Andreas DewesQuantronics Group. Advisors: Denis Vion, Patrice Bertet, Daniel Esteve

Demonstrating Quantum Speed-Up with a Two-Transmon Quantum Processor

Ph.D. defense, UPMC / CEA, 15/11/2011

2Outline

Realizing a Two-Qubit Processor

Realizing a Two-Qubit Gate

Demonstrating Quantum Speed-Up

Introduction & Motivation

2

3Why Research on Quantum Computing?

Example: Quantum Spin Models

Quantum Simulation: Not efficient on a classical computer.

3

image removed due to copyright

4Why Research on Quantum Computing?

problem size – n

Run

time

class

ical a

lgorithm –n

Grover algorithm – n

Database SearchInteger Factorization

problem size – n

Run

time

classical algorithm –

exp(1.9log[N]1/3 log[log[n]]2/3)

Shor algorithm – log(n)3

Quantum Algorithms: More efficient for certain complex problems.

this

wor

k

4

4

5

images removed due to copyright

Why Superconducting Qubits

1. Quantum behavior demonstrated in 1980s2. Since 1999 qubits with increasingly long

coherence times.3. Potentially as scalable as other integrated

electrical circuits

CEA Saclay ETH Zurich UC Santa Barbara

5

6DiVincenzo Criteria

1. Robust, resettable qubits2. Universal set of:

• Single-qubit gates• Two-qubit gates

3. Individual readout (ideally QND)

0 1?

U1

1

0U2

U1

0 1?

1

0

Realizeany unitaryevolution.For quantumspeed-up

6

7gg

Schoelkopf Lab, Yale UniversityDiCarlo et.al., Nature 460 (2009)

Two-Qubit Grover Search

Joint Qubit Non-Destructive Readout

Martinis Lab, UC Santa BarbaraYamamoto et.al. , PRB 82 (2010)

Two-Qubit Deutsch-Josza Algorithm

Individual Qubit Destructive Readout

This WorkDewes et. al., PRL 108 (2012), PRB Rapid Comm 85 (2012)

Two-Qubit Grover Search Algorithm (with quantum speed-up)Individual-Qubit Non-Destructive Readout

7

images removed due to copyright

Superconducting Two-Qubit Processors

8Outline

Realizing a Two-Qubit Gate

Demonstrating Quantum Speed-Up

Introduction & Motivation

Realizing a Two-Qubit Processor

8

9The Cooper Pair Box

200 nm

1

0

EC

Cg

EJ

9

10The Cooper Pair Box

)ˆcos(ˆ

ˆ2

2

JC EEH

EC

E

f

|0>

|1>

|2>

Cg

01

1

0

EJf

0112

10

11The Cooper Pair Box

zH ˆ2

ˆ01

EC

E

f

Cg

01

1

0

EJf

|0>

|1>

11

12The Qubit: A Transmon

EJ EC

Cg

1

0

CJ EE

Wallraff et al., Nature 431 (2004) Koch et al., Phys. Rev. A 76 (2007)

J.A. Schreier, Phys. Rev. B 77, 180502 (2008)

12

13The Qubit: A Transmon

EJ EC

Vfl(t)

E

f

|0>

|1>

|2>

01

Cg

GHz 8401 MHz 400200

1

0

01ˆ ˆ( )

2 ext zH

13

14Qubit Dispersive Readout

readout

50 W

4K

d

Wallraff et al., Nature 431 (2004)

d

0 1?

drive frequency

14

refle

cted

pha

se

LCr

1

CL

15Qubit Dispersive Readout0 1?

readout qubit

j0 or j1

50 W

4K

refle

cted

pha

se

d

j1

j0

|0>

|1>

j

|1> |0>

readout errors

ddrive frequency

15

16Qubit Dispersive Readout

readout qubit

j0 or j1

50 W

4K

d

j1

j0

|0>

|1>

Siddiqi et al., PRL 93 (2004); Mallet et al., Nat. Phys. 5 (2009)

drive frequency d

0 1?

