Download Parallel algorithms for 3D Reconstruction of Asymmetric

Quantum Parallelism and the Exact
Simulation of Physical Systems
Dan Cristian Marinescu
School of Computer Science
University of Central Florida
Orlando, Florida 32816, USA
Frontier(s)…from Webster’s unabridged


dictionary.
The part of a settled or civilized country nearest
to an unsettled or uncivilized region.
Any new or incompletely investigated field of
learning or thought.
Computing Frontiers, Ischia, April 14,
2004
2
What is a Quantum computer?



A device that harnesses quantum physical
phenomena such as entanglement and
superposition.
The laws of quantum mechanics differ radically
from the laws of classical physics.
The unit of information, the qubit can exist as a
0, or 1, or, simultaneously, as both 0 and 1.
Computing Frontiers, Ischia, April 14,
2004
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Does quantum computing represent the
frontiers of computing?




Is it for real? Can we actually build quantum
computers? - Very likely, but it will take some time….
If so, what would a quantum computer allow us to do
that is either unfeasible or impractical with today’s
most advanced systems? –
Exact simulation of
physical systems, among other things.
Once we have quantum computers do we need new
algorithms? –
Yes, we need quantum algorithms.
Is it so different from our current thinking that it
requires a substantial change in the way we educate
our students? –
Yes, it does.
Computing Frontiers, Ischia, April 14,
2004
4
Quantum computers: now and then


All we have at this time is a 7 (seven) qubit
quantum computer able to compute the prime
factors of a small integer, 15.
To break a code with a key size of 1024 bits
requires more than 3,000 qubits and 108
quantum gates.
Computing Frontiers, Ischia, April 14,
2004
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Approximate computer simulation of
physical systems



Eniac and the Manhattan project. The first
programs to run, simulation of physical processes.
Computer simulation – new approach to scientific
discovery, complementing the two well established
methods of science: experiment and theory.
Approximate simulation – based upon a model that
abstracts some properties of interest of a physical
system.
Computing Frontiers, Ischia, April 14,
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Exact simulation of physical systems




How far do we want to go at the microscopic level?
Molecular, atomic, quantum? - All of the above.
What about cosmic level? - Yes, of course.
Is it important? - Yes (Feynman, 1981) .
Who will benefit? –


Natural sciences  physics, chemistry, biology,
astrophysics, cosmology,….
Application  nanotechnology, smart materials, drug
design,…
Computing Frontiers, Ischia, April 14,
2004
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Large problem state space

From black hole thermodynamics – a system enclosed by
a surface with area A has a number of observable states:
c = 3 x1010 cm/sec
h = 1.054 x 10-34 Joules/second
G = 6.672 x 10-8 cm3 g-1 sec-2
For an object with a radius of 1 Km  N(A) = e80
Computing Frontiers, Ischia, April 14,
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Acknowledgments

Some of the material presented is from the book
Approaching Quantum Computing
by Dan C. Marinescu and Gabriela M. Marinescu
to be published by Prentice Hall in June 2004

Work supported by National Science Foundation grants
MCB9527131, DBI0296107,ACI0296035, and EIA0296179.
Computing Frontiers, Ischia, April 14,
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Contents







Computing and the Laws of Physics
Quantum Mechanics & Computers
Qubits and Quantum Gates
Quantum Parallelism
Deutsch’s Algorithm
Virus Structure Determination and Drug Design
Summary
Computing Frontiers, Ischia, April 14,
2004
10
The limits of solid-state technologies


For the past two decades we have enjoyed Gordon
Moore’s law. But all good things may come to an end…
We are limited in our ability to increase



the density and
the speed of a computing engine.
Reliability will also be affected


to increase the speed we need increasingly smaller circuits
(light needs 1 ns to travel 30 cm in vacuum)
smaller circuits  systems consisting only of a few particles
are subject to Heissenberg uncertainty
Computing Frontiers, Ischia, April 14,
2004
11
Power dissipation and circuit density


The computer technology vintage year 2000 requires
some 3 x 10-18 Joules per elementary operation.
In 1992 Ralph Merkle from Xerox PARC calculated that
a 1 GHz computer operating at room temperature, with
1018 gates packed in a volume of about 1 cm3 would
dissipate 3 MW of power.


