Download Quantum Process Tomography: Theory and Experiment

DECOHERENCE AND
QUANTUM PROCESS
TOMOGRAPHY
JUAN PABLO PAZ
Quantum Foundations and Information @ Buenos Aires
QUFIBA: http://www.qufiba.df.uba.ar
Departamento de Fisica Juan José Giambiagi, FCEyN, UBA, Argentina
PARATY SCHOOL ON
QUANTUM INFORMATION
August 2011
PLAN
Lecture 1: Decoherence and the quantum origin of the classical
world. Evolution of quantum open systems. Quantum Brownian
motion as a paradigm. Derivation of master equation for QBM.
How to use it to characterize decoherence. Decoherence timescales.
Pointer states.
Lecture 2: Decoherence and disentanglement. Decoherence in
quantum information processing. How to fight against
decoherence? How to characterize decoherence? Quantum process
tomography (QPT).
Lecture 3: New methods for Quantum Process Tomography.
“Selective and efficient QPT”. Theory and experimental
implementation with single photons.
HOW TO FIGHT AGAINST
DECOHERENCE
This
talk
QUANTUM PROCESS TOMOGRAPHY (QPT):
• What is QPT? Why is it hard? Standard QPT.
• A new method: Selective and Efficient Quantum Process Tomography (SEQPT).
• First experiments @ Buenos Aires
Para ver esta película, debe
disponer de QuickTime™ y de
un descompresor .
HOWIS
TOQUANTUM
FIGHT AGAINST
DECOHERENCE
WHAT
PROCESS
TOMOGRAPHY?
in
out  in 
out
A QUANTUM PROCESS IS A LINEAR MAP
(PRESERVING HERMITICITY, TRACE AND
POSITIVITY)
ANY LINEAR MAP IS DEFINED BY ITS „CHIMATRIX‟
out in    mn E n in E m
P0  I, P1  X, P2 Y, P3  Z
E m  Pm1  Pm2  ........ Pmn

nm
• MAP PRESERVES TRACE
E 0  I1  I2  ........ In
TrE E   D
mn  nm*
• MAP IS HERMITIAN


E
mn m E n  I
m
n
mn
mn


MAP  IS COMPLETELY POSITIVE (CP)


MATRIX
 mn
IS POSITIVE
HOW
FIGHT AGAINST
DECOHERENCE
WHY
ISTO
QUANTUM
PROCESS
TOMOGRAPHY HARD?
out  in    mn E n in E m ,

nm


mn
Em En  I
mn
•1) THERE ARE EXPONENTIALLY MANY COEFFICIENTS
 mn
(i.e. There are D2  D2 of them where D  2n)
• 2) TO FIND OUT ANY ONE OF THEM WE NEED EXPONENTIAL RESOURCES


QUANTUM PROCESS TOMOGRAPHY
STANDARD
(SQPT)
Chapter 10, Nielsen & Chuang’s book
Pik  Trk i 

• EXPERIMENTALLY DETERMINE “TRANSITION PROBABILITIES”
Pik    mn Tr k E n i E m
nm


• FIND
 mn INVERTING (HUGE) LINEAR SYSTEM.
“STANDARD QUANTUM PROCESS TOMOGRAPHY”
(NIELSEN & CHUANG, CHAPTER 10)
HOW TO FIGHT
AGAINST
DECOHERENCE
QUANTUM
PROCESS
TOMOGRAPHY
IS HARD!
QUANTUM RESOURCES (qbits, operations, state preparations, measurements, etc):
• prepare
D2 initial states and detect D2 final states
• repeat experiments (exponentially) many times to geta fixed precision in  mn
CLASSICAL RESOURCES (data processing in classical computers):

large (linear) system
• invert an exponentially
• THIS IS THE METHOD UDED IN PRACTICE NOWADAYS

Average fidelity: 0.7
X
X
 
X
Realization of the quantum Tofoli gate with
trapped ions” T Monz, K. Kim, W. Hänsel, M.
Riebe, A. S. Villar, P. Schindler, M. Chwalla, M.
Hennrich, and R. Blatt, Physical Review
Letters 102, 040501 (2009)
X
64x64 matrix. Obtained after inverting a
4096x4096 linear system formed with all the
probabilities measured after perfirnubg 4096
experiments (prepare each of 64 independent
states and measure each of 64 independent
transition probabilities.

