Download Quantum Random Walks

ICT Information Day
26.02.2009
Vladimír Bužek
Projects
• QUBITS
• Quantum gates for information processing (QGATES)
• Quantum applications (QAP) – 10MEURO
• EQUIP
• Entanglement in distributed systems (QUPRODIS)
• Hibrid Information processors (HIP)
• QUEST
• Controlled quantum coherence and entanglement in systems of trapped
particles (CONQUEST) – coordination 2.5 MEURO
• QUIPROCONE
• QUROPE
• ERA-Pilot QIST
• Quantum technologies (INTAS) – coordination
QGATES & CONQUEST
TRAPPED IONS (IN QED CAVITY)
NEUTRAL ATOMS IN CAVITIES
ATOMS & PHOTONS
z
Bx / J
NOBEL PRIZE 2005
Prof. Ted Hänsch
Prof. Ted Hänsch received the Nobel prize for his contribution to laser
spectroscopy, and in particular the frequency comb technique. The frequency
comb cleverly uses pulsed lasers to realize frequency "ruler", which allows one to
measure optical frequencies with extreme precision. For the first time, the
frequency (that is, the colour) of light emitted by atoms and ions can now be
directly measured in terms of the fundamental SI unit of frequency, which is
realized in atomic clocks.
Applications range from the measurement of fundamental constants all the way to
higher-bandwidth optical fibre communications. In particular, the frequency comb
opens the door for use of trapped atoms and ions (the object studied in
CONQUEST) as a clockwork in optical clocks, which are expected to be more
than 100 times more precise than the best clocks existing today. The fundamental
property of light waves which enables the frequency comb is its coherence - the
same property which is now being studied in CONQUEST for matter waves.
Príncipe de Asturias de
Investigación Científica 2006
Prof. Ignacio Cirac
The Prince of Asturias Foundation was formed in 1980 in the City of
Oviedo, the capital of the Principality of Asturias, in a ceremony presided
over by His Royal Highness the Prince of Asturias, Heir to the Throne of
Spain, accompanied by his parents, King Juan Carlos I and Queen Sofía.
The Prince of Asturias Awards symbolize the main objectives of the
Foundation: to contribute to upholding and promoting all those scientific,
cultural and humanistic values that form the heritage of humanity.
One goal of reasearch of Prof. Cirac is to propose and analyze
experiments that aim at observing and discovering interesting quantum
phenomena in atomic systems. Under certain conditions e.g. atomic
gases can take on exotic properties once they reach very low
temperatures. Another focus is to investigate, how atomic systems can
be controlled and manipulated at the quantum level using lasers.
Professor Cirac is also leading in the development of a theory of
Quantum Information which will be the basis of several applications in the
world of communication and computation once microscopic systems can
be completely controlled at the quantum level. The concepts developed
in the field of Quantum Optics and Quantum Information are also applied
to other fields, in particular to Condensed Matter Physics
CONTENT
I. Reconstruction of quantum channels from incomplete data
- from non-physical to physical maps via Max-Likelihood
- reconstruction of photon states in the cavity-QED
II. De-coherence in information processing
- q-decoherence from first principles
III. Quantum random walks
- QRW on a hypercube: scattering model
IV. Universal Quantum Machines
- universal quantum entangler
V. Programmable processors
- realization of POVMs via programmable devices
- general theory
VI. Graphs of entanglement, Ising model & QIT
I. Black box Problem
• How can we determine properties of unknown q-channel (black box
with no memory)? We can use qubits as probes and from
correlations between in and out states we can determine the map.
?
Maximum Likelihood
• ML works with finite sets of data, not with infinite ensembles
• In case of quantum operations, the related data are
• Input state specification i
• Measurement direction 
i
• Measurement outcome (binary) pi
• We build a functional
F     i   i  
i
i
pi  1   i   i  
i
 1  p 
• The numerical task is to find the  , for which this functional
reaches the maximum (using the logarithm of functional)
• Trace-preservation is obtained automatically from the
parameterization, CP has to be checked in the algorithm
i
Experimental Data
•
•
Data from the group of Ch. Wunderlich were analyzed
Depolarization channel was expected
1