A small city with 1,000 homes each using 3 KW would require
the same amount of power;
A 500 MW nuclear reactor could only power some 166 such
circuits.
Computing Frontiers, Ischia, April 14,
2004
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Energy consumption of a logic circuit
E
S
Speed of individual logic gates
Heat removal for a circuit with densely packed
logic gates poses tremendous challenges.
(b)
(a)
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Contents







Computing and the Laws of Physics
Quantum Mechanics & Computers
Qubits and Quantum Gates
Quantum Parallelism
Deutsch’s Algorithm
Virus Structure Determination and Drug Design
Summary
Computing Frontiers, Ischia, April 14,
2004
14
A happy marriage…

The two greatest discoveries of the 20-th century


quantum mechanics
stored program computers
led to the idea of
quantum computing and
quantum information theory
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Quantum; Quantum mechanics


Quantum  Latin word meaning some quantity. In
physics it is used with the same meaning as the
word discrete in mathematics.
Quantum mechanics  a mathematical model of
the physical world.
Computing Frontiers, Ischia, April 14,
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Heissenberg’s uncertainty principle

“... Quantum Mechanics shows that not only the
determinism of classical physics must be
abandoned, but also the naive concept of reality
which looked upon atomic particles as if they were
very small grains of sand. At every instant a grain
of sand has a definite position and velocity. This is
not the case with an electron. If the position is
determined with increasing accuracy, the possibility
of ascertaining its velocity becomes less and vice
versa.'‘ (Max Born’s Nobel prize lecture on December 11, 1954)
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Milestones in quantum computing








1961 - Rolf Landauer decrees that computation is physical and
studies heat generation.
1973 - Charles Bennet studies the logical reversibility of
computations.
1981 - Richard Feynman suggests that physical systems including
quantum systems can be simulated exactly with quantum computers.
1982 - Peter Beniof develops quantum mechanical models of Turing
machines.
1984 - Charles Bennet and Gilles Brassard introduce quantum
cryptography.
1985 - David Deutsch reinterprets the Church-Turing conjecture.
1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters discover
quantum teleportation.
1994 - Peter Shor develops a clever algorithm for factoring large
integers.
Computing Frontiers, Ischia, April 14,
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Deterministic versus probabilistic photon
behavior
D1
D3
Incident beam of light
D5
Detector D1
D7
Reflected beam
Beam splitter
Transmitted beam
Detector D2
D2
(a)
Computing Frontiers, Ischia, April 14,
2004
(b)
19
|
0
|
1
(a)
E
S
S
(b)
(c)
intensity = I
A
B
S
E
intensity = I/2
A
C
B
E
S
intensity = 0
(d)
E
A
intensity = I/8
(e)
Computing Frontiers, Ischia, April 14,
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Contents






Computing and the Laws of Physics
Quantum Mechanics & Computers
Qubits and Quantum Gates
Quantum Parallelism
Virus Structure Determination and Drug Design
Summary
Computing Frontiers, Ischia, April 14,
2004
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One qubit



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)
Computing Frontiers, Ischia, April 14,
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State description
0
0
V
q0 = q
1
45o
O
q1 = q
q0
1
1
O
(a)
V
30o
1
q1
(b)
Computing Frontiers, Ischia, April 14,
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|0>
z

|ψ 

r
x

y
b
|1>
Computing Frontiers, Ischia, April 14,
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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



  0 0  1 1
can be in state | 0 or in state | 1 or in any other state
that is a linear combination of the basis state
When we measure the qubit we find it


in state | 0
in state | 1
with probability
with probability
|  0 |2
| 1 | 2
Computing Frontiers, Ischia, April 14,
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0
0
Superposition states
1
(a) One bit
1
Basis (logical) state 0
Basis (logical) state 1
(b) One qubit
Computing Frontiers, Ischia, April 14,
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Qubit measurement
0
p0
p1
1
Possible states of one qubit before
the measurement
The state of the qubit after
the measurement
Computing Frontiers, Ischia, April 14,
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Two qubits

Represented as vectors in a 2-dimensional Hilbert
space with four basis vectors
00 , 01 , 10 , 11

When we measure a pair of qubits we decide that the
system it is in one of four states
00 , 01 , 10 , 11

with probabilities
|  00 | , |  01 | , | 10 | , | 11 |
2
2
Computing Frontiers, Ischia, April 14,
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2
2
28
Two qubits
  00 00  01 01  10 10  11 11
|  00 |  |  01 |  | 10 |  | 11 |  1
2
2
2
Computing Frontiers, Ischia, April 14,
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2
29
Measuring two qubits