Average fidelity: 0.67
Measured Chi-matrix shows the same
“fingerprint” of the ideal one (Tofoli)
HOW
FIGHT(continuation)
AGAINST DECOHERENCE
QPT
IS TO
HARD…
• STANDARD QUANTUM PROCESS (NIELSEN & CHUANG) IS EXPONENTIALLY
HARD EVEN TO ACHIEVE PARTIAL CHARACTERIZATION!!
ARE THERE OTHER METHODS? DCQP („DIRECT CARACTERIZATION OF A
QUANTUM PROCESS). D. Lidar and M. Mohseni, Phys. Rev. A 77, 032322 (2008)
• DIAGONAL MATRIX ELEMENTS nn ARE SURVIVAL PROBABILITIES OF SYSTEM
PLUS ANCILLA (A VERY EXPENSIVE RESOURCE!)
 Bell



 Bell

Bell
1 D

j j
D j 1
00 Bell    I Bell  Bell  Bell

• BUT OFF DIAGONAL ELEMENTS ARE STILL EXPONENTIALLY HARD TO CALCULATE


• NEED AN EXPENSIVE RESOURCE: CLEAN ANCILLA THAT INTERACTS WITH OUR
SYSTEM
HOW
TO FIGHT AGAINST
DECOHERENCE
AN
ALTERNATIVE
APPROACH
FOR QPT
• 1) SELECT A COEFFICIENT  mn (OR A SET OF THEM)
• 2) DIRECTLY MEASURE THEM WITHOUT DOING FULL QUANTUM PROCESS
TOMOGRAPHY

QUANTUM AND CLASSICAL RESOURCES Poly(Log(D))
METHOD BASED ON INTERESTING PROPERTY OF CHI-MATRIX
ALL MATRIX ELEMENTS ARE SURVIVAL PROBABILITIES OF A CHANNEL
Fmn  
d
 E m   E n  
1
Fmn  
D mn  mn 
D  1
Estimate Fmn  
• A simple consequence
following identity
of the
Estimate  mn
1
 d   A   B   DD 1 TrAB  TrATrB
THE IMPORTANCE OF BEING SELECTIVE…
QUESTION: How close are we approaching a “target” operation?
(T )
UT UT   mn
En  Em
nm
Example: C-NOT (chi-matrix has only 16 non-vanishing elements)
X
1
1
U

0
0

I

1
1

X

I

Z

I



I  Z   X
 T
2
2
AVERAGE FIDELITY PROVIDES A GOOD WAY TO QUANTIFY THIS

F
d
 UT  UT  

)
F   (T
mn Fnm
nm

1 
(T )
F
D mn nm 1
D 1  nm

MORAL: DON‟T NEED THE FULL MATRIX TO COMPUTE F (ONLY 16 ELEMENTS!,
INDEPENDENT
OF D!!)

HOW
TO FIGHTMAKE
AGAINST
TWO
PROBLEMS
NEWDECOHERENCE
METHOD LOOK IMPOSSIBLE!
PERFORMING
Fmn  
d
Fmn  
1
D mn  mn 
D  1
 E m   E n  
d
IMPLEMENTING E m   E n
IN THE LABORATORY???
IN THE LABORATORY???


TWO PROBLEMS!