0


0

0
0 0 0

x 0 0
0 x 0

0 0 1
Physical approximation of non-physical maps
 



exp
• Nonlinear polarization rotation  
i z
 2




 z   exp  i


2
z


z 

Reconstruction of Wigner functions of Fock
States in Cavities – ENS experiment
1,0
0,5
0,0
0,0
0,5
1,0
1,5
2,0
-0,5
-1,0
• MaxEnt scheme – up to 5 orders more reliable
than pattern-function or inverse Radon schemes,
requires just 3 distributions for rotated
quadratures,
The Wigner function of Fock states of cavity fields
from the experimental data obtained at the ENS,
Paris obtained from the measurement of a parity
operator [P.Bertet et al., PRL. 89, 200402 (2002)]
II. Decoherence due to Flow of
q-Information
• Q-Homogenization is the process in
which an open system interacts with
a reservoir. The original state of the
open system is transformed into the
state of reservoir particles.
• Theorem 1: Q-H is a contractive map
that can be realized only by a partialswap operation
• Theorem 2: Original information
encoded in the state
is transferred
into correlations between the system
and reservoir particles. This
information can be recovered iff
classical info about the order of
interaction is know.
At the output of the homogenizer
all qubits are in a vicinity of the
state .
Continuous version of discrete dynamical
semigroup
• Simulation of the discrete process of collision-like interaction between a
system qubit nad 25 000 reservoir
• Lindblad master equation
continuous interpolation of the
discrete process – one can determine
from the “first” principle decay time
and decoherence

 i  / 2  3 ,    1/(4T1 )  1  1   2  2  2  
t
[1/(2T2 )  1/(4T1 )]  3  3   
i /(2T1 )  1  2   2  1  i  3  i 3  
III. Quantum Random Walks
“Quantization” of classical discrete random walks (Markov)
Quadratic/exponential
improvement in
mixing/hitting properties
Implementation by
means of optical
multiports which “flip”
the coin.
Multidimensional QRW:
d
: d  direction 
Quantum coin: flipped
at every step
Direction of the next step
depends on |d>
Quantum Random Walks
• Recurrent probability:
• Coins (legend):
• Classical:
• Grover:
• Fourier:
• Analytic solution of recurrent
probability:
P ( d , n) 
• where:
1
( 2d ) 2 n 1
 (a ,..., a
1
recurrent paths ( a j ), ( a j ')
a1  a1 '
n 1
n
)( a1 ' ,..., an ' )
(a1 ,..., an )   (2d a j ,a j1  2)
j 1
IV. Universal quantum entangler


1









• No-go theorem:
2
• Best possible CP approximation – optimal UQE
V. Programmable Quantum Processors
t
• Quantum control of dynamics, e.g. C-NOT
data
program
• Quantum information distributors control via
input states of two ancillas (prorgam), e.g
assymetric cloners or Universal NOT gate
(specific processor)
c
• NO-GO Theorem (Nielsen & Chuang) –
Universal quantum processors implementing
arbitrary program encoded in program registers
and applied to data registers do not exist
• Probabilistic quantum processors measurement of program register realizes
arbitrary map on data register
• Deterministic processors – realize specific
classes maps
data
program
measurement
Universal Probabilistic Processor
- Quantum processor Udp
“universal”
Example: processor
- Data register rd,
D2
Data register
= qudit, program
1 register = 2 qudits
dim Hd = D
- Quantum programs Uk = program
register rp, dim Hp = N  D 2
• Nielsen & Chuang:
- N programs Þ N orthogonal states
- Universal quantum processors do not

• Hillery-Ziman-Buzek:
- Probabilistic implementation
- {Uk} operator basis,
U   kU k ,  k 
k
- program state
1
TrU kU
D
 U   k  k
k
•Error-correting schemes
- U(1) programmable rotations
U dp   U k   k  k ,
TrU k Ul   k  l   kl
D
D 1
k 1
 2ism 
 mn 
U k yes/no
U
measurement
exp  
 sn s
projective

N 

2
s 0
1 D
M  yes    no  I     ,  p    k
1 D 1
 2 ism  D k 1
 k   mn 
exp   1
 s sn
probability
of success: D 
N


P

s  0 success
2
D
VI. Ising Model
• Linear chain of qubits in a
magnetic field
the cyclic condition
• Interaction energy level
shifts
• Interaction Q entanglement
• In the interacting Universe
factorized states are more
exotic than entangled states
(N=2n+1)-qubit Ising X-state
• Level crossing
2n degeneracy such that at
additional
also at
at
;
this means
degeneracy
• X-state
• sum over all states with even number of
•
is the shortest distance between
(u is even)
Super Entanglement
• Bounds on shared entanglement
• Ising model provides miraculously entangled states
www.quniverse.sk