quantum simulation with superconducting...

Quantum simulation with

superconducting qubits

Hans MooijKavli Institute of Nanoscience

Delft University of Technology

KITP Santa Barbara - Workshop on Quantum Information Science

November 4, 2009

1. introduction superconducting qubits

2. flux qubits

3. quantum simulation with quantum vortices and flux qubits

superconducting order parameter

2o

h

eΦ = loop

2oV

t

ϕπ

Φ ∂=∂

2 2ioi

nϕ π πΦ= +Φ∑

ie ϕΨΨ =

Josephson junction

CIo

100 nm

2242

ooJ

C

E I

eEC

πΦ=

≡

2

(1 cos )2

sin

JC

o

QU E Ee

dQI Idt

ϕ

ϕ

= + −

= +

U

ϕϕϕϕ

circuit with Josephson junctions

ϕ1..ϕn phase differences+constraints

U(ϕi.. ϕn): potential energy

Ci(dϕi/dt)2 terms: kinetic energy

set of conjugate variables: Lagrangian, Hamiltonian

quantization

set of islands with discrete numbers of Cooper pairs

junctions provide off-diagonal coupling of charge states

charge qubit

charge qubit

0.5ΦΦΦΦoe

fluxoid qubit

flux qubit

oscillation qubit

phase qubittransmon

δE

0.5 ng

oscEδ ω=ћδE

0.5 f

ng=CVg/2e f=ΦΦΦΦ/ΦΦΦΦ0

2( ) cosg JCH E n n E ϕ= − −

2ˆ ˆ( ) cosˆ g JCH E n n E ϕ= − −

ˆ,n̂ iϕ

=

(classical Hamiltonian)

0 1 ng→

↑↑↑↑E/EC

0

1

EJ

0

1

EC/EJ>>1

around ng=0.5

other levels extremely high

ˆ { ( 0.5) }/2zx gJ CH E E nσ σ=− + −

quantum Cooper pair box: charge qubit

-0.2 0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

f

Uind/EL

n=1 n=0

0 1f

E/EL

0

1

∆0

1

around f=0.5

ˆ { ( 0.5) }/2zx LH E fσ σ=− ∆ + −

exp JJ C

C

Ea E E b

E

∆ = −

L fΦΦΦΦo EJEC

∆

E

ϕϕϕϕ

flux qubit EJ>>EC

Phase Qubitplasma oscillation of a current-biased single Josephson junction

2 4 6

-15

-10

-5

5

0.5

0.95

I/Io=0E/EJ

ϕ/2π

2.4 2.5 2.6 2.7

-6.48

-6.46

-6.44

3

2

1

Martinis group Santa Barbara

Nature 459, 546 (2009)

Nature 461, 504 (2009)

Koch et al. PRA 76, 042319 (2007), Schreier et al., PRB 77, 180502 (2008)

Yale group: Schoelkopf, Girvin

transmon: Cooper pair box with strongly reduced EC

coupled to high Q harmonic oscillator

external capacitive shunt

EJmax/h= 18 GHz

ECeff/h= 1.4 Ghz

νpl~ 6 GHz

increasing EJ leads to

- gradually decreasing anharmonicity

- exponentially decreasing dependence of δE on ng

circuit quantum electrodynamics

Nature 460, 240 (2009)

sources of decoherence

driving circuits: gate, bias flux, microwave lines, measurement

engineering design, calculation possible

microscopic defects

1/f noise due to many fluctuators

parasitic two-level systems with same energy splitting

smaller qubits, fewer defects

dephasing times > 1 µs reached in most qubit types

at optimal conditions

level separation 3-15 GHz, operation time 5-50 ns

1 µm

Φ~Φo/2

↓↓↓↓ ↑↑↑↑flux bias Φo/2

currents ± Ip

0

-1

0

1

0.5

↑↑↑↑Icirc

↑↑↑↑E

Φ/Φo→

∆

+Ip

-Ip

0

ground stateexcited state

( )z xH εσ σ= + ∆½( / 0.5)2o o pIε = Φ Φ − Φ

Mooij et al. Science 285, 1036 (1999), Orlando et al. PRB (1999)Van der Wal et al. Science 290 1140 (2000)

