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Production and detection of few-qubit
entanglement in circuit-QED processors
Leo DiCarlo
(in place of Rob Schoelkopf)
Department of Applied Physics, Yale University
Portland Convention Center
Sunday, March 14
8:30 a.m. - 12:30 p.m.
Outline
• What is a quantum processor, and why must it produce
(and why must we detect) highly-entangled qubit states?
• What is quantum entanglement?
• How to detect it?
the complete way: quantum state tomography
the scalable way: entanglement witnesses
• One specific example:
production and detection of two-qubit entanglement in
a circuit-QED processor
two-qubit conditional phase gate
joint qubit readout
• Going beyond two qubits
• Outlook
Defining a quantum processor
Related MM 2010 talks:
W6.00003 & T29.00012
What is a quantum processor?
Quantum processor: programmable computing device using
quantum superposition and entanglement in a qubit register.
The program it executes is compiled into a sequence of oneand two- (perhaps more-) qubit gates, following an algorithm.
2009 model
2 qubits
quantum processors based on circuit QED
2010 model
4 qubits
Anatomy of quantum algorithm
Register
qubits
Us
Uf
Ua
M
Ancilla
qubits
initialize
create
superposition
encode function
in a unitary
involves
entanglement
between qubits
analyze the
function
measure
involves
disentangling
the qubits
Maintain quantum coherence
1) Start in superposition: all values at once!
2) Build complex transformation out of one- and two-qubit gates
3) Make the answer we seek result in an eigenstate of measurement
Example: The Grover quantum search
1, x  x0
f ( x)  
0, x  x0
Problem: Find the unknown root x0 of f
Given: a quantum oracle that implements the unitary
U f x  (1) f ( x )0 x
Quantum circuit diagram of the algorithm
0
Ry /2
Ry /2
ij
0
Ry /2
00
Ry /2
quantum
oracle
Ry /2
Ry /2
Quantum debugging
Half-way along the algorithm, the two-qubit state ideally maximally
entangled!
The quality of a quantum processor relies on producing nearperfect multi-qubit entanglement at intermediate steps of the
computation.
A quantum computer engineer needs to detect this entanglement
as a way to benchmark or debug the processor.
oracle
Ry /2
0
b
0
 /2
Ry
Ry /2
c
10
d
e
Ry /2
00
 /2
Ry
f
 /2
Ry
g
 ideal
1

00  11

2

But what is quantum entanglement?
`Entanglement is simply Schrodinger’s name for superposition
in a multi-particle system.`
GHZ,
Physics Today 1993
Wavefunction description of pure two-qubit states
for N=2 qubits:
  c00 00  c01 01  c10 10  c11 11
2
2 complex numbers -
• normalization
  1
• irrelevant global phase
A pure 2-qubit pure state is fully described by 6 real #s
When are two qubits entangled?
Two qubits are entangled when their joint wavefunction cannot
be separated into a product of individual qubit wavefunctions
  1   2
vs
   1a   2a   1b   2b
Some common terms:
Unentangled = separable = product
state
Entangled = non-separable = non-product
state
When are two qubits entangled?
Entangled?
State
  1  1  11
1
   01  10
2
no

The singlet
1
   00  01  10  11 
2
1
1

 0  1   0  1 
2
2
1
   00  01  10  11 
2
yes
no
yes
Quantifying entanglement
Two qubits in a pure state
  c00 00  c01 01  c10 10  c11 11
are entangled if they have nonzero concurrence C
C ( )  2 c00c11  c01c10
  11
1
   01  10 
2
1
   00  01  10  11 
2
1
   00  01  10  11 
2
C 0
C 1
C 0
C 1
Quantifying entanglement – pure states
The concurrence is an entanglement monotone:
0  C ( )  2 c00c11  c01c10  1
If C ( a )  C ( b ) , we say state a is more entangled than state b
If C ( a )  1, we say state a is maximally entangled
Example:
1
 01  10  is maximally entangled.
The Bell state  
2
1
1
1
The state  
00 
10 
11 , with C  2 / 3 , is
3
3
3
entangled, but less entangled than a Bell state.
Reality check #1 : quantum states never pure!
Density-matrix description of mixed states
  
