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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 148:30 a.m. - 12:30 p.m.
Outline
• What is quantum entanglement?
• What is a quantum processor, and why must it produce (and why must we detect) highly-entangled qubit states?
• Going beyond two qubits
• Outlook
• How to detect it?the complete way: quantum state tomographythe scalable way: entanglement witnesses
• One specific example: production and detection of two-qubit entanglement in a circuit-QED processor
two-qubit conditional phase gatejoint qubit readout
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 one- and two- (perhaps more-) qubit gates, following an algorithm.
quantum processors based on circuit QED
2009 model2 qubits
2010 model4 qubits
Anatomy of quantum algorithm
Registerqubits
Ancillaqubits
M
createsuperposition
encode functionin a unitary
analyze thefunction
initializemeasure
involves entanglementbetween 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
involves disentanglingthe qubits
UaUfUs
Example: The Grover quantum search
/2yR /2
yR
/2yR
0
0
ij/2
yR
/2yR
/2yR
Quantum circuit diagram of the algorithm
00
quantumoracle
0
0
1,( )
0,
xf x
x x
x
Problem: Find the unknown root of f
Given: a quantum oracle that implements the unitary( ) 0( 1) f x
fU x x
0x
ideal
111
200
Quantum debugging
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00
oracle
b c d f
e
g
Half-way along the algorithm, the two-qubit state ideally maximallyentangled!
The quality of a quantum processor relies on producing near-perfect 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.
But what is quantum entanglement?
`Entanglement is simply Schrodinger’s name for superpositionin a multi-particle system.`
GHZ, Physics Today 1993
for N=2 qubits:
A pure 2-qubit pure state is fully described by 6 real #s
00 01 10 110 10 0 10 11c c c c
Wavefunction description of pure two-qubit states
• normalization
• irrelevant global phase
1 22 complex numbers -
When are two qubits entangled?
21
Two qubits are entangled when their joint wavefunction cannotbe separated into a product of individual qubit wavefunctions
Some common terms:
Entangled = non-separable = non-product state
Unentangled = separable = product state
2 21 1a b ba vs
1 11 1
0 11
21 0
0 00 112
11 10
0 00 112
11 10
When are two qubits entangled?
0 11
20 1
1
2
State Entangled?
no
yes
no
yes
The singlet
0 11
21 0
0 00 112
11 10
0 00 112
11 10
Quantifying entanglement
00 11 01 102)(C c c c c
00 01 10 110 10 0 10 11c c c c
Two qubits in a pure state
are entangled if they have nonzero concurrence
11 0C
C
0C
1C
1C
Quantifying entanglement – pure states
00 11 01 100 ( 2 1)C c c c c
The concurrence is an entanglement monotone:
If , we say state a is more entangled than state b ( () )a bC C
If , we say state a is maximally entangled)( 1aC
Example:
The Bell state is maximally entangled.
The state , with , is
entangled, but less entangled than a Bell state.
0 11
21 0
1 1 1
3 30 0 1
30 1 1 2 / 3C
Reality check #1 : quantum states never pure!
Tr[ ] 1
Density-matrix description of mixed states
†
Hermitian
Unity trace
for a pure state
i i ii
p [0,1], 1i i
i
p p for a mixed state
Fully describing a 2-qubit mixed state requires 15 real #s
2
Dim[ ] 2 2
2
N N
N
complex #s -
The decomposition is generally not unique!
Warning:
i i ii
p
Example:
1 1
1, 00
2p
2 2
1, 11
2p
1 1
1 1, 0 10
21
2p
2 2
1 1, 0 1
20 1
2p
1
0
-100
0110
11 00 01 10 11
Re( ) Im( )
and
Give the same
City-scape (Manhattan plot)
Quantifying entanglement – mixed states
The concurrence of a mixed state is given by
min ,i ii
C p C
where the minimization is over all possible decompositions of
The are the eigenvalues of the matrix in decreasing order, and
1 2 3 4) max 0,(C
i *YY YY
Can show:
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
• 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
• 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!
