1. Andreas Dewes demonstrated quantum speed-up using a two-transmon quantum processor. 2. The processor realized a universal set of gates including single-qubit rotations and a two-qubit entangling gate through tunable coupling between transmon qubits. 3. Quantum algorithms like Grover's search were implemented on the processor, exhibiting quantum speed-up over classical algorithms for the same tasks.

1. Andreas Dewes
Quantronics 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

2. 2Outline
Realizing a Two-Qubit Processor
Realizing a Two-Qubit Gate
Demonstrating Quantum Speed-Up
Introduction & Motivation
2

3. 3Why Research on Quantum Computing?
Example: Quantum Spin Models
Quantum Simulation: Not efficient on a
classical computer.
3
image removed due to copyright

4. 4Why Research on Quantum Computing?
problem size – n
Runtime
Database SearchInteger Factorization
problem size – n
Runtime
Quantum Algorithms: More efficient for
certain complex problems.
thiswork
4
4

5. 5
images removed due to copyright
Why Superconducting Qubits
1. Quantum behavior demonstrated in 1980s
2. 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

6. 6DiVincenzo Criteria
1. Robust, resettable qubits
2. Universal set of:
• Single-qubit gates
• Two-qubit gates
3. Individual readout (ideally QND)
0 1
?
U1
1
0
U2
U1
0 1
?
1
0
Realize
any unitary
evolution.
For quantum
speed-up
6

7. 7gg
Schoelkopf Lab, Yale University
DiCarlo et.al., Nature 460 (2009)
Two-Qubit Grover Search
Joint Qubit Non-Destructive Readout
Martinis Lab, UC Santa Barbara
Yamamoto et.al. , PRB 82 (2010)
Two-Qubit Deutsch-Josza Algorithm
Individual Qubit Destructive Readout
This Work
Dewes 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

8. 8Outline
Realizing a Two-Qubit Gate
Demonstrating Quantum Speed-Up
Introduction & Motivation
Realizing a Two-Qubit Processor
8

9. 9The Cooper Pair Box
200 nm
1
0
EC
Cg
EJ
9

10. 10The Cooper Pair Box
)ˆcos(
ˆ
ˆ
2
2


JC EEH 



EC
E

|0>
|1>
|2>
Cg
01
1
0
EJ

  0112
10

11. 11The Cooper Pair Box
zH  ˆ
2
ˆ
01


EC
E

Cg
01
1
0
EJ

|0>
|1>
11

12. 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

13. 13The Qubit: A Transmon
EJ EC
Vfl(t)
E

|0>
|1>
|2>
01
Cg
GHz8401 
MHz400200
1
0
01
ˆ ˆ( )
2
ext zH   

13

14. 14Qubit Dispersive Readout
readout
50 W
4K
d

Wallraff et al., Nature 431 (2004)
d
0 1?
drive frequency
14
reflectedphase
LC
r
1

C
L

15. 15Qubit Dispersive Readout
0 1?
readout qubit
0 or 1
50 W
4K
reflectedphase
d
1
0
|0>
|1>

|1> |0>
readout errors
ddrive frequency
15

16. 16Qubit Dispersive Readout
readout qubit
0 or 1
50 W
4K
d
1
0
|0>
|1>
Siddiqi et al., PRL 93 (2004);
Mallet et al., Nat. Phys. 5 (2009)
drive frequency d
0 1?
High
Low
16
switching
reflectedphase
|1> outcome 1 (High)
|0> outcome 0 (Low)

|1> |0>

17. 17Qubit Dispersive Readout
readout qubit
50 W
4K
d
Siddiqi et al., PRL 93 (2004);
Mallet et al., Nat. Phys. 5 (2009)
0 1?
switchingprobability
drive power
p(0)
Ad
p(1)
|1> |0>
1
17
|2>
p(2)