for a pure state
   pi  i  i
for a mixed state
i
pi  [0,1],  pi  1
i
Dim[ ]  2 N  2 N
 22 N complex #s -
  †
Tr[ ]  1
Hermitian
Unity trace
Fully describing a 2-qubit mixed state requires 15 real #s
Warning:
The decomposition
   pi  i  i
is generally not unique!
i
Example:
1
p1  ,
2
1
p2  ,
2
City-scape (Manhattan plot)
1  00
 2  11
1
0
-1
00
01
10
11
and
1
p1  ,
2
1
p2  ,
2
Im( )
Re( )
1
1 
 00  11 
2
1
2 
 00  11 
2
00 01
10
11
Give the same 
Quantifying entanglement – mixed states
The concurrence of a mixed state is given by


C     min   pi C  i   ,
 i

where the minimization is over all possible decompositions of

Can show:
C (  )  max 0, 1  2  3  4 
The i are the eigenvalues of the matrix
in decreasing order, and
*

  YY  YY
Yes, it’s completely non-intuitive!
A very non-linear function of 
Hill and Wootters, PRL (2007)
Wootters, PRL (2008)
Horodecki4, RMP (2009)
Getting

: Two-qubit state tomography
• A quantum debugging tool
• Knowing all there is to know about the 2-qubit state
• Necessary to extract C
• Drawback: it is not scalable diagnostic tool!
• Review: the N=1 qubit case
• Can we generalize the Bloch vector to N>1?
• Answer: the Pauli set
• Extracting usual metrics from the Pauli set
Geometric visualization for N=1: The Bloch sphere

1
j j

2 j{i , x , y , z}
z
I 1
  ( X , Y , Z )  ( X ,Y , Z )
2 2
v Bloch
v Bloch
Bloch vector
(NMR)
y
x
v Bloch  1
pure state
v Bloch  1
mixed state
Is there a similarly intuitive description for N=2 qubits?
State tomography of qubit decay
Steffen et al., PRL (2006)
Related MM talks:
Tomography of qutrits:
Z26.00013 (phase qubits)
Z26.00012 (transmon qubits)
Generalizing the Bloch vector: The Pauli set

1
1 2
1 2




j k
j k
4 j ,k{i , x , y , z}
The Pauli set
P
= the set of expectation values of the 16
2-qubit Pauli operators.
 gives a full description of the 2-qubit state
 is the extension of the Bloch vector to 2 qubits
 generalizes to higher N
One of them,
II  1 always
The two-qubit Pauli set can be divided into three sections:
• Polarization of Qubit 1
• Polarization of Qubit 2
P1   XI , YI , ZI
P2   IX , IY ,

IZ 
• Two-qubit correlations
P12   XX , XY , XZ , YX , YY , YZ , ZX , ZY , ZZ

Visualizing N=2 states: product states
  00
Re( )
1
0
P1
P2
P12
-1
00
01
10
11
10 11
00 01
1
   0  1  0  1 
2
1
0
-1
00
01
10
11
10 11
00 01
Im( )
Visualizing N=2 states: Bell states
1
 
