Charting the cQED Design LandscapeUsing Optimal Control Theory

ih ∂∂t|Ψ〉 = H(t) |Ψ〉

|00〉 O |00〉

|01〉 O |01〉

|10〉 O |10〉

|11〉 O |11〉

E(1)(t) E(0)(t)

t0 T

∆E(t)

t

Quantum Dynamics and Control

Michael Goerz1,2,3, Felix Motzoi4,5, Birgitta Whaley5, Christiane P. Koch1

1Institut fur Physik, Universitat Kassel, Germany 2Ginzton Lab, Stanford University 3Army Research Lab, Adelphi, MD4Department of Physics and Astronomy, Aarhus University, Denmark 5Department of Chemistry, UC Berkeley

AbstractWith recent improvements in coherence times, superconducting transmon qubits have be-come a promising platform for quantum computing. They can be flexibly engineered over awide range of parameters, but also require us to identify an efficient operating regime. Usingstate-of-the-art quantum optimal control techniques, we exhaustively explore the landscapefor creation and removal of entanglement over a wide range of design parameters. Weidentify an optimal operating region outside of the usually considered strongly dispersiveregime, where multiple sources of entanglement interfere simultaneously, which we name thequasi-dispersive straddling qutrits (QuaDiSQ) regime. At a chosen point in this region, auniversal gate set is realized by applying microwave fields for gate durations of 50 ns, witherrors approaching the limit of intrinsic transmon coherence. Our systematic quantum opti-mal control approach is easily adapted to explore the parameter landscape of other quantumtechnology platforms.

arXiv: 1606.08825 (accepted in npj Quantum Information)

1 Two Transmon Qubits Coupled via Cavity

resonator !Sec. V". This type of approach was also discussedin Refs. #14,29$ and here we will present various mecha-nisms to tune this type of interaction. The dispersive regimethat is the basis of the schemes relying on virtual excitationsof the resonator can also be used to create probabilistic en-tanglement due to measurement. This is discussed in Sec. VI.Next, we consider gates that are based on real photon popu-lation of the resonator !Sec. VII". For these gates, selectionrules will set some constraints on the transitions that can beused. Finally, we discuss a gate based on the geometric phasewhich was first introduced in the context of ion trap quantumcomputation #46,47$.

Before moving to two-qubit gates, we begin in Sec. IIwith a brief review of circuit QED and, in Sec. III, with adiscussion of single-qubit gates. A table summarizing the ex-pected rates and quality factors for the different gates is pre-sented in the concluding section.

II. CIRCUIT QED

A. Jaynes-Cummings interaction

In this section, we briefly review the circuit QED archi-tecture first introduced in Ref. #14$ and experimentally stud-ied in Refs. #3,15–17$. Measurement-induced dephasing wastheoretically studied in Ref. #18$. As shown in Fig. 1, thissystem consists of a superconducting charge qubit #1,48,49$strongly coupled to a transmission line resonator #50$. Nearits resonance frequency !r, the transmission line resonatorcan be modeled as a simple harmonic oscillator composed ofthe parallel combination of an inductor L and a capacitor C.Introducing the annihilation !creation" operator a!†", the reso-nator can then be described by the Hamiltonian

Hr = !ra†a , !2.1"

with !r=1/%LC and where we have taken "=1. Using thissimple model, the voltage across the LC circuit !or, equiva-lently, on the center conductor of the resonator" can be writ-ten as VLC=Vrms

0 !a†+a", where Vrms0 =%"!r /2C is the rms

value of the voltage in the ground state. An important advan-tage of this architecture is the extremely small separation b&5 #m between the center conductor of the resonator andits ground planes. This leads to a large rms value of theelectric field Erms

0 =Vrms0 /b&0.2 V/m for typical realizations

#3,15–17$.

Multiple superconducting charge qubits can be fabricatedin the space between the center conductor and the groundplanes of the resonator. As shown in Fig. 1, we will considerthe case of two qubits fabricated at the two ends of the reso-nator. These qubits are sufficiently far apart that the directqubit-qubit capacitance is negligible. Direct capacitive cou-pling of qubits fabricated inside a resonator was discussed inRef. #29$. An advantage of placing the qubits at the ends ofthe resonator is the finite capacitive coupling between eachqubit and the input or output port of the resonator. This canbe used to independently dc bias the qubits at their chargedegeneracy point. The size of the direct capacitance must bechosen in such a way as to limit energy relaxation anddephasing due to noise at the input-output ports. Some of thenoise is however still filtered by the high-Q resonator #14$.We note, that recent design advances have also raised thepossibility of eliminating the need for dc bias altogether #17$.

In the two-state approximation, the Hamiltonian of the jthqubit takes the form

Hqj= !