Geometric visualization for N=1: The Bloch sphere
1( , , ) ( , , )
2 2I
X Y Z X Y Z
Bloch vector (NMR)
Blochv
x
y
z
Blochv
1Blochv
1Blochv
pure state
mixed state
{ , , , }
1
2 j yj j
i x z
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)
One of them, always
1
, { ,
2
,
1
, }
21
4 kj k i x
j jy
kz
Generalizing the Bloch vector: The Pauli set
1 , ,P X YI I IZ• Polarization of Qubit 1
• Polarization of Qubit 2
• Two-qubit correlations
2 , ,I IP X Y ZI
12 , , , , , , , ,X Y Z X Y Z XX X X Y Y ZP Y ZY ZZ
The two-qubit Pauli set can be divided into three sections:
The Pauli set = 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
1II
P��������������
Visualizing N=2 states: product states
00
0 112
0 1
1
0
-1
1
0
-100
0110
1100 01 10 11
Re( ) Im( )
1P 2P 12P
0001
1011
00 01 10 11
Visualizing N=2 states: Bell states
0 11
20 1
0 00 11
211 10
1
0
-1
Re( ) Im( )
1
0
-100
0110
1100 01 10 11
0001
1011
00 01 10 11
Extracting usual metrics from the Pauli Set
2 1Tr[ ]
4P P ����������������������������
T T T
14
F P P ����������������������������
12 12 1( )
2P P
C ����������������������������
State purity:
Fidelity to a target state :T
Concurrence:
Warning:for pure states only
0C 1C
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
1 ns resolution
transmon qubits
DC - 2 GHz
A two-qubit quantum processor
T = 13 mK
flux bias lines
RV
LV
HV
Cf
Cf
Lf
Rf
cavity: “entanglement bus,” driver, & detector
Two-qubit joint readout via cavityC
avity
tra
nsm
issi
on
Frequency“Strong dispersive cQED”
11 10 01 00
Cf
CfHV
0
11
1
2 L2 R
2 2L R
Schuster, Houck et al., Nature (2007)
00 01
10 11
Prepareand measure
Qubit-state dependent cavity resonance
Chow et al., arXiv 0908.1955
Two qubit joint readout via cavityC
avity
tra
nsm
issi
on
Frequency
11 10 01 00
Cf
CfHV
Measure cavity transmission:“Are qubits both in their
ground state?”
00 00M
Direct access to qubit-qubit correlations with a single measurement channel!
Z II ZZ Z
Direct access to qubit-qubit correlations
How to reconstruct the two-qubit state from an ensemble measurement of the form
?1 2 12H Z II ZV ZZ
Answer: Combine joint readout with one-qubit pre-rotations
It is possible to acquire correlation info. with one measurement channel!
+Apply & , then measure:
/2xR
0,xR
Joint DispersiveReadout
1 2
12
Z II
Z
Z
Z
1 2 12Y II YZ Z
1 2 12Y II ZYZ
122 ZY
Example: How to extract YZ
/2xR
Apply , then measure:
/2xR
xR
All Pauli set components are obtained by linear operations on raw data.
Producing entanglement with the circuit QED processor
Related MM 2010 talks: W6.00003 & T29.00012
First demonstration of 2Q entanglement in SC qubits: Steffen et al., Science (2006)
0.55C
Spectroscopy of qubit2-cavity system
Qubit-qubit swap interactionMajer et al., Nature (2007)
cavity
left qubit
right qubit
Cavity-qubit interactionVacuum Rabi splittingWallraff et al., Nature (2004)
RV
Preparation1-qubit rotationsMeasurement
cavityQ
One-qubit gates: X and Y rotations
Rf
RV
sin(2 )qf t
x
y
z
Preparation1-qubit rotationsMeasurement
cavityI
One-qubit gates: X and Y rotations
RV
cos(2 )qf t
Rf
x
y
z
Preparation1-qubit rotationsMeasurement
cavityQ
One-qubit gates: X and Y rotations
Rf
RV
sin(2 )qf t
Lf
x
y
z
Preparation1-qubit rotationsMeasurement
cavityI
One-qubit gates: X and Y rotations
J. Chow et al., PRL (2009) Fidelity = 99%RV
cos(2 )qf t
Lf
x
y
z
cavity
Two-qubit gate: turn on interactions
Conditionalphase gate
Use control lines to push qubits near a resonance:
A controlled interaction, a la NMR
RV
z z
RV
20
11
02 Two-excitation manifold
Two-excitation manifold of system
Transmon “qubits” have multiple levels…
Strauch et al. PRL (2003): proposed using interactions with higher levels for computation in phase qubits
• Avoided crossing (160 MHz)
11 20
0211
Two-excitation manifold
11 1e1 11 i
01 e01 01i
10 0e1 10 i
0
2 ( )ft
a a
t
f t dt
Adiabatic phase gate
10
01
11
2-excitationmanifold
1-excitationmanifold
0
11 10 01 2 ( )ft
t
t dt
0201+10
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
U
00 1001 11
00
10
01
11
Adjust timing of flux pulse so that only quantum amplitude of acquires a minus sign:
11
01
10
11
1 0 0 0
0 0 0
0 0 0
0 0 0
i
i
i
e
e
e
U
00 1001 11
00
10
01
11
Implementing C-Phase with 1 fancy pulse
30 ns
11
1 1
2 20 1 0 0 10 1 1 0 110
1 0 0 0
0 1 0 0 1
0 0 1 0 2
0 0
0 0 10 1 1
0
0 1
1
Apply C-Phase entangler:
rotation on each qubit yields a maximal superposition:/2
yR
No longer a product state!
