cannot be sustained indefinitely; sooner or later
one will hit a wall.
Revolutionary rather than evolutionary approach to information
processing and to communication:
 Quantum computing and communication  quantum information.
 DNA Computing  biological information.
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Quantum information processing
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Quantum information Information encoded as the state of atomic or
sub-atomic particles.
A happy marriage between three of the greatest scientific achievements
of the 20th century:

quantum mechanics
 stored program computers
 information theory

In 1985 Richard Feynman wrote: “..it seems that the laws of physics
present no barrier to reducing the size of computers until bits are the
size of atoms and quantum behavior holds sway.”
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Information can be encoded as the photon polarization

x

Light is a form of electromagnetic radiation.
The electric and magnetic field

oscillate in a plane perpendicular to the
direction of propagation and
 are perpendicular to each other.
v
z

A photon is characterized by its

vector momentum (the vector momentum
determines the frequency) and
 polarization.
y

(a)
x
z
h
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y
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(b)
In the classical electromagnetic theory
light is described as having an electric field
which oscillates either
vertically, the light is x-polarized,or
horizontally, the light is y-polarized in a
plane perpendicular to the direction of
propagation, the z-axis.
The two basis vectors are |h> and |v>
Information encoding:
0  vertically polarized photon
1  horizontally polarized photon
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Information can be encoded as the spin of the electron

Sz

Rn(0)
1
h
2
-
1
h
2
Rn(180)

(a)
(b)

Spin  the intrinsic angular
momentum; it takes discrete values
(the spin quantum number s.)
Two classes of quantum particles:
 fermions - spin one-half
particles (e.g., electrons).
 s=+1/2 and s=-1/2
 bosons - spin one particles
(e.g., photons).
 s=+1, s=0, and s=-1
The electron has spin s = ½; the
spin projection can assume the
values + ½ referred to as spin
up, and - ½ referred to as spin
down.
The information can be encoded
0  spin up
1  spin down
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Quantum systems

Quantum concepts such as:

Uncertainty,
 Superposition,
 Entanglement,
 No-cloning
do not have a correspondent in classical physics.

Heisenberg’s Uncertainty Principle: the position and the momentum of
a quantum particle cannot be determined with arbitrary precision.
X  PX  h / 4
h=6.6262 x 10-34 Joule x second  Planck’s constant
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Non-determinism is a basic tenet of quantum physics
“Liebe Gott würfelt nicht”
(Dear God does not play
dice)
- Albert Einstein
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The qubit
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Qubit a unit of quantum information, the correspondent of a bit of
classical information.
Information is physical  a qubit must be encoded as some property of
a quantum particle.
There are multiple physical embodiments of a qubit
Photon – information encoded in the polarization
 Electron – information encoded in the spin
 Electron – information encoded in electric charge (quantum dots)
 Trapped ions in a cavity

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The physics of each embodiment is different: optics, condensed matter
physics, electromagnetic field theory, etc.
We need a mathematical model to describe quantum information
regardless of the physical embodiment of a qubit.
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The mathematical model of a qubit
A mathematical model of a physical system should describe:

1.
2.
3.
4.
5.

The state of the system
The means to observe the system
The dynamics of the system, the evolution from one state to another
The measurement process
How to describe a system consisting of two sub-systems
The postulates of quantum mechanics establish a correspondence
between the physical system and mathematical objects of the model

The state postulate
 The dynamics postulate
 The measurement postulate

The model
1.
2.
3.
4.
State of a quantum system  vector in an n-dimensional Hilbert space
Observables  the means to observe the system, linear operators
System dynamics  Hermitian (self-adjoint) operators
System measurements  measurement operators
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More about the measurement postulate
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The numerical outcome of a measurement of observable A of a
quantum system in state |   H n is an eigenvalue i of the operator
A used to measure the observable A; immediately after the
measurement the quantum state of the system is an eigenvector |ai>
coresponding to the eigenvalue i of the operator A.
A measurement is a projection of the current state.
In a projective measurement the measurement operators are selfadjoint and idempotent. The number of such operators is equal to
the dimension of the Hilbert space. Since orthogonal measurement
operators commute, they correspond to simultaneous observables.
von Neumann gave an axiomatic definition of the measurement. A
von Neumann measurement is a conditional expectation onto a
maximal Abelian subalgebra of the algebra of all bounded operators
acting on the Hilbert space.
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The state of a quantum system after a measurement
 Consider a quantum system in state |  

