Andreas Dewes
Quantronics Group. Advisors: Denis Vion, Patrice Bertet, Daniel Esteve
Demonstrating Quantum Speed-Up
with a...
2Outline
Realizing a Two-Qubit Processor
Realizing a Two-Qubit Gate
Demonstrating Quantum Speed-Up
Introduction & Motivati...
3Why Research on Quantum Computing?
Example: Quantum Spin Models
Quantum Simulation: Not efficient on a
classical computer...
4Why Research on Quantum Computing?
problem size – n
Runtime
Database SearchInteger Factorization
problem size – n
Runtime...
5
images removed due to copyright
Why Superconducting Qubits
1. Quantum behavior demonstrated in 1980s
2. Since 1999 qubit...
6DiVincenzo Criteria
1. Robust, resettable qubits
2. Universal set of:
• Single-qubit gates
• Two-qubit gates
3. Individua...
7gg
Schoelkopf Lab, Yale University
DiCarlo et.al., Nature 460 (2009)
Two-Qubit Grover Search
Joint Qubit Non-Destructive ...
8Outline
Realizing a Two-Qubit Gate
Demonstrating Quantum Speed-Up
Introduction & Motivation
Realizing a Two-Qubit Process...
9The Cooper Pair Box
200 nm
1
0
EC
Cg
EJ
9
10The Cooper Pair Box
)ˆcos(
ˆ
ˆ
2
2


JC EEH 



EC
E

|0>
|1>
|2>
Cg
01
1
0
EJ

  0112
10
11The Cooper Pair Box
zH  ˆ
2
ˆ
01


EC
E

Cg
01
1
0
EJ

|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. ...
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
14Qubit Dispersive Readout
readout
50 W
4K
d

Wallraff et al., Nature 431 (2004)
d
0 1?
drive frequency
14
reflectedpha...
15Qubit Dispersive Readout
0 1?
readout qubit
0 or 1
50 W
4K
reflectedphase
d
1
0
|0>
|1>

|1> |0>
readout errors
...
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., N...
17Qubit Dispersive Readout
readout qubit
50 W
4K
d
Siddiqi et al., PRL 93 (2004);
Mallet et al., Nat. Phys. 5 (2009)
0 1?...
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. Ith...
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 I...
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 01...
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...
22Schematic of the Full Processor
50 W
4K
readout I qubit I readout IIqubit II
outcome 00, 01, 10, or 11
0 or 1
( ) ( )0...
23
a)
100 m
1 mm
Realization of the Processor
1m
23
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
...
25Qubit Spectroscopy
fd,A(t)
time
readout pulsedrive pulse
f01
f02/2
1 us
25
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....
27Single-Qubit Gate Characterization
x
y
z
|0>
|1>
27
28Performing Rabi Oscillations
x
y
z
|0>
|1>
WRabi=85 MHz
28
29Characterizing Energy Relaxation (1)
x
y
z
|0>
|1>
1=(456 ns)-1
WRabi=85 MHz
29
30
x
y
z
|0>
|1>
Characterizing Dephasing ()
 =(764 ns)-1 2 =(416 ns)-1
1=(456 ns)-1
WRabi=85 MHz
30
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]
reado...
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>
|...
33Choice of Processor Working Points
Qubit I Qubit II
33
readout
single-qubitgate
two-qubitgate
Readout Relaxation Model
34Characterizing Qubit-Qubit Interaction
time
f01,A(t)
f01
II
f01
I
readout I readout II
drive pulse
1 us
34
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
36Processor: Operating Principle
time
f01[f(t)],a(t)
Δfm
(150 MHz)
x/y rotations two-qubit readoutz rotations
5.1 GHz
6.2 ...
37Outline
Demonstrating Quantum Speed-Up
Introduction & Motivation
Realizing a Two-Qubit Processor
Realizing a Two-Qubit G...
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...
39Two-Qubit Density Matrix & Pauli Set
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|0
/2
3/2
Density Matrix Pauli Set

 ...
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 ...
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 Tomogra...
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

Expe...
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 acqui...
44Observing the coherent swapping
|00>
|01>
|10>
|11>
<00| <01| <10| <11|
swap duration [ns]
stateoccupationprobability
44...
45Observing the coherent swapping
|00>
|01>
|10>
|11>
<00| <01| <10| <11|
swap duration [ns]
stateoccupationprobability
45...
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 ...
47Characterizing the iSWAP Gate
in
out
47
 
2
0 , 1 , 0 1 , 0 1i
in i

  
48
†
( )out i i
i
i
j
i jn j nE E   
 
2
, , ,i X Y ZE i  

 
Process Tomography of the  iSWAP

(element...
49Fidelity & Error Budget of the Gate
90
8%
2%
90%
Error Budget
fidelity
decoherence
(mainly relaxation)
unitary errors
1 ...
50Outline
Introduction & Motivation
Realizing a Two-Qubit Processor
Realizing a Two-Qubit Gate
Demonstrating Quantum Speed...
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 Su...
53gg
H D
0 1
0 1
Oracle (O)


