Before a measurement the state of the system
consisting of two qubits is uncertain (it is given by the
previous equation and the corresponding probabilities).
After the measurement the state is certain, it is
00, 01, 10, or 11 like in the case of a classical two bit
system.
Computing Frontiers, Ischia, April 14,
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Measuring two qubits (cont’d)


What if we observe only the first qubit, what
conclusions can we draw?
We expect 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 |  |  01 |
1 with probability
| 10 |  | 11 |
2
2
2
2
Computing Frontiers, Ischia, April 14,
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Measuring two qubits (cont’d)



 0I
Call
measure
I
Call  1
measure
I
0

the post-measurement state when we
the first qubit and find it to be 0.
the post-measurement state when we
the first qubit and find it to be 1.
 00 00   01 01
|  00 |  |  01 |
2
2

I
1

Computing Frontiers, Ischia, April 14,
2004
10 10  11 11
| 10 |  | 11 |
2
2
32
Measuring two qubits (cont’d)



II
0
 0II
Call
measure
II
Call  1
measure

the post-measurement state when we
the second qubit and find it to be 0.
the post-measurement state when we
the second qubit and find it to be 1.
00 00  10 10
|  00 |  | 10 |
2
2

II
1

Computing Frontiers, Ischia, April 14,
2004
01 01  11 11
|  01 |  | 11 |
2
2
33
Bell states - a special state of a pair of qubits

If
1
 00  11 
2
and
 01  10  0
When we measure the first qubit we get the post
measurement state
I
I
 1 | 11
 0 | 00
When we measure the second qubit we get the post
mesutrement state  II | 00  1II | 11
0
Computing Frontiers, Ischia, April 14,
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This is an amazing result!


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.
Computing Frontiers, Ischia, April 14,
2004
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Entanglement (Verschrankung)



Discovered by Schrodinger.
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.
Einstein called entanglement “Spooky action at
a distance".
Computing Frontiers, Ischia, April 14,
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Classical gates



Implement Boolean functions.
Are not reversible (invertible). We cannot recover the
input knowing the output.
This means that there is an irretrievable loss of
information.
Computing Frontiers, Ischia, April 14,
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NOT gate
x
x
0
1
y
1
0
z = (x) AND (y)
x
0
0
1
1
y
0
1
0
1
z
0
0
0
1
z = (x) NAND (y)
x
0
0
1
1
y
0
1
0
1
z
1
1
z = (x) OR (y)
x
0
0
1
1
y
0
1
0
1
z
0
1
1
1
z = (x) NOR (y)
x
0
0
1
1
y
0
1
0
1
z
1
0
0
0
y
0
1
0
14,1
z
0
1
1
0
y = NOT(x)
x
AND gate
y
x
NAND gate
y
x
OR gate
y
x
NOR gate
y
x
XOR gate
y
Computing
x
0
z = (x) XOR (y)
0
1
1
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2004
1
0
38
   0  1
   0 0  1 1
'
0
'
1
One-qubit gate
 g11 g12 

G  
 g 21 g 22 
  G
 0   g11
   
 1   g 21
Computing Frontiers, Ischia, April 14,
2004
g12  0 
 
g 22  1 
39
One qubit gates





I  identity gate; leaves a qubit unchanged.
X or NOT gate transposes the components
of an input qubit.
Y gate.
Z gate  flips the sign of a qubit.
H  the Hadamard gate.
Computing Frontiers, Ischia, April 14,
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Identity transformation, Pauli matrices, Hadamard
1 0

 0  I  
0 1
0 1

1  X  
1 0
0  i

 2  Y  
i 0 
1 0 

 3  Z  
 0  1
1 1 1 


H
2 1  1
  0 0  1 1
  1 0  0 1
  i1 0  i0 1
  0 0  1 1
  0
Computing Frontiers, Ischia, April 14,
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0 1
2
 1
0 1
2
41
CNOT a two qubit gate

Two inputs


Control
Target
Control input


Target input


+
O


+ 
O
addition modulo 2
The control qubit is transferred to the output as is.
The target qubit