Eelj

2$xj

!EJj

2$zj

, !2.2"

where Eelj=4ECj

!1!2ngj" is the electrostatic energy and EJj

=EJj

maxcos!%& j /&0" is the Josephson coupling energy. Here,ECj

=e2 /2C'jis the charging energy with C'j

the total boxcapacitance. ngj

=CgjVgj

/2e is the dimensionless gate chargewith Cgj

the gate capacitance and Vgjthe gate voltage. EJj

max isthe maximum Josephson energy and & j the externally ap-plied flux with &0 the flux quantum. Throughout this paper,the j subscript will be used to distinguish the different qubitsand their parameters.

With both qubits fabricated close to the ends of the reso-nator !antinodes of the voltage", the coupling to the resonatoris maximized for both qubits. This coupling is capacitive anddetermined by the gate voltage Vgj

=Vgj

dc+VLC, which con-tains both the dc contribution Vgj

dc !coming from a dc biasapplied to the input port of the resonator" and a quantum partVLC. Following Ref. #14$, the Hamiltonian of the circuit ofFig. 1 in the basis of the eigenstates of Hqj

takes the form

H = !ra†a + '

j=1,2

!aj

2$zj

! 'j=1,2

gj!# j ! cj$zj+ sj$xj

"!a† + a" ,

!2.3"

where !aj=%EJj

2 + #4ECj!1!2ng,j"$2 is the transition fre-

quency of qubit j and gj =e!Cg,j /C',j"Vrms0 /" is the coupling

strength of the resonator to qubit j. For simplicity of nota-tion, we have also defined # j =1!2ng,j, cj =cos ( j and sj=sin ( j, where ( j =arctan#EJj

/ECj!1!2ng,j"$ is the mixing

angle #14$.When working at the charge degeneracy point ng,j

dc =1/2,where dephasing is minimized #5$, and neglecting fast oscil-lating terms using the rotating-wave approximation !RWA",the above resonator plus qubit Hamiltonian takes the usualJaynes-Cummings form #77$

FIG. 1. !Color online" Layout and lumped element version ofcircuit QED. Two superconducting charge qubits !green" are fabri-cated inside the superconducting 1D transmission line resonator!blue".

BLAIS et al. PHYSICAL REVIEW A 75, 032329 !2007"

032329-2

superconducting qubits [1] inside atransmission line resonator, Fig. from [2]

Parameters:

• ω1 = 6.0 GHz• ω2 = 5.0 – 7.5 GHz (vary)• ωc = 4.5 – 11.0 GHz (vary)• α1 = −290 MHz• α2 = −310 MHz• g = 70 MHz• τc = 3.2 µs [3];• τq = 13.3 µs [4]

Hamiltonian in the rotating wave approximation (δq = ωq − ωd):

H = ~δca†a +∑q=1,2

[~δqb

†qbq +

αq2b†qb†qbqbq + g(b

†qa + bqa

†)]

+~2

(ε∗(t)a + ε(t)a†

)avoid treating only dispersive regime by numerical simulation of full Hamiltonian!

relevant parameters to describe landscape: ∆2/α = (ω2 − ω1)/α, ∆c/g = (ωc − ω1)/g

field-free properties of the Hamiltonian:

�20 �10 0 10 20 30 40�c/g

�3�2�1

0123

�2/↵

10�1

100

101

102

�20 �10 0 10 20 30 40�c/g

12

3

45

67

8

910

101

102

103

�20 �10 0 10 20 30 40�c/g

1.01.21.41.61.82.02.2

⇣ (MHz) T (C0 = 1) (ns) �dressed/�bare

a b c!c 6200

6235 MHz

!1 60005982 MHz

!2 59005882 MHz

�80

�40

0

40

80

ener

gysh

ift(M

Hz)

5400 5600 5800 6000 6200 6400 6600!c (MHz)

�8 �6 �4 �2 0 2 4 6 8�c/g

�80

�40

0

40

80

ener

gysh

ift(M

Hz) ⇣

!1 ⌘ 6000 MHz!2 ⌘ 5900 MHz

�80

�40

0

40

80

ener

gysh

ift(M

Hz)

�Ecav

�E01

�E10

�E01

�E10

5400 5600 5800 6000 6200 6400 6600!2 (MHz)

�2 �1 0 1 2�2/↵

�80

�40

0

40

80

ener

gysh

ift(M

Hz)

!1 ⌘ 6000 MHz!c ⌘ 6200 MHz

�E01 +�E10

�E11

d

e

f

g

field-free entanglement ζ = E00 + E01 + E10 − E11 from interfering dressed energy shifts.ζ varies by order of magnitudes, while dressed decay only varies up to factor 2.3!

2 Entanglement creation and destructionFor each point (ω2, ωc): find pulse to maximize entanglement (two-qubit gate) and pulse toimplement local gate ∈ SU(2)⊗ SU(2)), using multi-stage optimization scheme [5].

