Quantum Information
A Glimpse at the Strange and
Intriguing Future of Information
Dan C. Marinescu
School of Computer Science
University of Central Florida
Orlando, Florida 32816, USA
dcm@cs.ucf.edu
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Acknowledgments
The material presented is based on the books
Approaching Quantum Computing
ISBN 013145224X, Prentice Hall, March 2004
Approaching Quantum Information Theory
(in preparation)
by Dan C. Marinescu and Gabriela M. Marinescu
Work supported by National Science Foundation
grants MCB9527131, DBI0296107,ACI0296035,
and EIA0296179.
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Information
• 2,450,000,000 Google hits for the word “information”.
• The earliest historical meaning of the word information
in English was related to the act of informing, or giving
form or shape to the mind, as in education, instruction,
or training. A quote from 1387: "Five books come
down from heaven for information of mankind."
(Oxford English Dictionary)…..
Amazon.com was established later….
• 2667?
( Japanese imperial year based on the mythical founding of Japan by Emperor Jimmu in 660 BC)
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Information (cont’d)
• Information is a primitive concept (like matter or energy).
• Information abstracts properties of and allows us to
distinguish objects/entities/phenomena.
• There is a common expression of information, strings of
bits, regardless of the object/entity/process it describes.
Bits are independent of their physical embodiment.
• Information is transformed using logic operations. Gates
implement logic operations and allow for automatic
processing of information. The usefulness of information
increases if the physical embodiments of bits and gates
become smaller and we need less energy to process,
store, and transmit information.
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Classical Information
• Can be copied without altering it.
• Deterministic; the result of measuring/observing it
is deterministic (unless affected by noise).
• Cannot travel faster than light or backward in time.
• It is processed by conventional computers using
irreversible gates. During processing we experience
an irretrievable loss of information.
• Information Theory was developed by Shannon for
macroscopic bodies at a time when microscopic
systems carrying information were not known.
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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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SENSORS
DIGITAL CAMERAS
2000s
WORLD WIDE WEB
1990s
GOOGLE, YouTube
2000s
DISSEMINATE
MICROPROCESSORS
1980s
COLLECT
MILESTONES OF
INFORMATION
PROCESSING
1850 – 2007
PROCESS
BOOLEAN ALGEBRA
1854
COMMUNICATE
FIBER OPTICS
1990s
WIRELESS
2000s
DIGITAL
COMPUTERS
1940s
QUANTUM
COMPUTING
QUANTUM
COMMUNICATION
STORE
OPTICAL STORAGE
HIGH DENSITY
SOLID-STATE
1990s
SPINTRONICS
2000s
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Moore’s Law
Microprocessor
Year
# transistors
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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Limits of Solid-State Technology
• 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.
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The heat generated by densely packed solid-state
devices in a sphere of radius R is proportional to
the volume thus to R3; the heat can be removed
trough the surface of the sphere, proportional to R2
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Hitting a Wall…
• 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
A happy marriage between two of the greatest scientific
achievements of the 20th century:
quantum mechanics
stored program computers.
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.”
Quantum information Information encoded as
the state of atomic or sub-atomic particles.
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Richard Feynman
I think I can fairly say that
nobody understands
QuantumMechanics
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Light
• Light  electromagnetic radiation.
• The electric and magnetic field
•
– oscillate in a plane perpendicular to the direction of
propagation and
– are perpendicular to each other.
The dual, wave and corpuscular, nature of light:
– Diffraction phenomena  wave-like behavior
– Photoelectric effect
 corpuscular/granular The
light consists of quantum “particles” called photons.
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Polarization of Light
Polarization is given by the electric field vector
– Linearly polarized (vertical/horizontal)  the tip of the electric field
vector oscillates along any straight line in a plane perpendicular to
the direction of propagation.
– Circularly polarized (right- /left-hand)  the tip of the electric field
vector moves along a circle in a plane perpendicular to the direction
of propagation:
– Elliptically polarized light  the tip of the electric field vector moves
along an ellipse in a plane perpendicular to the direction of
propagation.