Unaltered if the control qubit is 0
Flipped if the control qubit is 1.
Computing Frontiers, Ischia, April 14,
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CNOT
WCNOT  GCNOT VCNOT
VCNOT    
GCNOT
1

0

0

0

0
1
0
0
0
0
0
1
0

0
1

0 
Computing Frontiers, Ischia, April 14,
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43
The two input qubits of a two qubit gates
  0 0  1 1
  0 0  1 1
VCNOT
 0  0 


  0    0    0 1 
           





 1  1  1 0
  
 1 1
Computing Frontiers, Ischia, April 14,
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44
State space dimension of classical and quantum
systems

Individual state spaces of n particles combine quantum
mechanically through the tensor product. If X and Y are
vectors, then



their tensor product X 
Y is also a vector, but its dimension is:
dim(X) x dim(Y)
while the vector product X x Y has dimension
dim(X)+dim(Y).
For example, if dim(X)= dim(Y)=10, then the tensor
product of the two vectors has dimension 100 while the
vector product has dimension 20.
Computing Frontiers, Ischia, April 14,
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45
Parallelism and Quantum computers


In quantum systems the amount of parallelism
increases exponentially with the size of the
system, thus with the number of qubits (e.g. a
21 qubit quantum computer is twice as powerful
as a 20 qubit quantum computer).
A quantum computer will enable us to solve
problems with a very large state space.
Computing Frontiers, Ischia, April 14,
2004
46
Contents







Computing and the Laws of Physics
Quantum Mechanics & Quantum Computers
Qubits and Quantum Gates
Quantum Parallelism
Deutsch’s Algorithm
Virus Structure Determination and Drug Design
Summary
Computing Frontiers, Ischia, April 14,
2004
47
A quantum circuit

Given a function f(x) we can construct a reversible
quantum circuit consisting of Fredking gates only,
capable of transforming two qubits as follows
|x>
|x>
Uf
|y>

| y o+ f(x )>
The function f(x) is hardwired in the circuit
Computing Frontiers, Ischia, April 14,
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48
A quantum circuit (cont’d)

If the second input is zero then the transformation
done by the circuit is
|x>
|x>
Uf
| f(x )>
|0>
Computing Frontiers, Ischia, April 14,
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49
A quantum circuit (cont’d)

Now apply the first qubit through a Hadamad gate.
0 1
|0>
H
2
|0>
Uf
|0>
0
f (0)  1 f (1)
2



The resulting sate of the circuit is
0 1
0 f(
)
2
The output state contains information about f(0) and f(1).
Computing Frontiers, Ischia, April 14,
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50
Quantum parallelism



The output of the quantum circuit contains information
about both f(0) and f(1). This property of quantum circuits
is called quantum parallelism.
Quantum parallelism allows us to construct the entire truth
table of a quantum gate array having 2n entries at once.
In a classical system we can compute the truth table in
one time step with 2n gate arrays running in parallel, or we
need 2n time steps with a single gate array.
We start with n qubits, each in state |0> and we apply a
Walsh-Hadamard transformation.
Computing Frontiers, Ischia, April 14,
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|x>
(m-dimensional)
|x>
Uf
|y>
(k-dimensional)
|yO
+ f(x) >
(n=m+k)-dimensional)
Computing Frontiers, Ischia, April 14,
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52
H0 
0 1
2
( H  H  H ) 000 

Uf (
1
2n
2 n 1
 x,0 ) 
x 0
1
2n
1
2
1
n
2
n
[( 0  1 )  ( 0  1 )    ( 0  1 )]
2 n 1
x
x 0
2 n 1
U
x 0
f
( x,0 ) 
Computing Frontiers, Ischia, April 14,
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1
2 n 1
2n
x 0
 x, f ( x ) )
53
Contents







Computing and the Laws of Physics
Quantum Mechanics and Computers
Qubits and Quantum Gates
Quantum Parallelism
Deutsch’s Algorithm
Virus Structure Determination and Drug Design
Summary
Computing Frontiers, Ischia, April 14,
2004
54
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)  Computing
f (Frontiers,
1) Ischia, April 14,
2004
55
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)
Computing Frontiers, Ischia, April 14,
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A quantum circuit to solve Deutsch’s problem
|0>
H
|x>
|x>
H
Uf
|1>
H
0
|y>
1
| y > +O f(x)
2
Computing Frontiers, Ischia, April 14,
2004
3
57
 0
 