The state of the system after we apply the projector Pi the is
|   Pi |  

The completeness of the set of projectors leads to a normalization
condition and after the mesaurement the state becomes a normaized pure
state:
| 
Pi |  
|  

  | Pi |  
  | Pi |  
'
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Observable A
Hermitian operator A with n
eigenvectors: |ei>
eigenvalues: λ
Quantum system in state
| φ H n
i
A 
n
λ P
i 1
i
i
n projectors
{P1, P2 ...Pi ...Pn }
Pi = |e
i><ei |
n
 Pi =1
P1
Pi
i 1
Pn
| φ 
Hn
New state: P1
Outcome of the measurement:
1
Probability of the outcome: |P1
λ
| φ |2
| φ 
New state: Pi
Hn
Outcome of the measurement:
Probability of the outcome: |Pi
λi
| φ |2
| φ 
New state: Pn
Hn
Outcome of the measurement:
Probability of the outcome: |Pn
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λn
| φ |2
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Two stage von Neumann-type
measurement. We measure
the observable Q of the
quantum system S using a
macroscopic apparatus M. At
time t the quantum system is in
a superposition state and the
apparatus is in state |m>.
qi is the eigenvalue
corresponding to the
eigenvector ai of observable Q.
At the end of the first phase the
composite system S + M is in
state shown at the left.
Then, as a result of the
measurement, the system
collapses.
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Quantum states
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Quantum states are represented as vectors in an n-dimensional Hilbert
space Hn  a vector space over C the set of complex numbers.
A canonical orthonormal basis in Hn can be expressed as the ket vectors
|0>, |1>, |2>, ……|n-1> or as the bra vectors <0|, <1|, <2|, ……<n-1|
The matrix representations of these vectors are:
1
0
0
0
 
 
 
 
0
1
0
0
| 0   0 , | 1   0 , | 2   1 ,....., | n  1   0 
 
 
 
 
 ... 
 ... 
 ... 
 ... 
0
0
0
1
 
 
 
 
 0 | (1,0,0,...,0),  1 | (0,1,0,...,0),  2 | (0,0,1,...,0),  n  1 | (0,0,0,...,1)
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One qubit
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Mathematical abstraction
Vector in a two dimensional complex vector space (Hilbert space)
Dirac’s notation
ket 
bra 

 |
column vector
row vector
bra  dual vector (transpose and complex conjugate)
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Orthonormal bases
| 1
| 0

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0 1
 
0 1
2
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One qubit
   0 0  1 1
|  0 , 1 |
are complex numbers
|  0 |  | 1 |  1
2
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The Boch sphere representation of one qubit


A qubit in a superposition state is represented as a vector connecting
the center of the Bloch sphere with a point on its periphery.
The two probability amplitudes can be expressed using Euler angles.
|0>
z
|ψ 


r
x

y
b
|1>
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A bit versus a qubit

A bit

Can be in two distinct states, 0 and 1
 A measurement does not affect the state.

A qubit
 can be in state | 0 or in state | 1 or in any other state that is
a linear combination of the basis state
   0 0  1 1

when we measure the qubit we find it
 in state | 0
with probability |  0
 in state | 1
with probability | 
|2
2
|
1
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0
0
Superposition states
1
(a) One bit
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Basis (logical) state 0
Basis (logical) state 1
(b) One qubit
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Other possible states of a qubit
1
1
0 
1
2
2
1
3
0 
1
2
2
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A different basis for one qubit
 
0 1
  0
 
0 1
2
2
 
2
 1
 
2
 0  1
 0  1
 
 

2
2
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The measurement of a qubit
0
p0
p1
1
Possible states of one qubit before
the measurement
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The state of the qubit after
the measurement
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A measurement is a projection of the surrent state
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Two qubits

Represented as a vector in a 2-dimensional Hilbert space with four
basis vectors 00 , 01 , 10 , 11
  00 00  01 01  10 10  11 11
with
|  00 |2  |  01 |2  | 10 |2  | 11 |2  1

When we measure a pair of qubits we decide that the system it is
in one of four states 00, 01, 10, and 11 with probabilities
|  00 | , |  01 | , | 10 | , | 11 |
2
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A measurement of two qubits
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Before a measurement the state of the two qubits is uncertain.
If we measure both qubits the state can be either 00, 01, 10, or 11.
What if we observe only the first qubit, what conclusions can we draw?
We expect that the system to be left in an uncertain sate, because we did
not measure the second qubit that can still be in a continuum of states.
The first qubit can be
 0 with probability
|  00 |2  |  01 |2
 1 with probability
|  |2  |  |2
10
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The measurement of the first qubit of the pair only