18. 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
1
2
sin0
2
cos 
 i
e













19. 19Two-Qubit Gate: Principle
qubit I qubit II
01 01
01 01
01 01
01 01
0 0 0
2
0 0
2/
0 0
2
0 0 0
2
qq
qq
I II
I II
I II
I II
H
g
g
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 

00
01
10
11
Cqq
qq qqg C
U2
19

20. 20Two-Qubit Gate: Principle
U2
time
01
II
01
I
01
II
01
I
2
1001 

2
1001 

|10>
|01>
|10>
|01>
qq
III
g 0101 
20
III
0101  

21. 21Two-Qubit Gate: Principle
U2
time
/4gqq
01
II
01
I
01
II
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 t
t e
i g t g t
U 
   
 
 
  
 
 
 
1( ) ( )
1 0 0 0
0 0 0
( ,0) SWAP
0 0 02
0 0 0 1
z zi
qq
ii
e 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
i
i
U
g
ie
e i
i
 
  

 
 
 
  
 
 
 
1
1
|10>
|01>
|10>
|01>
01
21
III
0101  

22. 22Schematic of the Full Processor
50 W
4K
readout I qubit I readout IIqubit II
outcome 00, 01, 10, or 11
0 or 1
( ) ( )01 01 2
2
I II
ext e
I II I II
Z Z
I
Y Y
I
xt qq
I
H g       

22

23. 23
a)
100 m
1 mm
Realization of the Processor
1m
23

24. 24
50Ω
50Ω
ADC
card
4-8
GHz
readout
I
Q
LO
Vc
20 mK
4 K
300 K
600 mK
20dB
1.35
GHz
Eccosorb
filter
processor
chip
dc flux
fast flux10 MHz clock
50W
Measurement Setup
20dB
1.4-20
GHz
23dB
20dB
DC-7.2
GHz
dB
drive
I Q
24

25. 25Qubit Spectroscopy
fd,A(t)
time
readout pulsedrive pulse
f01
f02/2
1 us
25

26. 26Flux Dependence of Qubit Frequencies
01
I = 8 GHz
01
I = -240 MHz
dI = 0.2
01
II = 8.4 GHz
01
II = -230 MHz
dII = 0.35
III
Qubit I Qubit II
26

27. 27Single-Qubit Gate Characterization
x
y
z
|0>
|1>
27

28. 28Performing Rabi Oscillations
x
y
z
|0>
|1>
WRabi=85 MHz
28

29. 29Characterizing Energy Relaxation (1)
x
y
z
|0>
|1>
1=(456 ns)-1
WRabi=85 MHz
29

30. 30
x
y
z
|0>
|1>
Characterizing Dephasing ()
 =(764 ns)-1 2 =(416 ns)-1
1=(456 ns)-1
WRabi=85 MHz
30

31. 31Characterizing Register Readout
-5 -4 -3
0.0
0.2
0.4
0.6
0.8
1.0
-3 -2 -1
readout2
switchingprobability
power [dB]
readout1
|0>
|1> |1>
|0>
31
errors
17 %
7 %
16 %
13 %
|00> |01> |10> |11>
00
01
10
11
71%
76%
74%
80%
readout matrix R
),,,(),,,( 11100100
1
11100100
pppppppp  
R

32. 32Characterizing Register Readout
-5 -4 -3
0.0
0.2
0.4
0.6
0.8
1.0
-3 -2 -1
readout2
power [dB]
readout1
|1>
|2>
|0>
|2>
|1>
|0>
32
errors
14 %
3 %
11 %
6 %
|0>
|1>
|2>
12

33. 33Choice of Processor Working Points
Qubit I Qubit II
33
readout
single-qubitgate
two-qubitgate
Readout Relaxation Model

34. 34Characterizing Qubit-Qubit Interaction
time
f01,A(t)
f01
II
f01
I
readout I readout II
drive pulse
1 us
34

35. 35Characterizing Qubit-Qubit Interaction
time
2gqq=8.7 MHz
f01,A(t)
f01
II
f01
I
readout I readout II
drive pulse
1 us
35