1 0 1
 0  0 1         
 0 1  0
 0
 
1 1 1 1 


1
1
1
1
1

1
1

1




1
1
1


 

  
G1  H  H 
2 1  1
2 1  1 2 1 1  1  1


1  1  1 1 


1 1 1 1  0 
1

 
 
1 1  1 1  1 1  1   1
1  G1 0  
  



2 1 1 1 1 0
2 1

 
 
1  1  1 1  0 
  1

 
 
 0  1  0  1 
1
1  ( 00  01  10  11 )  


2
2 
2 

x 
0 1
2
y 
0 1
2
Computing Frontiers, Ischia, April 14,
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58
y  f ( x) 
0 1
2

y  f ( x)  (1)
 f ( x)
0  f ( x)  1  f ( x)
2
f ( x)

f ( x)  1  f ( x)
2
0 1
2
 0 1

2
y  f ( x)  
 0  1

2
if
f ( x)  0
if
f ( x)  1
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x 
y 
0 1
2
0 1
2
2
2

1
 





0

1
0

1
1
  1


if


 
2 
2  2 1 
 

  1

 
 x  ( y  f ( x))  
1

 
  0  1  0  1 
1   1



 


 1  if
2
2
2


 
 
  1

 


1
 





0

1
0

1
1   1


if


 
2 
2  2   1
 

1

 
 x  ( y  f ( x))  
1

 
  0  1  0  1 
1   1



 


  1 if
2
2
2


 
 
1

 

Computing Frontiers, Ischia, April 14,
2004
f (0)  f (1)  0
f (0)  f (1)  1
f (0)  0, f (1)  1
f (0)  1, f (1)  0
60
2

1
 

 1   1 if
 2 1 
 

  1

 
 x  ( y  f ( x))  
1

 
 1   1
   if
 2   1
1

 

f (0)  f (1)
f (0)  f (1)
Computing Frontiers, Ischia, April 14,
2004
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1

1 1 1   1 0  1  0



G3  H  I 
2 1  1  0 1 
2 1

0







 3  G3 2  






1

1 0
2 1

0

1

1 0
2 1

0

0
1
0
1
0
1
0
1
 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
0
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
 
Computing Frontiers, Ischia, April 14,
2004
if
f (0)  f (1)
if
f (0)  f (1)
62
Evrika!!

By measuring the first output qubit qubit we are able to
determine
f (0)  f (1) performing a single evaluation.
3   f (0)  f (1)
0 if

f (0)  f (1)  
1 if
0 1
2
f (0)  f (1)
f (0)  f (1)
Computing Frontiers, Ischia, April 14,
2004
63
Contents







Computing and the Laws of Physics
Quantum Mechanics and Computers
Qubits and Quantum Gates
Quantum Parallelism
Deutsch’s Algorithm
Virus Structure Determination and Drug Design
Summary
Computing Frontiers, Ischia, April 14,
2004
64
Sindbis virus reconstruction and pseudo-atomic modeling. The reconstruction computed at
~11Å (top half of first two panels) and ~22Å (bottom half of first two panels), viewed along
a two-fold axis and represented as a surface-shaded solid (left panel) and as a thin, nonequatorial section (middle panel). The 11Å reconstruction, which shows significantly
greater detail compared to that in the 22Å map available approximately one year ago,
provides more accurate data for fitting atomic models as illustrated in the right panel, which
is an enlarged view of the boxed area in the middle panel.
Computing Frontiers, Ischia, April 14,
2004
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Final remarks


A tremendous progress has been made in quantum
computing and quantum information theory during
the past decade.
Motivation  the incredible impact this discipline
could have on how we store, process, and transmit
data and knowledge in this information age.
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2004
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Final remarks (cont’d)

Computer and communication systems using quantum
effects have remarkable properties.




Quantum computers enable efficient simulation of the most
complex physical systems we can envision.
Quantum algorithms allow efficient factoring of large integers with
applications to cryptography.
Quantum search algorithms speedup considerably the process of
identifying patterns in apparently random data.
We can improve the security of our quantum communication
systems because eavesdropping on a quantum communication
channel can be detected with high probability.
Computing Frontiers, Ischia, April 14,
2004
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Summary


Quantum computing and quantum information theory
is truly an exciting field.
It is too important to be left to the physicists alone….
Computing Frontiers, Ischia, April 14,
2004
68