Transcript title

School of Chemistry, Tel Aviv University
The World of Quantum
Information
Marianna Safronova
Department of Physics and Astronomy
November 16, 2008
Outline
 Quantum Information: fundamental principles
(and how it is different from the classical one).
 Bits & Qubits
 Quantum weirdness: entanglement, superposition & measurement
 Logic gates & Quantum circuits
 Cryptography & quantum information
 A brief introduction to quantum computing
 Real world: what do we need to build a quantum
computer/quantum network?
 Current status & future roadmap
Why quantum information?
Information is physical!
Any processing of information
is always performed by physical means
Bits of information obey laws of classical physics.
Why quantum information?
Information is physical!
Any processing of information
is always performed by physical means
Bits of information obey laws of classical physics.
Why Quantum Computers?
Computer technology is
making devices smaller
and smaller…
…reaching a point where classical
physics is no longer a suitable model for
the laws of physics.
Bits & Qubits
Fundamental building
blocks of classical
computers:
BITS
STATE:
Definitely
0 or 1
Fundamental building
blocks of quantum
computers:
Quantum bits
or
QUBITS
Basis states: 0 and 1
Superposition:
  0   1
Bits & Qubits
Fundamental building
blocks of classical
computers:
BITS
STATE:
Definitely
0 or 1
Fundamental building
blocks of quantum
computers:
Quantum bits
or
QUBITS
Basis states: 0 and 1
Qubit: any suitable two-level
quantum system
Bits & Qubits:
Superposition
primary differences
  0   1
Bits & Qubits:
primary differences
Measurement

Classical bit: we can find out if it is in state 0 or 1 and the
measurement will not change the state of the bit.

Qubit: Quantum calculation:
number of parallel processes
due to superposition
QO
FR
Bits & Qubits:
 Superposition
 Measurement
primary differences
  0   1

Classical bit: we can find out if it is in state 0 or 1 and the
measurement will not change the state of the bit.

Qubit: we cannot just measure  and  and thus determine
its state! We get either 0 or 1 with corresponding
probabilities ||2 and ||2.
2
2
   1

The measurement changes the state of the qubit!
Hilbert space is a big place!
Multiple qubits
- Carlton Caves
Hilbert space is a big place!
Multiple qubits


- Carlton Caves
Two bits with states 0 and 1 form four definite states 00,
01, 10, and 11.
Two qubits: can be in superposition of four
computational basis set states.
   00   01   10   11
2 qubits
4 amplitudes
3 qubits
8 amplitudes
10 qubits
1024 amplitudes
20 qubits
1 048 576 amplitudes
30 qubits
1 073 741 824 amplitudes
500 qubits More amplitudes than our estimate of
number of atoms in the Universe!!!
Entanglement
 
00  11
2
  
Results of the measurement
First
qubit
Second qubit
0
0
Entangled
states
1
1
Quantum cryptography
Classical cryptography
Scytale - the first known mechanical device to implement
permutation of characters for cryptographic purposes
Classical cryptography
Private key cryptography
How to securely transmit a private key?
Key distribution
A central problem in cryptography:
the key distribution problem.
1)
2)
Mathematics solution: public key cryptography.
Physics solution: quantum cryptography.
Public-key cryptography relies on the computational
difficulty of certain hard mathematical problems
(computational security)
Quantum cryptography relies on the laws of quantum
mechanics (information-theoretical security).
Quantum key distribution
A quantum communication channel:
physical system capable delivering quantum systems
more or less intact from one place to another.



Quantum mechanics: quantum bits cannot be copied or
monitored.
Any attempt to do so will result in altering it that can not
be corrected.
Problems
 Authentication
 Noisy channels
Quantum logic gates
Logic gates
Classical NOT gate
A
NOT A
Quantum NOT gate
(X gate)
 0  1
X
 1  0
A
NOT A
Matrix form representation
0
1
1
0
0 1 
X 

1
0


The only non-trivial
single bit gate
    
X  
    
More single qubit gates
Any unitary matrix U will produce a quantum gate!
1 0 
Z 

0 1
 0  1
Hadamard gate:
 0  1
H
 0  1
Z
1 1 1 
H


2 1 1

0 1
2

0 1
2
Single qubit gates,two-qubit gates,
three-qubit gates …

How many gates do we need to make?