1. Random Search

2. Gradient-free optimization of analytical pulse parameters

Use Nelder-Mead downhill simplex to minimize/maximize entanglement

3. Gradient-based optimization (Krotov’s method) for fine-tuning

Use Krotov’s method [6] to continue optimization of pulse for arbitrary perfect entangler [7]and arbitrary local gate ∈ SU(2)⊗ SU(2), based on Cartan decomposition [8]

U = k1 exp

[i

2

(c1σxσx + c2σyσy + c3σzσz

)]k2; k1,2 ∈ SU(2)⊗ SU(2)

c1/⇡0

1 c2/⇡

0

0.5

c 3/⇡

0.5

Q

P

A3

A2

A1ML

O

N

c1/⇡0

1 c2/⇡

0

0.5

c 3/⇡

0.5

Q

P

A3

A2

A1ML

O

N

c1/⇡0

1 c2/⇡

0

0.5

c 3/⇡

0.5

Q

P

A3

A2

A1ML

O

N

T = 200 ns T = 50 ns T = 10 nsa b c

optimization result for maximizing/minimizing entanglement:

�3�2�1

0123

�2/↵

�3�2�1

0123

�2/↵

�3�2�1

0123

�2/↵

�20 �10 0 10 20 30 40�c/g

�3�2�1

0123

�2/↵

�20 �10 0 10 20 30 40�c/g

0.0

0.2

0.4

0.6

0.8

1.0

C0

0.0

0.2

0.4

0.6

0.8

1.0

CS

Q

0.0

0.2

0.4

0.6

0.8

1.0

CP

E

�20 �10 0 10 20 30 40�c/g

0.0

0.2

0.4

0.6

0.8

1.0

CP

E⇥

(1�

CS

Q)

T = 200 ns T = 50 ns

field-free

minimization

maximization

combined

T = 10 nsa

d

g

b

e

h

c

f

i

k m n

optimal point for universal gate implementation in quasi-dispersive “straddling qutrit” regime(“QuaDiSQ”)

quantum speed limit (QSL):

• decay of qubit imposes limit on thelowest reachable gate error

•QSL at 10 ns when perfect entanglercan no longer be realized at minimum error

• at point X: QSL for universal set at 50 ns

10�3

10�2

gate

erro

r

QSL PE"0

avg

"PEavg

101 102gate time (ns)

10�3

10�2

gate

erro

r

QSL XH1

"0,Xavg

"H1,Xavg

a

b

3 A universal set of gatesoptimize for all gates of the universal set at point X in QuaDiSQ regime

−500 0 500δ (MHz)

0

50

1002 1

0 10 20 30 40time (ns)

−300−100

1000

100

200

300

0 10 20 30 40time (ns)

0.00.10.2

−500 0 500δ (MHz)

2 1

0 10 20 30 40time (ns)

0 10 20 30 40time (ns)

−500 0 500δ (MHz)

2 1

0 10 20 30 40time (ns)

0 10 20 30 40time (ns)

−500 0 500δ (MHz)

2 1

0 10 20 30 40time (ns)

0 10 20 30 40time (ns)

−500 0 500δ (MHz)

2 1

0 10 20 30 40 50time (ns)

0 10 20 30 40 50time (ns)

Hadamard (1)

εno diss.avg = 6.3× 10−4

εdiss.avg = 4.2× 10−3

Im[Ψ

(t)]

|00〉|0

1〉|1

0〉|1

1〉

Re[Ψ(t)]

Hadamard (2)

εno diss.avg = 9.1× 10−4

εdiss.avg = 4.6× 10−3

Re[Ψ(t)]

Phasegate (1)

εno diss.avg = 9.0× 10−4

εdiss.avg = 4.6× 10−3

Re[Ψ(t)]

Phasegate (2)

εno diss.avg = 3.7× 10−4

εdiss.avg = 4.0× 10−3

Re[Ψ(t)]

BGATE

εno diss.avg = 6.5× 10−4

εdiss.avg = 4.3× 10−3

Re[Ψ(t)]

|F(ε

)|(a

rb.

un.)

dφ dt

(MH

z)|ε|(

MH

z)P

outs

ide

For longer gate duration (T = 100 ns) spectral width can be constrained to ±200 MHz, andpulses can be made robust w.r.t. 1% fluctuation in pulse amplitude (see preprint).

References

[1] J. Koch et al., Phys. Rev. A 76, 042319 (2007), J. Majer et al., Nature 449, 443 (2007)

[2] A. Blais et al., Phys. Rev. A 75, 0332329 (2007)

[3] M. J. Peterer et al., Phys. Rev. Lett. 114, 010501 (2015)

[4] A. W. Cross and J. M. Gambetta, Phys. Rev. A 91 032325 (2015)

[5] M. H. Goerz, K. B. Whaley, and C. P. Koch, EPJ Quantum Technology 2, 21 (2015)

[6] D. M. Reich, M. Ndong, and C. P. Koch, J. Chem. Phys. 136, 104103 (2012).

[7] P. Watts et al., Phys. Rev. A 91, 062306 (2015); M. H. Goerz et al., Phys. Rev. A 91,062307 (2015)

[8] J. Zhang et al., Phys. Rev. A 67, 042313 (2003).