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

ϕϕϕϕ1111/π/π/π/π

ϕϕϕϕ2222/π/π/π/π

ϕϕϕϕ1111/π/π/π/π

α=1.α=1.α=1.α=1. α=0.75α=0.75α=0.75α=0.75αααα

exp JJ C

C

Ea E E b

Eα

∆ = −

αααα

f

Ip= 428 nA∆= 1 GHz

Ip= 360 nA∆= 4.6 GHz

Ip=281 nA∆= 8 GHz

Pswitch

α α α α =0.71 αααα=0.64 α α α α =0.60

20 mΦo

F.G. Paauw et al. PRL 102, 090501 (2009)

( )z xH εσ σ= + ∆½( / 0.5)2o o pIε = Φ Φ − Φ

tunable ∆∆∆∆

controlled-NOT gate

Plantenberg et al. Nature 447,836 (2007)

|11⟩⟩⟩⟩

|01⟩⟩⟩⟩

|10⟩⟩⟩⟩

|00⟩⟩⟩⟩

ππππ

2o

ko

LIπ

Φ=

1,2 1,5 1,8

0

5

10

Vsw (V)

dPH

/dV

sw (

V -

1 )

0 pi/2 pi

A. Lupascu et al. Nature Physics 3, 119 (2007)

oscillator state h ⇔⇔⇔⇔qubit ground stateoscillator state l ⇔⇔⇔⇔ qubit excited state

P(h)P(hh)P(ll)

∆t1 (ns)2 64

∆∆∆∆t2 set to give 2ππππ pulse

P(%)

100

0

50 µµµµm

Pieter de Groot - 2-qubit sample with bifurcative readout

cross-talk between readout systems < 0.1 %

tunable-αααα qubit coupled to low Q oscillator

vacuum Rabi oscillation around symmetry point

Arkady Federov

vortex in 2D Josephson junction array: particle moving in 2D potential EJmass determined by junction capacitance 1/EC

quantum vortices

in 2D or quasi-1D array of Josephson junctions

Bose Einstein condensationPRL 69, 2971 (1992)

quantum interference around charge

PRL 71, 2311 (1993)Anderson localizationPRL 77, 4257 (1996)

Mott insulatorPRL 76, 4947 (1996)

Berezinskii, Kosterlitz,Thouless transition

PRB 35, 7291 (1987)

superconductor-insulator transition EJ/EC

PRL 63, 326 (1989)

arXiv:cond-mat/0108266 (2001)

New J of Phys 7, 181 (2005)

chain of flux qubits: excitation at one end, readout at the other end

1D coupled chains of flux qubitsFloor Paauw, Delft

alfa-qubits

coupling

αααα loop

1i iz xσ σ +

4 1

23

4 degenerate states

1 2

2 degenerate states

trapped vortex in junction array

A B

CD

Iv

Ih

4 degenerate states

A B

Iv

2 degenerate states

ϕϕϕϕi

17 junctions

(12-1) loop equations

6 phase variables

EJ>>EC

if vertical symmetry is assumed: 3 variables

6 charge variables

EC>>EJ

trapped fluxoid: two positions

central junction q times larger

0.25 0.5 0.75 1 1.25 1.5 1.75

0.1

0.2

0.3

0.4

potential energy in phase landscape, symmetric solutionphase difference across central junction imposed, rest optimized

q=1.3

ϕcentral/π

potential energy/EJ

0.25 0.5 0.75 1 1.25 1.5 1.75

q=1.5

1.4

1.3

1.2

1.1

1.0

(U-Umin)/EJ

0.2

0.0

0.4

1.00 0.5 1.5

1.1 1.2 1.3 1.4 1.5 1.61.0 1.2 1.4 1.6

q

Ubar/EJ

0.0

0.2

0.4

0.25 0.5 0.75 1 1.25 1.5 1.75

-0.1

0

0.1

0.2

0.3

0.4

flux tilt

I

quantum calculation (Jos Thijssen et al.)