 00  11 
2
Re( )
1
0
-1
00
01
10
11
 
10 11
00 01
1
 00  01  10  11 
2
1
0
-1
00
01
10
11
10 11
00 01
Im( )
Extracting usual metrics from the Pauli Set
State purity:
1
Tr[  ]  P  P
4
2
Fidelity to a target state
T
F  T  T
:
1
 P  PT
4
Concurrence:
P12  P12  1
C ( ) 
2
C0
Warning:
for pure states only
C 1
Measuring the Pauli set with a joint qubit readout
Related MM 2010 talks:
W6.00003 & T29.00012
Filipp et al., PRL (2009)
DiCarlo et al., Nature (2009)
Chow et al., arXiv 0908.1955
A two-qubit quantum processor
flux bias lines
fR
1 ns resolution
VR
DC - 2 GHz
fC
VH
fL
fC
T = 13 mK
VL
cavity: “entanglement bus,”
driver, & detector
transmon qubits
Two-qubit joint readout via cavity
 1
1
fC
VH
Cavity transmission
fC
11
10
01
 1
0
00
2 R
2 L
2 L  2 R
“Strong dispersive cQED”
Schuster, Houck et al., Nature (2007)
Frequency
Qubit-state dependent cavity resonance
Prepare 00
and measure
01
10
11
Chow et al., arXiv 0908.1955
Two qubit joint readout via cavity
fC
Cavity transmission
fC
11
10
01
00
VH
Measure cavity transmission:
“Are qubits both in their
ground state?”
M  00 00
 ZI  IZ  ZZ
Frequency
Direct access to qubit-qubit correlations with a single measurement channel!
Direct access to qubit-qubit correlations
How to reconstruct the two-qubit state from an ensemble measurement of
the form
VH  1 ZI  2 IZ  12 ZZ
?
Answer: Combine joint readout with one-qubit pre-rotations
Example: How to extract
YZ
Rx /2
Rx0,
Apply
Apply
  /2
, then measure:
x

 /2 +
&
, then measure:
x
x
R
R
R
Joint Dispersive
Readout
1 ZI   2 IZ
 12 ZZ
1 YI  2 IZ  12 YZ
1 YI  2 IZ  12 YZ
212 YZ
It is possible to acquire correlation info. with one measurement channel!
All Pauli set components are obtained by linear operations on raw data.
First demonstration of 2Q entanglement in SC
qubits:
Steffen et al., Science (2006)
C  0.55
Producing entanglement with the circuit QED processor
Related MM 2010 talks:
W6.00003 & T29.00012
Spectroscopy of qubit2-cavity system
right qubit
Qubit-qubit swap interaction
Majer et al., Nature (2007)
left qubit
Cavity-qubit interaction
Vacuum Rabi splitting
Wallraff et al., Nature (2004)
cavity
VR
One-qubit gates: X and Y rotations
fR
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
Q
VR
sin(2 f q t )
One-qubit gates: X and Y rotations
fR
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
I
VR
cos(2 f q t )
One-qubit gates: X and Y rotations
fR
fL
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
Q
VR
sin(2 f q t )
One-qubit gates: X and Y rotations
fL
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
I
VR
J. Chow et al., PRL (2009)
Fidelity = 99%
cos(2 f q t )
Two-qubit gate: turn on interactions
VR
Use control lines to push
qubits near a resonance:
Conditional
phase gate
A controlled  z   z
interaction, a la NMR
cavity
VR
Two-excitation manifold of system
02
Two-excitation
manifold
11
02
11
Two-excitation
manifold
• Avoided crossing (160 MHz)
11  02
20
Transmon “qubits” have multiple levels…
Strauch et al. PRL (2003): proposed using interactions with higher levels for
computation in phase qubits
Adiabatic phase gate
tf
02
a  2   f a (t )dt
11
t0
2-excitation
manifold

11  ei11 11
tf
11  10  01  2   (t )dt
t0
01
1-excitation
manifold
10
01  e
10  e
i01
i10
01
10
Implementing C-Phase with 1 fancy pulse
00
01
1 0
 0 ei01
U 
0 0

0 0
10
0
0
ei10
0
11
30 ns
 00
 01

 10
i11  11
e 
0
0
0
00 01 10 11
Adjust timing of flux pulse so that
only quantum amplitude of 11
acquires a minus sign:
1
0
U 
0

0
0
1
0
0
0 0  00
0 0  01

1 0  10

0 1 11
11
How to create a Bell state using C-Phase
  0 0
Ry /2
rotation on each qubit yields a maximal superposition:
 
1
1
0

1

0

1

00  01  10  11





2
2
Apply C-Phase entangler:
 1 0 0 0 


0

1
0
0

   1  00  01  10  11
 0 0 1 0 
2


 0 0 0 1
No longer a product state!