• In a beam of linearly polarized light each photon has a
random orientation of the polarization vector.
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The Spin of Atoms and Sub-atomic Particles
• Spin  the intrinsic angular momentum; it takes
discrete values (the spin quantum number s.)
• Two classes of quantum particles:
– fermions - spin one-half (e.g., electrons).
• s=+1/2 and
• s=-1/2
– bosons - spin one particles (e.g., photons).
• s=+1,
• s=0, and
• s=-1
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The spin of the electron:
+ ½  spin up,
- ½  spin down.
Sz
Rn(0)
1
h
2
-
1
h
2
Rn(180)
(a)
(b)
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Quantum Mechanics
Quantum mechanics  mathematical model of the
physical world.
Quantum concepts such as:
– Uncertainty,
– Superposition,
– Entanglement,
– No-cloning,
do not have a correspondent in classical physics.
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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
• Non-determinism  basic tenet of quantum mechanics.
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Max Born’s Nobel Prize Lecture, Dec. 11, 1954
“... 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.”
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“Liebe Gott würfelt nicht”
(Dear God does not play
dice)
- Albert Einstein
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Superposition Principle
States of a quantum system:
- Orthogonal.
- Non-orthogonal.
- Superposition – a weighted sum (some elements
appear with a – sign because the phases are
negative). Schrödinger’s cat.
In a quantum system, in addition to reliably
distinguishable states there are states that cannot
be reliably distinguishable.
Incomplete distinguishability is one of the tenets of
quantum mechanics.
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From Lewis Carroll to…. Incomplete
Distinguishability in Quantum Physics
“I have a very long and sad tale’’ said the Mouse.
“I see that your tail is long, but why do you say it is
sad?’’ asked
Alice.
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V
H
135
45
(a) Orthogonal states of a photon can be
reliably distinguished from one another:
V (vertical) from H (horizontal);
135 deg from 45 deg.
-
(b) Non-orthogonal states of a photon cannot
be reliably distinguished from one another:
V (vertical) from 45 deg.
+
=
=
2
2
(c) Superposition states.
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Entanglement (Vërschränkung)
• Discovered by Schrödinger.
• 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.”
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EPR (Einstein, Podolski,Rosen) Effect
• A source generates two entangled particles e.g., two
•
photons entangled in polarization.
A measurement of one of them, say using a V-H
basis, produces a random result (V) or (H) and at the
same time forces the other particle to enter the same
state.
Measurement
Source
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No Cloning Principle
Monogamy of Entanglement
Quantum states cannot be cloned. Cloning
- would increase distinguishability of states,
- it is a non-linear transformation.
A
A
Quantum Copy
Machine
A’
B
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Charles Bennett and Peter Shor: “classical information
can be copied freely, but can only be transmitted forward
in time to a receiver in the sender's forward light cone.
Entanglement, by contrast cannot be copied, but can
connect any two points in space-time. Conventional
data-processing operations destroy entanglement, but
quantum operations can create it, preserve it and use it
for various purposes, notably speeding up certain
computations and assisting in the transmission of
classical data or intact quantum states (teleportation)
from a sender to a receiver.”
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Quantum Information
• Embodied by the state of atomic or sub-atomic
particles.
• Superposition - we cannot reliably recognize differences
between the states of a quantum system except under
special conditions.
• The state of a quantum system cannot be measured or
copied without disturbing it.
• Quantum state can be entangled. Two or more systems
have a definite state though neither has an identifiable
state of its own.
• Qubits – elementary units of quantum information.