High

Low

16

switching

refle

cted

pha

se

|1> outcome 1 (High)|0> outcome 0 (Low)

j

|1> |0>

17Qubit Dispersive Readout

readout qubit

50 W

4K

d

Siddiqi et al., PRL 93 (2004); Mallet et al., Nat. Phys. 5 (2009)

0 1?

switc

hing

pro

babi

lity

drive power

p(0)

Ad

p(1)

|1> |0>

1

17

|2>

p(2)

18Single-Qubit X,Y & Z Gates

EJ EC

σz

x

y

z|0>

|1>

01 0( ) cos([ ] )A t t

B(t)

0

Cg

Vd(t)

Vfl(t)

Cin

U1

G. Ithier, PhD Thesis (2005)

18

ϕ

θ

12

sin02

cos ie

19Two-Qubit Gate: Principle

qubit I qubit II

01 01

01 01

01 01

01 01

0 0 02

0 02/

0 02

0 0 02

qq

qq

I II

I II

I II

I II

Hg

g

00

01

10

11

Cqq

qq qqg C

U2

19

20Two-Qubit Gate: Principle

U2

time

01II

01I

01II

01

I

2

1001

2

1001

|10>

|01>

|10>

|01>

qqIII g 0101

20

III0101

21Two-Qubit Gate: Principle

U2

time

/4gqq

01II

01I

01II

01

I

/2gqq

01 ( )01 01

1 0 0 0

0 cos( ) sin( ) 0( , )

0 sin( ) cos( ) 0

0 0 0 1

z qq qqi tI II

qq qq

g t i g tt e

i g t g tU

1( ) ( )

1 0 0 0

0 0 0( ,0) SWAP

0 0 02

0 0 0 1

z zi

qq

iie e iU

ig

1 1( )

( )

2 0 0 0

0 1 0( ,0) SWAP

0 1 04 2

0 0 0 2

z

z

qq

iiU

g

iee i

i

11

|10>

|01>

|10>

|01>

01

21

III0101

22Schematic of the Full Processor

50 W

4K

readout I qubit I readout IIqubit II

outcome 00, 01, 10, or 11

j0 or j1

01 01 22

I IIext e

I II I IIZ Z

IY Y

Ixt qq

IH g

22

23

a)

100 m

1 mm

Realization of the Processor

1m

23

24

50Ω

50Ω

ADCcard

4-8GHz

readout

I

Q

LOVc

20 mK

4 K

300 K

600 mK

20dB

1.35GHz

Eccosorbfilter

processorchip

dc flux

fast flux10 MHz clock

50W

Measurement Setup

20dB

1.4-20GHz

23dB

20dB

DC-7.2GHz

dB

drive

I Q

24

25Qubit Spectroscopy

f d,

A(t

)

time

readout pulsedrive pulse

f01

f02/2

1 us

25

26Flux Dependence of Qubit Frequencies

01I = 8 GHz

01I = -240 MHz

dI = 0.2

01II = 8.4 GHz

01II = -230 MHz

dII = 0.35

III

Qubit I Qubit II

26

27Single-Qubit Gate Characterization

x

y

z|0>

|1>

27

28Performing Rabi Oscillations

x

y

z|0>

|1>

Rabi=85 MHz

28

29Characterizing Energy Relaxation (1)

x

y

z|0>

|1>

1=(456 ns)-1

Rabi=85 MHz

29

30

x

y

z|0>

|1>

Characterizing Dephasing ()

=(764 ns)-1 2 =(416 ns)-1

1=(456 ns)-1

Rabi=85 MHz

30

31Characterizing Register Readout

-5 -4 -30.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1

read

out 2

switc

hing

pro

babi

lity

power [dB]

read

out 1

|0>

|1> |1>

|0>

31

errors

17 %

7 %

16 %

13 %

|00> |01> |10> |11>

00

01

10

11

71 %

76 %

74 %

80 %

prepared register state

read

out o

utco

me

readout matrix R

),,,(),,,( 111001001

11100100 pppppppp R

32Characterizing Register Readout

-5 -4 -30.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1

read

out 2

switc

hing

pro

babi

lity

power [dB]