1111
1111
1111
1111
2
1
y

yx
xy
The Two-Qubit Grover ...
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...
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 i...
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
i...
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 > 2...
58Summary
Realized a Two-Qubit Processor following the
DiVincenzo criteria.
58
Characterized a
Universal 2-Qubit Gate
with...
59Outlook: Processor Scaling Problems
59
1. Hard / Impossible to switch off coupling
2. Frequency Crowding of Qubits
3. Ex...
60(Partial) Solution: New Architecture
60
cell 2
  
high Q
coupler
cell n
… …
readout
pulses
cell 1
Z drives,
function ...
61Acknowledgments
Thank you!
Special Thanks to Florian Ong & Romain Lauro as well as group
technicians: Pascal Senat, Thom...
62Supplementary Material
62
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 Pre...
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 c...
65gg
|0>
f
x
x
diffusion
operator
0 1
0 111 xx 
Oracle Function (R)


















1111
1111
1111
...
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 basi...
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]
reado...
68Experimental State Tomography
t = 0 ns t = 31 ns (iSWAP)
|11>
|00>
|01>
|10>
<00|
<01|
<10|
<11|
|11>
|00>
|01>
|10>
<0...
69Desired Process: iSWAP gate















2000
010
010
0002
2
1
SWAP
i
i
i
|11>
|00>
|01>
|10>
<00|
<01...
70Measurement of Pauli Set During SWAP
70
71Measurement of Pauli Set During SWAP
71
72gg
III) Towards more scalable elementary processors
« n+1 in line » architecture based on frequency agility, individual ...
Vous êtes cordialement invités à la soutenance ainsi
qu'au pot qui suivra.
Soutenance de Thèse
Andreas Dewes
Demonstrating...
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Demonstrating Quantum Speed-Up with a Two-Transmon Quantum Processor Ph.D. defense, UPMC / CEA, 15/11/2011

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The accompanying slides of my PhD defense presentation on experimental quantum computing, held at the CEA Saclay in November 2012.

Please not that some slides appear "broken" due to the animation sequences they contain, to get a correct view of the presentation, just download the PPTX.

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Demonstrating Quantum Speed-Up with a Two-Transmon Quantum Processor Ph.D. defense, UPMC / CEA, 15/11/2011

  1. 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. 2. 2Outline Realizing a Two-Qubit Processor Realizing a Two-Qubit Gate Demonstrating Quantum Speed-Up Introduction & Motivation 2
  3. 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. 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. 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. 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. 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. 8. 8Outline Realizing a Two-Qubit Gate Demonstrating Quantum Speed-Up Introduction & Motivation Realizing a Two-Qubit Processor 8
  9. 9. 9The Cooper Pair Box 200 nm 1 0 EC Cg EJ 9
  10. 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. 11. 11The Cooper Pair Box zH  ˆ 2 ˆ 01   EC E  Cg 01 1 0 EJ  |0> |1> 11
  12. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 23 a) 100 m 1 mm Realization of the Processor 1m 23
  24. 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. 25. 25Qubit Spectroscopy fd,A(t) time readout pulsedrive pulse f01 f02/2 1 us 25
  26. 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. 27. 27Single-Qubit Gate Characterization x y z |0> |1> 27
  28. 28. 28Performing Rabi Oscillations x y z |0> |1> WRabi=85 MHz 28
  29. 29. 29Characterizing Energy Relaxation (1) x y z |0> |1> 1=(456 ns)-1 WRabi=85 MHz 29
  30. 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. 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. 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. 33. 33Choice of Processor Working Points Qubit I Qubit II 33 readout single-qubitgate two-qubitgate Readout Relaxation Model
  34. 34. 34Characterizing Qubit-Qubit Interaction time f01,A(t) f01 II f01 I readout I readout II drive pulse 1 us 34
  35. 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. 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. 37. 37Outline Demonstrating Quantum Speed-Up Introduction & Motivation Realizing a Two-Qubit Processor Realizing a Two-Qubit Gate 37
  38. 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. 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. 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. 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. 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. 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. 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. 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. 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. 47. 47Characterizing the iSWAP Gate in out 47   2 0 , 1 , 0 1 , 0 1i in i    
  48. 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. 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. 50. 50Outline Introduction & Motivation Realizing a Two-Qubit Processor Realizing a Two-Qubit Gate Demonstrating Quantum Speed-Up 50
  51. 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. 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. 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. 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. 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. 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. 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. 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. 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. 60(Partial) Solution: New Architecture 60 cell 2    high Q coupler cell n … … readout pulses cell 1 Z drives, function selectors XY drives
  61. 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. 62. 62Supplementary Material 62
  63. 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. 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. 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. 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. 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. 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. 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. 70. 70Measurement of Pauli Set During SWAP 70
  71. 71. 71Measurement of Pauli Set During SWAP 71
  72. 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. 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

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