36. 36Processor: Operating Principle
time
f01[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
iSWAP
Z
Z
0 1
0 1
36

37. 37Outline
Demonstrating Quantum Speed-Up
Introduction & Motivation
Realizing a Two-Qubit Processor
Realizing a Two-Qubit Gate
37

38. 38
0 100 200 300 400
0,0
0,2
0,4
0,6
0,8
1,0
|10>
|00>
|11>
swap duration [ns]
statepopoulations
f01,A(t)
time
Y
readout
swap duration
f01
II
f01
I
Two-Qubit Gate Tune-Up
|01>
SWAPi SWAPi
1 10.44 0.52 2   I II I II
T µs T µs T T µs 
38

39. 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

40. 40Measuring the Full Pauli Set
f01,A(t)
time
Y
readout
f01
II
f01
I
x
y
z |0>
|1>
X -/2,Y/2
XI YI ZI
IX IY IZ
XX XY XZ
YX YY YZ
ZX ZY ZZ
single-qubit
operators
two-qubit
correlators
ZI
IZ
ZZ
swap duration
40

41. 41
Y
Yϕ,Y /2+ϕ
X -/2,Y/2
readout
f01,A(t)
f01
II
f01
I
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|
Experimental Tomography: iSWAP gate
41
01
time

42. 42
Y
Yϕ,Y /2+ϕ
X -/2,Y/2
31 ns
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|
f01,A(t)
f01
II
f01
I
01 01 10
2
i
e

Experimental Tomography: iSWAP gate
time
42

43. 43
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|
Y
Yϕ,Y /2+ϕ
X -/2,Y/2
31 ns
f01,A(t)
f01
II
f01
I
Compensating the acquired phase
43
01id  01 10
2
i
94 %id idF     85 %F 
time

44. 44Observing the coherent swapping
|00>
|01>
|10>
|11>
<00| <01| <10| <11|
swap duration [ns]
stateoccupationprobability
44
(no phase compensation, no frequency displacement)
|10>
|00>|01>

45. 45Observing the coherent swapping
|00>
|01>
|10>
|11>
<00| <01| <10| <11|
swap duration [ns]
stateoccupationprobability
45
(no phase compensation, no frequency displacement)
|10>
|00>|01>

46. 46Quantum Process Tomography
in
i i i
in in in   
2
0 , 1 , 0 1 , 0 1i
in i

  
†
( ) iou
ij
in it jnj iE E    
 
2
, , ,i x y zE I i  


out
46
map Operator basisprocess

47. 47Characterizing the iSWAP Gate
in
out
47
 
2
0 , 1 , 0 1 , 0 1i
in i

  

48. 48
†
( )out i i
i
i
j
i jn j nE E   
 
2
, , ,i X Y ZE i  

 
Process Tomography of the  iSWAP

(elements < 1 % not shown)
48

49. 49Fidelity & Error Budget of the Gate
90
8%
2%
90%
Error Budget
fidelity
decoherence
(mainly relaxation)
unitary errors
1 †
ij inide ja
ij
il E E  
  
post-error map
 ~
49
(elements < 1 % not shown)
1
( ) 0.90ig dTF r  
Dewes et. al., PRL 108 (2012)

50. 50Outline
Introduction & Motivation
Realizing a Two-Qubit Processor
Realizing a Two-Qubit Gate
Demonstrating Quantum Speed-Up
50

51. 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

52. 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 %

53. 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. 54
|0>
|0>
Y/2
Y/2
iSWAP
Z 1/2
Z2/2
iSWAP
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 Algorithm
54

55. 55
|0>
|0>
Y/2
Y/2
iSWAP
Z-/2
Z-/2
iSWAP
X/2
X/2
State Preparation
Y(/2)
50 100 150 200 ns
f01[(t)],a(t)
0
iSWAP iSWAP
Z(/2)
X(/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. 56
|0>
|0>
Y/2
Y/2
iSWAP
Z-/2
Z-/2
iSWAP
X/2
X/2
Readout
0 1
0 1
Y(/2)
readout
50 100 150 200 ns
f01[(t)],a(t)
0
iSWAP iSWAP
Z(/2)
X(/2)
X12()
State Preparation
Single-Run Success Probability
Oracle (O) Decoding (D)
56

57. 57
Dewes et. al., PRB Rapid Comm 85 (2012)
|0>
|0>
Y/2
Y/2
iSWAP
Z /2
Z /2
iSWAP
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 Probability
57
query, check and guess
query and check

58. 58Summary
Realized 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.