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Quantum Information
Classical
Information
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Quantum
Information
Measurement
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Classical
Information
33
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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Entropy
• Thermodynamic:
  the number of
S  k B log 
microstates
• Informational, Shannon’s:
X is a random variable
pXi -probability of outcome Xi
• Quantum, von Neumann’s:
 is the density matrix
H   p X i log p X i
i
S (  )  Tr (  log  )
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A Bit Versus a Qubit
0
0
Superposition states
1
(a) One bit
1
Basis (logical) state 0
Basis (logical) state 1
(b) One qubit
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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
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Quantum Gates
• One-qubit gates  X - transposes the components of a
•
•
qubit; Z - flips the sign of a qubit; Hadamard - creates a
superposition state.
Two-qubit gates  CNOT
Three-qubit gates  Toffoli
• Quantum gates are reversible  in principle no power
dissipation.
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Universal Quantum Gates
• Any Boolean expression can be written as a sum
(logical OR) of products (logical AND) of Boolean
variables and/or negation of Boolean variables.
Thus, any classical logic circuit can be
implemented using only AND, OR, and NOT gates.
• NAND and NOR are classical universal gates.
• Similarly, we can simulate any complex n-qubit
quantum circuit using a small set of one-qubit and
CNOT gates.
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Decoherence
Decoherence  randomization of the internal state of a
quantum computer due to interactions with the
environment.
Conceptually decoherence can be prevented using:
- Quantum fault-tolerant circuits.
- Quantum Error Correcting Codes.
- Entanglement Purification and Distillation extract a
subset of states of high entanglement and high purity
from a large set of less entangled states.
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Di Vicenzo’s Criteria for Physical
Implementation of a Quantum Computer
1.
2.
3.
4.
5.
Scalable physical system with well characterized qubits.
Initialize the qubits state as |000…00>.
Long decoherence times.
Universal set of quantum gates (operations).
Qubit specific measurements
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Entering the Quantum Wonderland ….
• We now have:
– quantum gates and quantum circuits
– quantum communication channels.
• What should we be excited about?
–
–
–
–
Quantum parallelism
Quantum teleportation
Communication with entangled particles
Quantum key distribution
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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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Bush
Kerry
Bush
Bush
Kerry
Bush
Balanced function f(0) = f(1)
Bush
Kerry
Bush
Kerry
Kerry
Bush
Unbalanced function
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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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Quantum Teleportation
• The process of transferring the state of a quantum
particle to possibly distant one.
• Based upon the entanglement.
• No cloning - the original state is destroyed in the
quantum teleportation process.
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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
X
Z
Z
Classical
Channel
Particle 2 is in the same
state as particle 3
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A Teleportation Experiment
• Francesco De Martini, University of Rome, 1997.
• Based upon an idea of Sandu Popescu.
• A UV laser beam interacts with a non-linear 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.
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Reflecting mirror
Reflecting mirror
A
D
P
o
l
a
r
I
z
e
r
Alice
h
v
Source
h
Bob
v
B
C
Reflecting mirror
Reflecting mirror
Carol
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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.
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Vertical
Horizontal
45 deg
Vertical/Horizontal (VH)
135 deg
Diagonal (DG)
(a)
(b)
Quantum communication channel
Source of
polarized
photons
Quantum wiretap
Photon
separation
system
Eve
Classical wiretap
Alice
Classical communication channel
Bob
(c)
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Information Encoding for Quantum Key
Distribution
• 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 45 deg polarization, and
– 0  a photon with a 135 deg polarization.
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Alice sends to Bob photons with X(V/H) and Z(D45/D135 ) polarization.
Bob chooses a base and measures incoming photons.
X Z X X Z X Z X Z Z X Z X Z X Z X X Z bases
results
Bob sends the basis he used for each photon over a classical channel.
XZ XXZ X Z XZ
Z XZ X
Z
X Z X X Z
Alice tells Bob which ones are correct over a classical channel.
X
XZ X Z XZ
X
X Z
X
Bob examines the ones they agree upon (if no eavesdropping).
Bob decodes the photons.
1
10 0
1 11
1
0
0
0
Alice sends Bob the parity of a selected subset.
1
10 0
1 11
1
0
0
0
Parity of
1,4,5,9,11..
=EVEN
Bob verifies the parity of a selected subset.