q=1.3, m=5, symmetric case (3 degrees of freedom)

classical : crosses

↑↑↑↑energy current tilt →→→→

9 degrees of freedom

square array with one trapped fluxoid

4 equivalent fluxoid positions

0.4 0.6 0.8 1.2 1.4 1.6

0.002

0.004

0.006

0.008

0.4 0.6 0.8 1.2 1.4 1.6

0.01

0.02

0.03

0.04

0.05

0.06

0.4 0.6 0.8 1.2 1.4 1.6

0.05

0.1

0.15

0.2

q=0. 6 q=1.0 q=1.4

barriers

center junctions q times larger

than other junctions

1.1 1.2 1.3 1.4 1.5qv=qh=q

0.05

0.1

0.15

0.2

0.25

0.3

Ubar êEJ

barrier as a function of central junction strength

1

3 2

4Ix

Iy

Ix >0: 1 and 4 lower energy

Iy >0: 1 and 2 lower energy

0.4 0.6 0.8 1.2 1.4 1.6

0.05

0.1

0.15

quantum calculation Jos Thijssen, 9 degrees of freedom

energy

tilt induced by current

classical

IV I

IIIII

Vg

W.J. Elion, J.J. Wachters, L.L. Sohn, J.E. Mooij, PRL 71, 2311 (1993)

Quantum interference of vortices around charge, Aharonov-Casher

-3 -2 -1 1 2 3

-4

-2

2

4

Ix

Ix=0, Iy=0 Ix=2, Iy=0

Iy

Ix14

23

1

2

3

4

00

00

e

e

e

e

∆ ∆∆ ∆

∆ ∆∆ ∆

E/∆

E/∆ eigenvector

2 (-1,1,-1,1)

0 (-1,0,1,0)

0 (0,-1,0,1)

-2 (1,1,1,1)

E/∆ eigenvector

3.2 (-1,4.2,-4.2,1)

1.2 (1,-4.2,-4.2,1)

-1.2 (-4.2,-1,1,4.2)

-3.2 (4.2,1,1,4.2)

.57 .95 .32 .27

.38

.38

.99

.63

.33

.59

.16

.63

.99

1.29

1.30

.60

.76

.30

.16

.30

.38

.59

.33

.76

.60

.46

.46

.38

.27 .32 .95 .57

.57

.95

.27

.32

.27

.11

.41

.79

.79 .41 .11 .27

.16

.59

.33

.63

.99

.38

.38

.30

.76

.60

1.30

1.30

.99

.63

.38

.30

.16

.38

.46

.46

.60

.76

.33

.59

.79 .41 .11 .27

.27

.32

.57

.95

.79

.41

.11

.27

.27 .32 .95 .57

.38

.38

.46

.30

.76

.16

.38

.30

.46

.60

.60

1.30

.33

.30

.59

.63

.38

.16

.76

.33

1.30

.99

.99

.38

.27 .11 .41 .79

.79

.41

.27

.11

.27

.32

.95

.57

.27

.11

.79

.41

.59

.16

.76

.30

.46

.38

.38

.63

.33

1.29

.60

.60

.46

.30

.38

.63

.59

.38

.99

.99

1.30

.33

.76

.16

.57 .95 .32 .27

.57

.95

.32

.27

.27 .11 .41 .79

1111

22223333

4444 1111

22223333

4444

a bclassical

nearest neighbor interaction

1.0 4.9 3.3 5.6

4.9 1.0 5.6 3.3

2.1 8.9 1.0 4.9

8.9 2.1 4.9 1.0

− − + + − − + + + + − − + + − −

a=1

b=1

2 3 4

2

3

4

+ +8.9+8.9+8.9+8.9 −−−−1.01.01.01.0

−−−−4.94.94.94.9+2.1+2.1+2.1+2.1

+ −−−−1.01.01.01.0 +5.6+5.6+5.6+5.6

+3.2+3.2+3.2+3.2−−−−4.94.94.94.9

++++

++++

++++

++++

++++

++++

++++

++++

++++

1D chain

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

2D array

++++

++++

++++

++++

++++

++++

++++

++++

++++

1D chain

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

++++

IV I

IIIII

Iy

Ix

4 degenerate statesVg

Berry phase

H(p1,p2)

make closed loop through parameter space

wave function picks up geometric phase and dynamical phase

retrace loop backwards to eliminate dynamical phase, but do something smart to double geometric phase

e-change on island charge?