How to create a Bell state using C-Phase
0 0
0 0 10 1 0 1
1 0 0 0
0 1 0 0 1
0 0 1 0 2
0 0 0 1
1
rotation on LEFT qubit yields:( / 2)YR
1Bell
21 00 1
How to create a Bell state using C-Phase
Two-qubit entanglement experiment
0 11
20 1
0 0 0 10 0 0 1 1 01 0 12
11
11
Ideally:
/2yR /2
yR
/2yR
0
0
wavefunction
density matrix
Re( )
Expt’l state tomography
10
Re( )
Bell state Fidelity Concurrence
00 1100 1101 1001 10
91% 88%
94% 94%
90% 86%
87% 81%
Entanglement on demand/2
yR /2yR
/2yR
0
0
ij
Switching to the Pauli set
Related MM 2010 talks: T29.00012 & W6.00003
10
0 11
20 1
0 00 11
211 10
0 00 11
211 10
Experimental N=2 Pauli sets
Pauli set movies
Look at evolutions of separable and entangled states
~98% visibility for separable states, ~92% visibility for entangled states
/2yR
yR
/2yR
0
0
10
yR0
0
• a test for systematic errors in tomography, such as offsets and amplitude errors in Pauli bars
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
Can we characterize entanglement without reliance on ?
Can we place lower bounds on without performing fullstate tomography?
C
C
C
Witnessing entanglement with a subset of the Pauli set
An entanglement witness is a unity-trace observable with a positive expectation value for all product states.
0 W state is entangled, guaranteed.
0 W witness simply doesn’t know
2 W C Witness also gives a lower bound on :
W
Witnesses require only a subset of the Pauli set!
An optimal witness of an entangled state gives the strictest lower bound on
opt 14
I X ZIW X Y ZY Example: Is the optimal
witnessfor the singlet
C
C
Witnessing entanglement
/2yR
yR
/2yR
0
0
10
Entanglement is witnessed (by some witness) at all angles inentangled-state movie!
Witnessing entanglement
Control: entanglement is not witnessed by any witnessin the separable movie!
yR0
0
Bell inequalities: another form of entanglement witness
x
x’z’z
state is clearly highly entangled!