1
10 0
10
1 11
1
11
1
0
0
0
0
Parity of
1,4,5,9,11.. = OK
Secret Key
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Physical Embodiment of a Qubit
• Photon  information encoded as the photon polarization;
•
•
e.g., horizontal and vertical.
Electron  information encoded as the electron spin;
two independent spin values, +1/2 and -1/2.
Quantum dots  information encoded as the
presence/absence of electrons
– Small devices that contain a tiny droplet of free electrons.
– Fabricated in semiconductor materials; typical dimensions between nanometers
to a few microns.
– The size and shape of these structures and therefore the number of electrons
they contain, can be precisely controlled; a quantum dot can have anything from
a single electron to a collection of several thousands.
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Physical Embodiment of a Qubit (cont’d)
• A two-level atom in an optical cavity.
• Two internal states of an ion in a trap.
• Others
– Liquid-state NMR.
– NMR spin lattices.
– Nitrogen vacancies in diamond.
– Josephson junctions.
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Milestones in Quantum Computing
• 1961 - Rolf Landauer computation is physical.
• 1973 - Charles Bennett  logical reversibility of
computations.
• 1981 - Richard Feynman  physical systems including
quantum systems can be simulated exactly with quantum
computers.
• 1982 - Peter Benioff  develops quantum mechanical
models of Turing machines.
• 1994 - Peter Shor algorithm for factoring large numbers.
• 1995 - Lov Grover  quantum database searching algorithm
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Milestones in Quantum Information Theory
• 1984 - Charles Bennett and Gilles Brassard 
quantum cryptography.
• 1985 - David Deutsch  reinterprets the ChurchTuring conjecture.
• 1993 - Bennett, Brassard, Crepeau, Jozsa, Peres,
Wootters  quantum teleportation.
• 1994 - Calderbank, Shor, Steane quantum error
correcting codes
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The point is:
We must make it as
simple as possible
… but not simpler !
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Final Remarks
• Building a quantum computer faces tremendous
technological and theoretical challenges.
• We are years, possibly decades away from actually
building a quantum computer. All we had on
February 13 2007 was a 7 qubit liquid NMR quantum
computer able to factor the integer 15.
• Applications of quantum cryptography seem ready
for commercialization. In 2003 a successful quantum
key distribution experiment over a distance of some
100 km has been announced.
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“Success is the ability
to go from failure to failure
with no loss of
enthusiasm.”
Sir Winston Churchill
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The Answer to the Puzzle
• The filter block of the 16 qubit quantum computer that
D-Wave Systems plan to unveil on Feb 13... The filter
block is an electronic interface between our world and
the quantum entangled world.
• An array of 128 lumped element filters, one for each
input line. The space is constrained because the filters
and wires need to fit into the dilution fridge cylinder.
The filters remove noise and crosstalk (opposite of an
antenna) from the signals that drop down to the heart
of a new quantum computer, cooled to 0.005 above
absolute zero…
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D-Wave Press Release - February 14
• Right now, Orion is a "proof of concept," a demonstration of what the final
product could look like. At the demonstration, Rose had the system come up
with answers to Sudoku problems and, in another demo, seek out similar
molecules to the active ingredient in the drug Prilosec in a chemical
database. The computer found several molecules that shared similar
structural elements with Prilosec, but the molecule that matched it closest
was the active ingredient in another drug called Nexium. Plucking out Nexium
demonstrated the system's accuracy, the company said. Nexium is actually a
mirror image of the molecule in Prilosec that AstraZeneca invented to extend
its patents.
• The computer itself--which is cooled down to 4 millikelvin (or nearly minus
273.15 degrees Celsius) with liquid helium--was actually in Canada.
Attendees only saw the results on a screen. Still, it was the largest
demonstration of a quantum computer ever, Rose said.
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D-Wave Press Release - February 14
• End of 2007  32 qubit quantum computer.
• In mid 2008  512 qubit quantum computer
• End of 2008  1024 qubit quantum computer
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