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Call  0 the post-measurement state when we measure the first
qubit and find it to be 0.
I
Call  1 the post-measurement state when we measure the first
qubit and find it to be 1.
I

I
0

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1


 00 00   01 01
|  00 |  |  01 |
2
2
10 10  11 11
| 10 |2  | 11 |2
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The measurement of the second qubit of the pair only


Call  0 the post-measurement state when we measure the second
qubit and find it to be 0.
II
Call  1 the post-measurement state when we measure the second
qubit and find it to be 1.
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II

II
0

II
1

00 00  10 10

|  00 |  | 10 |
2
2
01 01  11 11
|  01 |2  | 11 |2
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Bell states - a special state of a pair of qubits
1
 If
and  01  10  0
 00  11 
2
When we measure the first qubit we get the post measurement state
 0I | 00
 1I | 11
When we measure the second qubit we get the post measurement state
 0II | 00
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 1II | 11
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This is an amazing result!
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The two measurements are correlated, once we measure the first qubit
we get exactly the same result as when we measure the second one.
The two qubits need not be physically constrained to be at the same
location and yet, because of the strong coupling between them,
measurements performed on the second one allow us to determine the
state of the first.
An entangled pair is a single quantum system in a superposition of
equally possible states. The entangled state contains no information
about the individual particles, only that they are in opposite states.
The important property of an entangled pair is that the measurement of
one particle influences the state of the other particle. Einstein called
that “Spooky action at a distance".
Entanglement is an elegant, almost exact translation of the German
term Verschrankung used by Schrodinger who was the first to
recognize this quantum effect.
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Milestones in quantum physics
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1900 - Max Plank presents the black body radiation theory; the quantum
theory is born.
1905 - Albert Einstein develops the theory of the photoelectric effect.
1911 - Ernest Rutherford develops the planetary model of the atom.
1913 - Niels Bohr develops the quantum model of the hydrogen atom.
1923 - Louis de Broglie relates the momentum of a particle with the
wavelength
1925 - Werner Heisenberg formulates the matrix quantum mechanics.
1926 - Erwin Schrodinger proposes the equation for the dynamics of the
wave function.
1926 - Erwin Schrodinger and Paul Dirac show the equivalence of
Heisenberg's matrix formulation and Dirac's algebraic one with
Schrodinger's wave function.
1926 - Paul Dirac and, independently, Max Born, Werner Heisenberg, and
Pasqual Jordan obtain a complete formulation of quantum dynamics.
1926 - John von Newmann introduces Hilbert spaces to quantum
mechanics.
1927 - Werner Heisenberg formulates the uncertainty principle.
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Milestones in computing and information theory






1936 - Alan Turing dreams up the Universal Turing Machine, UTM.
1936 - Alonzo Church publishes a paper asserting that ``every function
which can be regarded as computable can be computed by an universal
computing machine''.
1945 - ENIAC, the world's first general purpose computer, the brainchild
of J. Presper Eckert and John Macauly becomes operational.
1946 - A report co-authored by John von Neumann outlines the von
Neumann architecture.
1948 - Claude Shannon publishes ``A Mathematical Theory of
Communication’’.
1953 - The first commercial computer, UNIVAC I.
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Milestones in quantum computing
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1961 - Rolf Landauer decrees that computation is physical and studies
heat generation.
1973 - Charles Bennett studies the logical reversibility of computations.
1982 - Richard Feynman suggests that physical systems including
quantum systems can be simulated exactly with quantum computers.
1982 - Peter Benioff develops quantum mechanical models of Turing
machines.
1984 - Charles Bennett and Gilles Brassard introduce quantum
cryptography.
1985 - David Deutsch reinterprets the Church-Turing conjecture.
1993 - Bennett, Brassard, Crepeau, Josza, Peres, Wooters discover
quantum teleportation.
1994 - Peter Shor develops a clever algorithm for factoring large
numbers.
1996 – Lov Grover develops the quantum search algorithm.
1997 – Demonstration of quantum teleportation; De Martini at University
of Rome; Zeillinger at Innsbruck
2001 – A group at IBM Research in San Jose build a quantum computer
with seven qubits and factors the number 15 using Shor’s algorithm.
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Next seminar – Tuesday June 4, 2012
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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
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