Frank K. Wilhelm Saarland University, Saarbrücken ... - ITP at UCSB

Tuesday, February 26, 2013 

Making optimal control 

work for 

superconducting qubits 

Frank K. Wilhelm 

Saarland University, Saarbrücken, Germany 

1


Topics 

• Superconducting qubits for control 

theorists 

• Control tools 

• Control tasks 

• Application: Controlled-Z gates 

• Closing the loop - control and tuneup 

2


Superconducting qubits 

for control theorists 

3


Superconducting artificial atoms 

Linear LC-oscillator 

Josephson junction: 

nonlinear inductor 

J. Clarke, FKW, Nature 2008 

4


Superconducting artificial atoms 

Linear LC-oscillator 

Phase qubit, e.g. 

Josephson junction: 

nonlinear inductor 

Anharmonic 

energy levels 

J. Clarke, FKW, Nature 2008 

4

Bus, memory ... 

A. Blais et al., PRA 2004 

Schoelkopf and Girvin, Nature 2008 


Circuit QED 

E. Lucero, UCSB 

Key element: 

Superconducting 

resonator at 

microwave 

frequencies 

5

• 

• 

• 

• 

• 

• 


Control implications 

Not a spin - higher levels 

Human made ... parameter 

uncertainty 

Cryogenic, heavily filtered setup 

Operated at microwave 

frequencies 

Strong inter-element coupling 

Aiming at unitary gates 

6


Control tools 

8

Challenge 

Change performance index J in the end in response to uj 

Avoid numerical gradients 

Back-propagation: In step j optimize Ûback = Û † (tj,T) Ûgate 


9

!"#$%"&'()*&+$,-.'/01 

Adapting to experiments 

Include linear filters using transfer matrices 

2+).'-,%+#3'*,&4.'/#41 

Motzoi et al., PRA 2011 


Filters, Wires, 

Antennae ... 

!"#$%"&'()*&+$,-.'/01 

2+).'-,%+#3'*,&4.'/#41 

12

!"#$%"&'()*&+$,-.'/01 



2+).'-,%+#3'*,&4.'/#41 




Antennae ... 

!"#$%"&'()*&+$,-.'/01 

!"#$%"&'()*&+$,-.'/01 

2+).'-,%+#3'*,&4.'/#41 

2+).'-,%+#3'*,&4.'/#41 

12

!"#$%"&'()*&+$,-.'/01 



2+).'-,%+#3'*,&4.'/#41 

For linear systems: 

Include filter transfer matrix 

into GRAPE 




Antennae ... 

!"#$%"&'()*&+$,-.'/01 

!"#$%"&'()*&+$,-.'/01 

2+).'-,%+#3'*,&4.'/#41 

2+).'-,%+#3'*,&4.'/#41 

12


Control tasks 

13

Transmon/Phase qubit 

Lucero et al., 2008; 

Chow et al., 2009 


Leakage 

Motzoi et al., PRL 2009 15





Leakage 

Solution: Envelope shaping 

Derivative Removal 

by Adiabatic Gate 






Leakage 

Spectral limitation: 

Duration/bandwith uncertainty 

S 

ω 

12 

∆2 

Solution: Envelope shaping 

Derivative Removal 

by Adiabatic Gate 

ω 

01 

tg 

ω 


Solution DRAG 

Theorists dream ... experimental reality 


u1(t)cos ωt + u2(t) sin ωt 

u2 = ˙u1 

∆2 

Gambetta et al., 2011; 



16

Solution DRAG 




u2 = ˙u1 

∆2 

• 

• 

• 

CAD of analytical 

scheme 

amenable to long 

pixels 

third control can 

be removed - 

family of DRAG 

solutions 




16

Solution DRAG 




u2 = ˙u1 

∆2 

• 

• 

• 

CAD of analytical 

scheme 

amenable to long 

pixels 

third control can 

be removed - 

family of DRAG 

solutions 




16

Classical: Why derivative? 