How to create a Bell state using C-Phase
 1

0
0

0
0
1
0
0
0
0
1
0
0

0
1
   00  01  10  11 
0
2

1
RY ( / 2) rotation on LEFT qubit yields:
1
Bell 
 10  01
2

Two-qubit entanglement experiment
Ry /2
0
Ry /2
10

Ry /2
0
Re(  )
Ideally:
wavefunction
1
 
 00  11 
2
density matrix
  
1
  00 00  00 11  11 00  11 11 
2
Expt’l state tomography
Entanglement on demand
0
Ry /2
Ry /2
ij
0
Ry /2
Re(  )
Bell state
Fidelity
Concurrence
00  11
91%
88%
00  11
94%
94%
01  10
90%
86%
01  10
87%
81%
Switching to the Pauli set
Related MM 2010 talks:
T29.00012 & W6.00003
Experimental N=2 Pauli sets
  10
1
   00  01  10  11
2
1
 
 00  11
2


1
   00  01  10  11
2

Pauli set movies
Look at evolutions of separable and entangled states
• a test for systematic errors in tomography, such
as offsets and amplitude errors in Pauli bars
Ry
0
0
0
Ry /2
Ry
10
0
Ry /2
~98% visibility for separable states,
~92% visibility for entangled states
Working around Concurrence
Related MM 2010 talks:
W6.00003 &Y26.00013
Drawbacks of Concurrence
 Requires full state tomography (knowing  )
 Is a very non-linear function of 
 It is difficult to propagate experimental errors in tomography
to error (bias and noise) in C
Can we characterize entanglement without reliance on C ?
Can we place lower bounds on
state tomography?
C
without performing full
Witnessing entanglement with a subset of the Pauli set
An entanglement witness is a unity-trace observable
a positive expectation value for all product states.
W 0
W 0


W
with
state is entangled, guaranteed.
witness simply doesn’t know
Witness also gives a lower bound on
C:
2 W  C
An optimal witness of an entangled state gives the strictest
lower bound on C
Witnesses require only a subset of the Pauli set!
Example:
W
opt
1
  II  XX  YY  ZZ 
4
Is the optimal
witness
for the singlet
Witnessing entanglement
Entanglement is witnessed (by some witness) at all angles in
entangled-state movie!
0
Ry /2
Ry
10
0
Ry /2
Witnessing entanglement
Control: entanglement is not witnessed by any witness
in the separable movie!
0
0
Ry
Bell inequalities: another form of entanglement witness
z’
z
CHSH operator = entanglement witness
x’
CHSH  XX   XZ '  ZX   ZZ '

x
Clauser, Horne,
Shimony & Holt (1969)
CHSH  XX   XZ '  ZX   ZZ '
Bell state
Separable bound:
CHSH  2
state is clearly
highly entangled!
not test of
hidden variables…
(loopholes abound)
Also UCSB group,
closing detection loophole, Ansmann et al., Nature (2009)
2.61  0.04
Beyond two qubits
Related MM 2010 talks:
W6.00003, Y26.00013 (with transmon qubits)
Z26.00003 (with phase qubits)
DiCarlo et al., (2010)
Reed et al., (2010)
The density matrix for N=3
1
1 2 3
1 2 3






j k l
j k  l
8 j ,k ,l{i , x , y , z}
Knowing the three-qubit state = expectation values of 63 Pauli operators
P1   XII , YII , ZII
P2   IXI , IYI , IZI
P3   IIX , IIY , IIZ



Polarization of qubit 1
Polarization of qubit 2
Polarization of qubit 3
2-qubit correlations
P12   XXI , XYI , XZI , YXI , YYI , YZI , ZXI , ZYI , ZZI
P13   XIX , XIY , XIZ , YIX , YIY , YIZ , ZIX , ZIY , ZIZ
P23   IXX , IXY , IXZ , IYX , IYY , IYZ , IZX , IZY ,


IZZ 
3-qubit correlations
P123   XXX , XXY , XXZ ,
...
, ZZX , ZZY , ZZZ

Example Pauli set for N=3: product state
  111
P1 P2 P3
P12
P13
P123
P23
Re( )
1
0
-1
000
111
111
000
Example Pauli set for N=3: 3-qubit entangled state
The GHZ state
 