Do we need three-qubit and four-qubit gates?

Where do we find such physical interactions?

Coming up with one suitable controlled interaction for
physical system is already a problem!
Universality:
classical computation
Only one classical gate (NAND) is needed to
compute any function on bits!
A
B
A AND B
A
NOT A
A
B
A NAND B
A
0
0
1
1
B
0
1
0
1
A
AND
0
0
0
1
B
A
NAND
1
1
1
0
B
Universality:
quantum computation
How many quantum gates do we need
to build any quantum gate?
Any n-qubit gate can be made from 2-qubit gates.
(Since any unitary n x n matrix can be decomposed to
product of two-level matrices.)
Only one two-qubit gate is needed!
Example: CNOT gate
Quantum parallelism
I can look at both
sides of my coin at
a single glance
Quantum parallelism
f ( x) : 0,1  0,1
Superposition
0 1
2
0
Uf
x, y  x, y  f ( x)
x
y
Uf
x
0, f (0)  1, f (1)
f(x)
Single circuit just evaluated f(x) for
both x=0 and 1 simultaneously!
2
Quantum parallelism:
a major problem



So we can evaluate functions for all values of x at the
same time using just one circuit!
Need only n+1 qubits to evaluate 2n values of x.
But we still get only one answer when we measure the
result: it collapses to x,f(x)!!!
QO
FR
Quantum parallelism:
a major problem





So we can evaluate functions for all values of x at the
same time using just one circuit!
Need only n+1 qubits to evaluate 2n values of x.
But we still get only one answer when we measure the
result: it collapses to x,f(x)!!!
We can’t read out the result for specific value of x!
With one qubit we get either f(0) or f(1) and we don’t
know which one in advance.
This is why we need quantum algorithms!
Quantum algorithms
Unique features of quantum computation



Superposition: n qubits can represent 2n integers.
Problem: if we read the outcome we lose the superposition
and we can’t know with certainty which one of the values
we will obtain.
Entanglement: measurements of states of different
qubits may be highly correlated.
Quantum algorithms
Strategy:
Use superposition to calculate 2n values
of function simultaneously and
do not read out the result until a useful
result is expected with reasonably high
probability.
Use entanglement
Current advantages of quantum
computation




Shor's quantum Fourier transform provides
exponential speedup over known classical algorithms.
Applications: solving discrete logarithm and factoring
problems which enables a quantum computer to break
public key cryptosystems such as RSA.
Quantum searching (Grover's algorithm) allows
quadratic speedup over classical computers.
Simulations of quantum systems.
How to factor 15?


Pick a number less then 15: 7
Calculate 7n mod 15:
n
1
2
3
4


7n
7
49
343
2401

Calculate gcd 7
15×A 7n mod 15
1
7
45
4
330
13
2400
1
R/2
 1,15
gcd 48,15  3, gcd 50,15  5
R=4
Shor’s algorithm for N=15








Choose n such as 2n<15: n=4
Choose y: y=7
Initialize two four-qubit register  0  0000 0000
Create a superposition of states of the first register
Compute the function f(k)=7k mod 15 on the second
register.
Operate on the first register by a Fourier transform
Measure the state of the first register: u=0, 4, 8, 12
are only non-zero results.
Two cases give period R=4, therefore the procedure
succeeds with probability 1/2 after one run.
Back to the real world:
What do we need to build a quantum computer?



Qubits which retain their properties.
Scalable array of qubits.
Initialization: ability to prepare one certain state
repeatedly on demand. Need continuous supply of 0 .
Universal set of quantum gates. A system in which
qubits can be made to evolve as desired.

Long relevant decoherence times.