1) How to perform the integral over the
entire Hilbert space?
2) How to apply a non physical (non CP)
map?
THIS TALK: TWO SOLUTIONS!
HOWTO
TOINTEGRATE
FIGHT AGAINST
DECOHERENCE
HOW
IN HILBERT
SPACE?
• USE 2-DESIGNS!
• A SET OF STATES (S) IS A 2-DESIGN IF AND ONLY IF
1
 d   A   B   # S   j A  j
 J S
j B j
• 2-designs are powerful tools!! USEFUL RESULTS
a)
b)
c)
2-DESIGNS EXIST!
THEY HAVE AT LEAST D2 STATES
STATES OF (D+1) MUTUALLY UNBIASED BASIS FORM A 2-DESIGN
d)
EFFICIENT ALGORITHMS TO GENERATE 2-DESIGNS EXIST

HOW TO FIGHT
AGAINST DECOHERENCE
INTERLUDE
ON 2-DESIGNS
• IS THE EXISTENCE OF 2-DESIGN A SURPRISE?
1
1
1
 dx f x   dx a  bx  cx  2  f x1  f x 2 
0
0
2
1
1
x1  
 S  x1, x 2  2  design
2
12
2
• 2-DESIGNS FOR SPIN 1/2: ENABLE TO COMPUTE AVERAGES OF PRODUCTS
OF TWO EXPECTATION VALUES (integrals of functions that depend upon TWO
bras and TWO kets)

1
1
 d   A   B   6 x Ax x Bx  6 x Ax x Bx
1
1
 y Ay y By  y Ay y By
6
6
1
1
 z Az z Bz  z Az z Bz
6
6
HOWTO
TO
FIGHTAAGAINST
DECOHERENCE
HOW
APPLY
NON CP MAP?
TRANSFORM IT INTO THE DIFFERENCE BETWEEN CP MAPS…


1
Fmn  
 j  Em  j  j En  j

DD 1 j



1
F mn  
 j  E m  E n   j  j E m  E n   j

DD 1 j

1
ReFmn   F  mn   F  mn 
2
OPERATOR BASI
Em
CAN WE PREPARE
THOSE STATES?
2-DESIGN
SPLIT THE OPERATOR BASE INTO (D+1) COMMUTING SUBSETS
(each set contains the identity and D-1 commuting operators)


EACH COMMUTING SET OF E m OPERATORS DEFINES A BASIS
ALL SUCH (D+1) BASES ARE MUTUALLY UNBIASEDrs)
j
HOWTO
TO
FIGHTAAGAINST
DECOHERENCE
HOW
APPLY
NON CP MAP?
USE 2-DESIGN DEFINED BY THE (D+1) MUBs ASSOCIATED WITH THE SPLIT
OF THE OPERATOR BASIS E m
b 
b 
 j  bk ; b 1,...,D 1; k 1,...,D E m  k   k'
E m  E n  bk


 bk'  b k''
EFFICIENT PROCCEDURE FOR PREPARING SUCH STATES EXIST!


HOW
TO FIGHT
AGAINST DECOHERENCE
NEW
METHOD:
SELECTIVE
AND EFFICIENT Q.P.T.
FIRST EFFICIENT METHOD TO DETERMINE ANY ELEMENT OF CHI MATRIX OF
A QUANTUM PROCESS
Poly(Log(D)) QUANTUM GATES REQUIRED
Poly(Log(D)) CLASSICAL POST-PROCESSING REQUIRED
NO ANCILLARY RESOURCES (CLEAN QUBITS) ARE REQUIRED
j

