Production and detection of few-qubit entanglement in circuit-QED processors Leo DiCarlo (in place...

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


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


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


cavityI


RV

cos(2 )qf t

Rf

x

y

z


cavityQ


Rf

RV

sin(2 )qf t

Lf

x

y

z


cavityI


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


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


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

Production and detection of few-qubit entanglement in circuit-QED processors Leo DiCarlo (in place...

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