Excitation profile at weak drive: Fourier transform 

Drive envelope 

Simple integration by parts: 

Motzoi et al., in preparation 


Detuning δ 

Ω(t) =Ω1(t)+iΩ2(t) 

17

Wah-wah pulses 

Sideband modulation, optimized with Magnus expansion 

Amplitude [GHz] 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

0 

-0.01 

0 2 4 6 8 10 12 14 16 

Time [ns] 

Ω X 

ΩY 

Sideband modulation 

+ DRAG gets 

ˆX ⊗ exp(−iα ˆ -2.5 

Z) -3 


Achieved Error 

10 0 

10 -1 

10 -2 

10 -3 

10 -4 

phase 

10 20 30 40 50 60 70 80 

0 

-0.5 

-1 

-1.5 

-2 

Gate Time [ns] 

Φ 

Φ |*,1> 

Φ |*,0> 

Φ avg 

10 20 30 40 50 60 70 80 

Gate Time [ns] 

α 

γ 

20

Control amplitude [Ghz] 

Numerical optimization 

Long pulse At the limit 

0.02 

0.015 

0.01 

0.005 

-0.005 

-0.01 

-0.015 

-0.02 


0 

Ω x (t) 

Ω y (t) 

0 20 40 60 80 100 120 

Time [ns] 

= 3.60507e-05 

= 1.83674e-05 

Amplitude [GHz] 

No speed limit other than sufficient # of pixels 

0.03 

0.02 

0.01 

0 

-0.01 

-0.02 

0 0.5 1 1.5 2 2.5 

Time [ns] 

Ω X 

ΩY 

Schutjens, Abu Dagga, Egger, FKW, in preparation 

21

Robustness of optimal pulses 


Spörl et al., 2007 

Montangero et al., 2007 

Khani et al., 2011 

22

Robustness of optimal pulses 

Timing jitter 

Slow noise 

Extremum - flat 


Spörl et al., 2007 

Montangero et al., 2007 

Khani et al., 2011 

22


Making a CZ work 

23

CPHASE is symmetric 


Baseline 

26


Asymmetric sequence: 

Ginger 

Fred 

iSWAP - Strauch - iSWAP 


Baseline 

26





Ginger 

Fred 

Baseline 

• 

• 

• 

three sequential 

steps 

need to correct 

phases 

third-level errors 

26





Ginger 

Fred 

Baseline 

Filtering extreme 

• 

• 

• 

1ns 4 ns 

three sequential 

steps 

need to correct 

phases 

third-level errors 

26


GRAPE pulse, no filter 

Strauch+ 

phase comp 

Ginger 

Fred 

(iSWAP)2+ 

Stark-shift correction 

Palindromic! 

27


The challenge: Tuneup 

• 

Translation of 

control voltage 

into qubit 

frequency not 

known precisely 

• Enters 

everywhere: High 

sensitivity 

29


Pulse debugging 

Phase / leakage level error landscape 

Experiment Theory 

• Map out error landscape and minimize by 

hand 

• seems to work 

30


Lessons learned and 

potential solutions 

• GRAPE pulses can work in superconducting 

qubits, but don‘t right now 

• Precise system characterization is crucial, 

robust GRAPE does not suffice (yet) 

• Nonlinear transfer functions limiting factor 

So we work on these issues 

31

• 

• 

• 

• 


Include the 

unmeasurable? 

Way out: Genetic 

algorithms 

Self-learning 

Challenge: Easy 

measurement of 

performance (possible in 

specific cases) 

Note: These are GAs for 

unitaries, seeded by 

pretty good guesses 

fs-chemistry: 

Rabitz, Gerber ... 

32

• 

• 

• 

• 

Based on Schirmer, Oi, 

Langbein, Wiseman, Ferrie ... 

Hamiltonian learning 

Example: Find resonators and 

couplings 

Input: SWAP-spectroscopy = 

nonlocal FID 

Previous: 10000 shots / point 

speedup by nonadaptive Bayesian 

update 

Y.R. Sanders and FKW, in preparation 


33

Bayesian tomography 

• Measure chi- 

Matrix efficiently 

• diligent use of 

priors 

• adaptive 

measurement 


 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 2000 4000 6000 8000 10000 

M 

M. Stenberg and FKW, in preparation 

34


Conclusions 

• Superconducting qubits and optimal control 

mutually beneficial 

• Solved higher-level leakage issue 

• Ultrafast gates in frequency-crowded 3Dtransmons 

• challenge in application: closing the controlcharacterization 

loop 

35

Frank K. Wilhelm Saarland University, Saarbrücken ... - ITP at UCSB

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