E m  E n  j

M
 1 
1

Fmn    x i  O 
M i1
 M 


NO x  0
 j ?: 
YES  x 1
PHOTONIC IMPLEMENTATION: SECOND EXPERIMENT IN
OUR LAB IN BUENOS AIRES!
FULLY CHARACTERIZING A QUANTUM CHANNEL AFFECTING TWO QUBITS
ENCODED IN A SINGLE (HERALDED) PHOTON (2010 unpublished)
STATE
PREPARATION
j
E m  E n  j
STATE
DETECTION
j
PHOTONIC IMPLEMENTATION: SECOND EXPERIMENT IN
OUR LAB IN BUENOS AIRES!
FULL QUANTUM PROCESS TOMOGRAPHY
OUR METHOD GIVES PERFECT AGREEMENT WITH STANDARD QPT (alla NIELSEN
AND CHUANG) (2010 unpublished)
QuickTime™ and a
mpeg4 decompressor
are needed to see this picture.
FULL QUANTUM PROCESS TOMOGRAPHY
IMPORTANT: EACH MATRIX ELEMENT IS ESTIMATED BY SAMPLING OVER THE 2DESIGN. PRECISION INCREASES WITH SAMPLE SIZE (2010 unpublished)
QuickTime™ and a
mpeg4 decompressor
are needed to see this picture.
FULL QUANTUM PROCESS TOMOGRAPHY
IMPORTANT: EACH MATRIX ELEMENT IS ESTIMATED BY SAMPLING OVER THE 2DESIGN. PRECISION INCREASES WITH SAMPLE SIZE (2010 unpublished)
QuickTime™ and a
mpeg4 decompressor
are needed to see this picture.
POWER OF THE METHOD: SAMPLING & PARTIAL QPT
COMPUTE FIDELITY OF A QUANTUM GATE WITHOUT DOING FULL QUANTUM
PROCESS TOMOGRAPHY! (PRL 2011 in press)
(T )
UT UT   mn
En  Em
nm
F

X

(T )
d



U

U



 mn Fnm

T
T
nm
Z
UT  0 0  Z  1 1  X 

UT  I  I

)
(T
00  1
1
1
I  Z   Z  I  Z   X
2
2
HOWVERSION
TO FIGHTOF
AGAINST
DECOHERENCE
FIRST
SELECTIVE
AND EFFICIENT QPT
GENERAL RESULT (DIAGONAL AND OFF DIAGONAL COEFFICIENTS)
1
D mn  mn  
D  1
d
 E m    E n 
PROCEDURE TO MEASURE OFF-DIAGONAL COEFFICIENTS

AN (EXPENSIVE!) EXTRA RESOURCE: A CLEAN QUBIT
POLARIZATION OF EXTRA QUBIT (CONDITIONED ON SURVIVAL OF THE
STATE) REVEALS REAL AND IMAGINARY PARTS OF  mn
Method inspired in a known quantum algorithm
AN ALGORITHM TO MEASURE THE EXPECTATION VALUE OF A UNITARY OPERATOR U
USING AN EXTRA (ANCILLARY) QUBIT
“THE SCATTERING ALGORITHM”
THE BASIC INGREDIENT FOR DQC1 MODEL OF QUANTUM COMPUTATION
1
0  1

2
 y  ImTrU 
U


 x  ReTrU 
Polarization of ancillary qubit reveals the expectation value of U!



HOW TO FIGHT AGAINST
DECOHERENCE
SUMMARY
FULL QPT IS ALWAYS HARD. STANDARD METHODS FOR PARTIAL QPT ARE
ALSO EXPONENTIALLY HARD
• THERE IS AN ALTERNATIVE METHOD FOR EFFICIENT AND SELECTIVE PARTIAL
QUANTUM PROCESS TOMOGRAPHY
• IT INVOLVES ESTIMATION OF „SURVIVAL PROBABILITIES‟ OF A SET OF STATES
FORMING A 2-DESIGN (VERY USEFUL RESOURCE!)
• THE METHOD HAS BEEN EXPERIMENTALLY IMPLEMENTED @ BUENOS AIRES
Pa ra ve r e sta pe lícu la , de be
disp on er de Qu ic kTime ™ y de
un d es co mpre so r .
“Ancilla-less, selective and efficient QPT”, C. Schmiegelow,
A. Bendersky, M. Larotonda, J.P. Paz, Phys. Rev. Lett.
(2011) in press; arXiv:1105.4815