Ability to efficiently read out the result.
Real world strategy
“…If X is very hard it can be
substituted with more of Y.
Of course, in many cases both X and Y are beyond
the present experimental state of the art …”
David P. DiVincenzo
The physical implementation of quantum computation.
Quantum Computer
(Innsbruck)
P1/2
D5/2
„quantum
bit“
S1/2
Courtesy of the R. Blatt’s Innsbruck group
Experimental proposals

Liquid state NMR

Trapped ions

Cavity QED

Trapped atoms

Solid state schemes

And other ones …
The DiVincenzo criteria
Quantum Information Processing
with
Neutral Atoms
Quantum computing with neutral atoms
1. A scalable physical system with
well characterized qubits: memory
(a) Internal atomic state qubits:
ground hyperfine states of neutral trapped atoms
well characterized
Very long lived!
MF=-2,-1,0,1,2
F=2
5s1/2
1
87Rb:
6.8 GHz
F=1
0
Nuclear spin I=3/2
MF=-1,0,1
1. A scalable physical system with
well characterized qubits: memory
(b) Motional qubits : quantized levels in the trapping potential
also well characterized
1. A scalable physical system with
well characterized qubits: memory
(a)Internal atomic state qubits
(b) Motional qubits
Advantages: very long decoherence times!
Internal states are well understood: atomic spectroscopy & atomic clocks.
1. A scalable physical system with
well characterized qubits
Optical lattices: loading of one atom per site
may be achieved using Mott insulator transition.
Scalability: the properties of optical lattice
system do not change in the
principal way when the size of the system
is increased.
Designer lattices may be created
(for example with every third site loaded).
Advantages: inherent scalability and parallelism.
Potential problems: individual addressing.
2: Initialization
Internal state preparation: putting atoms in the ground hyperfine state
Very well understood (optical pumping technique is in use since 1950)
Very reliable (>0.9999 population may be achieved)
Motional states may be cooled to motional ground states (>95%)
Loading with one atom per site: Mott insulator transition and other
schemes.
Zero’s may be supplied during the computation (providing individual
or array addressing).
3: A universal set of quantum
gates
1. Single-qubit rotations: well understood and had been carried out
in atomic spectroscopy since 1940’s.
2. Two-qubit gates: none currently implemented
(conditional logic was demonstrated)
Proposed interactions for two-qubit gates:
(a) Electric-dipole interactions between atoms
(b) Ground-state elastic collisions
(c) Magnetic dipole interactions
Only one gate proposal does not involve moving atoms (Rydberg gate).
Advantages: possible parallel operations
Disadvantages: decoherence issues during gate operations
Rydberg gate scheme
Gate operations are mediated by excitation of Rydberg states
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
Why Rydberg gate?
Rydberg gate scheme
Gate operations are mediated by excitation of Rydberg states
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
Do not need to
move atoms!
FAST!
Local blockade of Rydberg excitations
Excitations to Rydberg
states are suppressed
due to a dipole-dipole
interaction or van der
Waals interaction
http://www.physics.uconn.edu/~rcote/
Rydberg gate scheme
FAST!
Rb
40p
D
R
1
0
Apply a series of laser pulses to
realize the following logic gate:
5s
1
2
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
00
10
01
11
 00
 01
 10
  11
Rb
Rydberg gate scheme
5s
40p
 1
D
R
00
01
10
11
1
0
1
2




00
01
R0
R1
2  2

00
  01

R0

R1
 1

00
  01
  10
  11
4. Long relevant decoherence times
Memory: long-lived states.
5s1/2
F=2
1
F=1
0
6.8 GHz
Fundamental decoherence mechanism for
optically trapped qubits: photon scattering.
Decoherence during gate operations:
a serious issue.
5: Reading out a result
“Quantum jump” method via cycling transitions.
Advantages: standard atomic physics technique, well understood and reliable.
Experimental status
 All five requirements for quantum computations are
implemented.
 Specifically two-qubit gates are demonstrated in
several implementations.
 Several-qubit systems and search algorithm with toy
data are demonstrated with NMR.
 Decoherence-free subspaces, noiseless subsystems,
and active control are demonstrated.
Future…
Whatever is the future of the
Quantum computation it is certainly
a fascinating new field
for human creativity, ingenuity and
Our strive to learn more
about the Universe.