Clauser, Horne,Shimony & Holt (1969)
2CHSH Separable bound:
Bell state
2.61 0.04
Also UCSB group, closing detection loophole, Ansmann et al., Nature (2009)
CHSH operator = entanglement witness
' 'X ZC X X Z ZXSH ZH
' 'X ZC X X Z ZXSH ZH
not test of hidden variables…(loopholes abound)
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)
1 , ,P X YII II IZI
3 , ,I IP X YI I ZII
Knowing the three-qubit state = expectation values of 63 Pauli operators
Polarization of qubit 1
Polarization of qubit 2
2-qubit correlations
2 , ,I II IP X Y IIZ
12 , , , , , , , ,I I I I IX YX X X Y Y Y ZP ZZ X Y ZI I IX Y IZZ
3-qubit correlations
Polarization of qubit 3
13 , , , , , , , ,X Y Z X YI IX X X Y Y Y ZP ZI I I IZ X YI I ZZI
23 , , , , , , , ,X Y Z X YX XI I I I I I IP IX Y Y YZ X YZ Z ZIZ
123 , , , ... , , ,X X X ZX Y Z ZX X X Z ZX Y ZP ZZ
2 21 1
, , { , , }
3
,
31
8 j k l i x ykjk l
zlj
The density matrix for N=3
Example Pauli set for N=3: product state
111
Re( )1
0
-1000
111111
000
1P 3P2P 12P 13P 123P23P
00 12
0 11
1 The GHZ stateExample Pauli set for N=3: 3-qubit entangled state
GreenbergerHorne
Zeilinger
Re( )1
0
-1000
111000
111
1P 3P2P 12P 13P 123P23P
Upgrading the processor: more qubits
Q3
Q2 Q1
Q4
DiCarlo et al., (2010)
Reed et al., (2010)
We have recently extendedC-Phase gates and joint qubit readout to produce and detect 3-qubit entanglement.
Invitation to MM talks:W6.00003 (w/ transmon qubits, and joint readout)Also,Z26.00003 (w/ phase qubits, and individual qubit readouts)
Measure cavity transmission:“Are all 3 qubits in the ground
state?”
000 000M
I Z I ZI I Z I Z ZI Z ZZ I I Z Z I Z Z
Cav
ity t
rans
mis
sion
Frequency
111 000
Joint 3-Qubit Readout
Same concept: Use cavity transmission to learn a collective property, now of the three qubits
The trick still works! Combining joint readout with one-qubit “analysis” gives access to all 3-qubit Pauli operators, only more rotations are necessary.
Example: extract
I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z
4 ZZZ
no pre-rotation:
on Q1 and Q2:
on Q1 and Q3:
on Q2 and Q3:
0,xR
0,xR
0,xR
Joint Readout
000 000
M
( )xR ( )xR ( )xR
3-Qubit state tomography
ZZZ
I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z
Outlook
N
123456789
10
22 1N
31563
2551,0234,095
16,38365,535
262,1431,048,575
Is full state tomography scalable?
In ion traps: Haffner, Nature (2005)
Steffen et al., Science (2006)Filipp et al., PRL (2009)
DiCarlo et al., Nature (2009)Chow et al., arXiv 0908.1955
Steffen et al., PRL (2007)M. Metcalfe, Ph.D. Thesis (2008)
Neeley et al., ????DiCarlo et al., (2010)
Number of qubits Parameters in density matrix
How far will we push full state tomography?
Up to N=8 qubits
10 hours of ensemble averaging!
Summary in pictures
Few-qubit processorsbbased on circuit QED
C-phase gate
Joint Readout
Generation and detection of entanglement
21
2 21 1a b ba
vs
0C
1C
Related MM 2010 talks by Yale cQED team
L. DiCarlo et al.Realization of Simple Quantum Algorithms with Circuit QEDW6.00003: Thursday 03/18, 12:27 PM, Room 253
J. M. Chow et al.Quantifying entanglement with a joint readout of superconducting qubitsT29.00012: Wednesday 03/17, 4:42 PM, Room C123
M. D. Reed et al.Eliminating the Purcell effect in cQEDNEW TITLE: ????Y26.00013: Friday 03/19, 10:24 AM, Room D136
Lev S. Bishop et al.Latching behavior of the driven damped Jaynes-Cummings model in cQEDT29.00012: Wednesday 03/17, 5:06 PM, Room C123
Acknowledgements
Expt: Jerry ChowMatthew Reed Blake Johnson Luyan SunJoseph Schreier Andrew Houck David Schuster Johannes Majer Luigi Frunzio
Theory: Jay Gambetta Lev BishopAndreas NunnenkampJens Koch Alexandre Blais
PI’s: Robert Schoelkopf, Michel Devoret, Steven Girvin
Funding sources:
Circuit QED team members ‘09
Eran Ginossar
Luigi Frunzio
David Schuster
BlakeJohnson
JensKoch
HanheePaik
AndreasNunnenkamp
MattReed
SteveGirvin
JerryChow
LeoDiCarlo
Lev Bishop
AdamSears
AndreasFragner
Luyan Sun
RobSchoelkopf
JayGambetta