59. 59Outlook: Processor Scaling Problems
59
1. Hard / Impossible to switch off coupling
2. Frequency Crowding of Qubits
3. Exponential Increase in Complexity with n

60. 60(Partial) Solution: New Architecture
60
cell 2
  
high Q
coupler
cell n
… …
readout
pulses
cell 1
Z drives,
function selectors
XY
drives

61. 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

62. 62Supplementary Material
62

63. 63Error Sources (still needs better visualiz!)
|0>
|0>
ϕ1
α1
ϕ2
α2
iSWAP(ε1,δ1)
Zβ1
Zβ2
iSWAP(ε2,δ2)
φ1
γ1
φ2
γ2
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

64. 64The Two-Bit Search Problem
f(x)=
1, x = y
0, x  y
}11,10,01,00{, yx
f
x1
x2
0 f(x)
x1
x2
Classical algorithm: Max. 3 calls of f needed
to find solution with certainty
64

65. 65gg
|0>
f
x
x
diffusion
operator
0 1
0 111 xx 
Oracle Function (R)


















1111
1111
1111
1111
2
1
00 xx 
readout


yx
xy 01 1y

yx
xy 01
=>1 call of f needed
=>Quantum Speed-Up
The Two-QubitGrover Search Algorithm
65

66. 66gg
0 45 90 135 180 225 270 315 360
-2
-1
0
1
2
10
5
10
6
1
2
3
4
5
6
X
Y
X
Y

rotation  of qubit II measurement basis ( )
N100
2 2
22
readout error
corrected
bare
Violation of CHSH Inequality
66

67. 67Characterizing Register Readout
-5 -4 -3
0.0
0.2
0.4
0.6
0.8
1.0
-3 -2 -1
readout2
probability,contrast
power [dB]
readout1
|0>
|1>
|2> |2>
|1>
power [dB]
|0>
67

68. 68Experimental State Tomography
t = 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

69. 69Desired Process: iSWAP gate















2000
010
010
0002
2
1
SWAP
i
i
i
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|
01 ( )1001
2
1
i
SWAPi
69

70. 70Measurement of Pauli Set During SWAP
70

71. 71Measurement of Pauli Set During SWAP
71

72. 72gg
III) Towards more scalable elementary processors
« n+1 in line » architecture based on frequency agility, individual readouts and multiplexing
cell 2
  
high Q
coupler
cell n
… …
readout
pulses
cell 1
Z drives,
function selectors
XY
drives
Difficulty of phase compensations for both single and 2-qubit gates !
6 7 8 9 10 11 12
readoutdrive
WR=50MHz
parkingcoupler
frequency (GHz)
JBA
r
cq,park
coupling
gqr = 60 MHz
qr = 4 MHz
1,r = 500 kHz1d = 20 kHzgqq = 20-5 MHz
gqc = 40 MHz
gqq,park = 1 MHz
aqq,park =1%
qq,park = 10°/µs
Residual couplings
- coupling gd ~ g2/
- amplitude a ~ gd/
- phase = 2 gd
2/t

73. Vous êtes cordialement invités à la soutenance ainsi
qu'au pot qui suivra.
Soutenance de Thèse
Andreas Dewes
Demonstrating Quantum Speed-Up with a
Two-Transmon Quantum Processor
Jeudi, 15.11 à 14:30h
Amphithéâtre Claude-Bloch, Bat. 772