read

out 1

|1>|2>

|0>

|2>|1>

|0>

32

errors

14 %

3 %

11 %

6 %

|0>

|1>

|2>12

33Choice of Processor Working Points

Qubit I Qubit II

33

read

out

sing

le-q

ubit

gate

two-

qubi

t gat

e

Readout Relaxation Model

34Characterizing Qubit-Qubit Interaction

time

f 01,

A(t

)

f01II

f01I

readout I readout II

drive pulse1 us

34

35Characterizing Qubit-Qubit Interaction

time

2gqq=8.7 MHz

f 01,

A(t

)

f01II

f01I

readout I readout II

drive pulse1 us

35

36Processor: Operating Principle

time

f 01[

f(t)

], a(

t)

Δfm

(150 MHz)

x/y rotations two-qubit readoutz rotations

5.1 GHz

6.2 GHz

5.25 GHz

|0>

|0>

Y π/2

Xπ/2

iSW

AP

Z

Z

0 1

0 1

36

37Outline

Demonstrating Quantum Speed-Up

Introduction & Motivation

Realizing a Two-Qubit Processor

Realizing a Two-Qubit Gate

37

38

0 100 200 300 4000,0

0,2

0,4

0,6

0,8

1,0

|10>

|00>

|11>

swap duration [ns]

stat

e po

poul

atio

ns

f 01,

A(t

)

time

Y

readout

swap duration

f01II

f01I

Two-Qubit Gate Tune-Up

|01>

SWAPi SWAPi

1 10.44 0.52 2 I II I IIT µs T µs T T µs

38

39Two-Qubit Density Matrix & Pauli Set

|11>

|00>|01>

|10>

<00|

<01|

<10|

<11|0

/2

3/2

Density Matrix Pauli Set

ji

jiji,4

1

},,,{, ZYXIji

39

40Measuring the Full Pauli Setf 0

1,

A(t

)

time

Y

readout

f01II

f01I

x

y

z |0>

|1>

X -/2,Y/2

XI YI ZI

IX IY IZ

XX XY XZ

YX YY YZ

ZX ZY ZZ

single-qubitoperators

two-qubitcorrelators

ZI

IZ

ZZ

swap duration

40

41

YYϕ,Y /2+ϕ

X -/2,Y/2

readout

f 01,

A(t

)

f01II

f01I

|11>

|00>|01>

|10>

<00|<01|

<10|<11|

Experimental Tomography: iSWAP gate

41

01

time

42

YYϕ,Y /2+ϕ

X -/2,Y/2

31 ns

|11>

|00>|01>

|10>

<00|<01|

<10|<11|

f 01,

A(t

)

f01II

f01I

01 01 10

2

ie

Experimental Tomography: iSWAP gate

time

42

43

|11>

|00>|01>

|10>

<00|<01|

<10|<11|

Y

Yϕ,Y /2+ϕ

X -/2,Y/2

31 ns

f 01,

A(t

)

f01II

f01I

Compensating the acquired phase43

01id 01 10

2

i

94 %id idF 85 %F

time

44Observing the coherent swapping

|00>

|01>

|10>

|11>

<00| <01| <10| <11|swap duration [ns]

stat

e oc

cupa

tion

prob

abili

ty

44

(no phase compensation, no frequency displacement)

|10>|00>|01>

45Observing the coherent swapping

|00>

|01>

|10>

|11>

<00| <01| <10| <11|swap duration [ns]

stat

e oc

cupa

tion

prob

abili

ty

45

(no phase compensation, no frequency displacement)

|10>|00>|01>

46Quantum Process Tomography

in

i i iin in in 2

0 , 1 , 0 1 , 0 1iin i

†( ) iouij

in it jnj iE E 2

, , ,i x y zE I i

out

46

map Operator basisprocess

47Characterizing the iSWAP Gate

in

out

47

20 , 1 , 0 1 , 0 1i

in i

48

†( )out i ii

ij

i jn j nE E

2

, , ,i X Y ZE i

Process Tomography of the iSWAP

c(elements < 1 % not shown)