Greenberger
Horne
Zeilinger
P1 P2 P3
P12
P13
1
 000  111 
2
P123
P23
1
Re( )
0
-1
000
111
111
000
Upgrading the processor: more qubits
Q2
Q1
Q3
Q4
 We have recently extended
C-Phase gates and joint qubit readout to
produce and detect 3-qubit entanglement.
DiCarlo et al., (2010)
Reed et al., (2010)
Invitation to MM talks:
W6.00003
(w/ transmon qubits, and joint readout)
Also,
Z26.00003 (w/ phase qubits, and
individual qubit readouts)
Joint 3-Qubit Readout
Cavity transmission
Same concept: Use cavity transmission to learn a collective property, now
of the three qubits
111
000
Frequency
Measure cavity transmission:
“Are all 3 qubits in the ground
state?”
M  000 000
 ZII  IZI  IIZ  ZZI  ZIZ  IZZ  ZZZ
3-Qubit state tomography
The trick still works!
Combining joint readout with one-qubit “analysis” gives access to all 3-qubit
Pauli operators, only more rotations are necessary.
Rx0,
Joint
Readout
Rx0,
M
Rx0,
000 000
Example: extract ZZZ

Rx ( ) on Q1 and Q2: 
Rx ( ) on Q1 and Q3: 
Rx ( ) on Q2 and Q3: 
no pre-rotation:
ZII
ZII
ZII
ZII




IZI
IZI
IZI
IZI




IIZ
IIZ
IIZ
IIZ




ZZI
ZZI
ZZI
ZZI




ZIZ
ZIZ
ZIZ
ZIZ




IZZ
IZZ
IZZ
IZZ




ZZZ
ZZZ
ZZZ
ZZZ
4 ZZZ
Outlook
Is full state tomography scalable?
Number of qubits
N
1
2
3
4
5
6
7
8
9
10
Parameters in density matrix
22 N  1
Steffen et al., PRL (2007)
3
M. Metcalfe, Ph.D. Thesis (2008)
15
Steffen et al., Science (2006)
63
Filipp et al., PRL (2009)
255
DiCarlo et al., Nature (2009)
1,023
Chow et al., arXiv 0908.1955
4,095
Neeley et al., ????
16,383
DiCarlo et al., (2010)
65,535
262,143 In ion traps:
Haffner, Nature (2005)
1,048,575
How far will we push full state tomography?
Up to N=8 qubits
10 hours of ensemble averaging!
Summary in pictures
C0
  1   2
C 1
vs
   1a   2a   1b   2b
C-phase gate
Joint Readout
Few-qubit processors
bbased on circuit QED
Generation and
detection of
entanglement
Related MM 2010 talks by Yale cQED team
J. M. Chow et al.
Quantifying entanglement with a joint readout of superconducting qubits
T29.00012: Wednesday 03/17, 4:42 PM, Room C123
Lev S. Bishop et al.
Latching behavior of the driven damped Jaynes-Cummings model in cQED
T29.00012: Wednesday 03/17, 5:06 PM, Room C123
M. D. Reed et al.
Eliminating the Purcell effect in cQED
NEW TITLE: ????
Y26.00013: Friday 03/19, 10:24 AM, Room D136
L. DiCarlo et al.
Realization of Simple Quantum Algorithms with Circuit QED
W6.00003: Thursday 03/18, 12:27 PM, Room 253
Acknowledgements
PI’s: Robert Schoelkopf, Michel Devoret, Steven Girvin
Expt:
Jerry Chow
Matthew Reed
Blake Johnson
Luyan Sun
Joseph Schreier
Andrew Houck
David Schuster
Johannes Majer
Luigi Frunzio
Theory:
Jay Gambetta
Lev Bishop
Andreas Nunnenkamp
Jens Koch
Alexandre Blais
Funding sources:
Circuit QED team members ‘09
Lev
Bishop
Matt
Rob
Leo
Reed
DiCarlo Hanhee Schoelkopf
Paik
Steve
Girvin
Jens
Koch Jerry
Chow
Andreas
Fragner
Blake
Johnson
Jay
Gambetta
Eran
Ginossar
Adam
Sears
Luyan
Sun
David
Schuster
Luigi
Frunzio
Andreas
Nunnenkamp