48

IXIY

IZXI

XXXY

XZYI

YXYY

YZZI

ZXZY

ZZ

II

49Fidelity & Error Budget of the Gate

90

8%2%

90%

Error Budget

fidelity

decoherence(mainly relaxation)

unitary errors

1 †ij inide ja

ijil E E

post-error map

c c~

49

(elements < 1 % not shown)

1

0.90ig dTF r

T1T

1T1

ZZZ

SSS

Dewes et. al., PRL 108 (2012)

IXIY

IZXI

XXXY

XZYI

YXYY

YZZI

ZXZY

ZZ

II

50Outline

Introduction & Motivation

Realizing a Two-Qubit Processor

Realizing a Two-Qubit Gate

Demonstrating Quantum Speed-Up

50

51The Two-Bit Search Problem

1, x = y

0, x y

}11,10,01,00{, yx

f01(00)=0 f01 (01)=1 f01(10)=0 f01 (11)=0

fy(x)=

51

52Benchmark: Classical Search Algorithms

x / f(x) f00 f01 f10 f11

00 1 0 0 0

01 0 1 0 0

10 0 0 1 0

11 0 0 0 1

Algorithm Success Probability

Query and Check 25 %

Query, Check and Guess 50 %

52

Algorithm Success Probability

Query and Check 25 %

53gg

H D

0 1

0 1

Oracle (O)

1111

1111

1111

1111

2

1

y

yx

xy

The Two-Qubit Grover Search Algorithm

Decoding (D)State Preparation Readout

x

x

Algorithm Success Probability

Query and Check 25 %

Query, Check and Guess 50 %

Grover Algorithm 100 %

53

( )1 yf x

x

x x|0>

|0>

Grover et. al., PRL 79, 1997

54

|0>

|0>

Y/2

Y/2

iSW

AP Z 1/2

Z2/2

iSW

AP X/2

X/2

State Preparation Oracle (O) Decoding (D)

1 2

f00 -1 -1

f01 -1 +1

f10 +1 -1

f11 +1 +1

Implementation of the Algorithm54

55

|0>

|0>

Y/2

Y/2

iSW

AP Z-/2

Z-/2

iSW

AP X/2

X/2

State Preparation

Y(p

/2)

50 100 150 200 ns

f 01[(ft)

], a(

t)

0

iSWAP iSWAP

Z(p

/2)

X(p

/2)

F=98% F=87% F=70%

f00

Implementation of the Algorithm

Similar to: DiCarlo et.al., Nature 460 (2009)

Oracle (O) Decoding (D)

55

56

|0>

|0>

Y/2

Y/2

iSW

AP Z-/2

Z-/2

iSW

AP X/2

X/2

Readout

0 1

0 1

Y(p

/2)

readout

50 100 150 200 ns

f 01[(ft)

], a(

t)

0

iSWAP iSWAPZ(p

/2)

X(p

/2)

X 12(p

)

State Preparation

Single-Run Success ProbabilityOracle (O) Decoding (D)

56

57

Dewes et. al., PRB Rapid Comm 85 (2012)

|0>

|0>

Y/2

Y/2

iSW

AP Z±/2

Z±/2

iSW

AP X/2

X/2

Readout

0 1

0 1

67 %

Fi > 25 % (50 %) for all oracles → Quantum Speed-Up achieved!

55 % 62 %52 %

f00 f01 f10 f11

State Preparation Oracle Function (R) Diffusion Operator (D)

Single-Run Success Probability57

query, check and guess

query and check

58SummaryRealized a Two-Qubit Processor following the DiVincenzo criteria.

58

Characterized a Universal 2-Qubit Gate with 90 % fidelity.

Ran the Grover search algorithm and demonstrated quantum speed-up.

59Outlook: Processor Scaling Problems59

not scalable!

1. Hard / Impossible to switch off coupling2. Frequency Crowding of Qubits3. Exponential Increase in Complexity with

n

60(Partial) Solution: New Architecture60

cell 2

high Qcoupler

cell n

… …readoutpulses

cell 1

Z drives,function selectors

XYdrives

61Acknowledgments

Thank you!

Special Thanks to Florian Ong & Romain Lauro as well as group technicians: Pascal Senat, Thomas David & Pief Orfila as well as members of the mechanical workshop!

F. OngR. Lauro

61

62Supplementary Material62

63Error Sources (still needs better visualiz!)

|0>

|0>

ϕ1α1

ϕ2α2

iSW

AP

(ε1,δ

1) Zβ1

Zβ2

iSW

AP

(ε2,δ

2) φ1

γ1

φ2γ2

State Preparation Oracle Function (R) Diffusion Operator (D) Readout

0 1

0 1

Rotation axis errors………………………………..Rotation angle errors …………………………….SWAP duration errors ……………………………SWAP frequency errors ……………………….Z-gate length errors ……………………………….Relaxation & Dephasing ………

=>quantitative explanation of data (Fmodel>97 %)

(max. 11 °)(max. 3.2 °)(max. xx ns)

(max xx MHz)(max xx ns)

(T1=400, 450 ns, T= 800 ns)

63

64The Two-Bit Search Problem

f(x)=1, x = y

0, x y

}11,10,01,00{, yx

fx1

x2

0 f(x)

x1

x2

Classical algorithm: Max. 3 calls of f neededto find solution with certainty

64

65gg

|0>

f

x

x diffusionoperator

0 1

0 111 xx

Oracle Function (R)

1111

1111

1111

1111

2

1

0 0 xx

readout

yx

xy 01 1y

yx

xy 01

=>1 call of f needed=>Quantum Speed-Up

The Two-QubitGrover Search Algorithm65

66gg

0 45 90 135 180 225 270 315 360

-2

-1

0

1

2

105 1061

2

3456

X

Y

Xj

Yj

j

rotation j of qubit II measurement basis (°)

N10

0 s

2 222s

readout errorcorrectedbare

Violation of CHSH Inequality66

67Characterizing Register Readout

-5 -4 -30.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1

read

out 2

prob

abili

ty, c

ontr

ast

power [dB]

read

out 1

|0>

|1>

|2> |2>|1>

power [dB]

|0>

67

68Experimental State Tomographyt = 0 ns t = 31 ns (iSWAP)

|11>

|00>|01>

|10>

<00|

<01|

<10|

<11|

|11>

|00>|01>

|10>

<00|

<01|

<10|

<11| F= xx %F= xx %

68

69Desired Process: iSWAP gate

2000

010

010

0002

2

1SWAP

i

ii

|11>

|00>|01>

|10>

<00|

<01|

<10|

<11|

|11>

|00>|01>

|10>

<00|

<01|

<10|

<11|

01 10012

1i

SWAPi

69

70Measurement of Pauli Set During SWAP70

71Measurement of Pauli Set During SWAP

71

72ggIII) Towards more scalable elementary processors

« n+1 in line » architecture based on frequency agility, individual readouts and multiplexing

cell 2

high Qcoupler

cell n

… …readoutpulses

cell 1

Z drives,function selectors

XYdrives

Difficulty of phase compensations for both single and 2-qubit gates !

6 7 8 9 10 11 12

readoutdriveWR=50MHz

parkingcoupler

frequency (GHz)

JBA

DrDcq,park

coupling

gqr = 60 MHzcqr = 4 MHzG1,r = 500 kHzG1d = 20 kHzgqq = 20-5 MHz

gqc = 40 MHz

gqq,park = 1 MHz

aqq,park =1%

jqq,park = 10°/µs

Residual couplings

- coupling gd ~ g2/D- amplitude a ~ gd/D- phase j= 2p gd

2/ D t

Vous êtes cordialement invités à la soutenance ainsiqu'au pot qui suivra.

Soutenance de ThèseAndreas Dewes

Demonstrating Quantum Speed-Up with a Two-Transmon Quantum Processor

Jeudi, 15.11 à 14:30hAmphithéâtre Claude-Bloch, Bat. 772