Superconducting Qubits

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c Nori Franco n r i o C c m r u t RIKEN, NSA, LPS, ARO, NSF, CREST NSF, ARO, LPS, NSA, RIKEN, e Ann Arbor, USA University of Michigan, n Advanced Science Institute, RIKEN, Japan RIKEN, Science Institute, Advanced p a

u u Q Yu-xi Liu, S. Ashhab, K. Maruyama, J.R. Johansson Yu-xi Liu, S. Ashhab, Main collaborators:

S members: Group Funding: Funding: J.Q. You, C.P. Sun, L.F. Wei, X.B. Wang, J.S. Tsai, Y. Nakamura, L.F. Wei, X.B. Wang, J.S. Tsai, J.Q. You, C.P. Sun, S. Savelev, M. Grajcar, A.M. Zagoskin, Y. Pashkin, T. Yamamoto, … F. Xue, H. Mooij, Quick overview (just few slides!) of several types of superconducting qubits

Short pedagogical review in:

You and Nori, Physics Today (November 2005)

Color reprints available online at dml.riken.jp t i b u q

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C Physics Today Figures from: You and Nori, t i b u q

x u l (November 2005) (November F Physics Today Figures from: You and Nori, Figures from: You and Nori, Physics Today (November 2005) End of the Quick overview of several types of superconducting qubits

For a short pedagogical overview, please see:

You and Nori, Physics Today (November 2005) Now we will focus on a few results from our group

No time today to cover other results s t n e t n o

m e t s Phys. Today

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Q Chiorescu et al, Reduced magnetic flux: Flux qubit: Symmetry and parity

In standard atoms, electric-dipole-induced selection rules for transitions satisfy : %lm&'%&' 1 and 0, 1

In superconducting qubits, there is no obvious analog for such selection rules.

Here, we consider an analog based on the

symmetry of the potential U((m, (p) and the interaction between: -) superconducting qubits (usual atoms) and the -) magnetic flux (electric field).

Liu, You, Wei, Sun, Nori, PRL (2005) Superconducting flux qutrits:

Symmetry of potential energy

Creating cyclic transitions (not allowed for normal atoms)

Turning on-and-off inter-level transitions

Using this for generating photons on demand e 0 ! ! & (1999)) f PRB 0 2 lowest lowest energy levels) ! i k p m three *+ P ) are(see, e.g., Orlando et al, / 1 pm

(( (,,) (here we consider the we consider (here

t I i 2

2 (()(, r m ,) J2 t P E /2 2 ; 2 (1 2 ) with capacitance junction the of u J2 e 0 1 2 1 2 q E ! 3 3 ( 3 3 (

J2 ) pm *+ *+ 2 E x p J p m J m & - & . & . 4 4 & &!&- MM P

u ( )/2; ( )/2; /( , ) p m p 22

l p m k k 2 1 cos cos 1 cos 2 2 MCMMC F &-- (

&.-.- 0 Phases and momenta (conjugate variables Effective masses

H U f U E E f 0 2 + + * * / 1 5

( /0 12 mp + ( +

((( (()( Symmetry Symmetry and parity m p m p (()(, ( Parity of U( , ) U

: t * * i J p m J m 95, 087001 (2005) 087001 95, J p m J m b 2 1cos cos 1cos2 1/2 ( , ) even function of and 2 1cos cos 1cos2 2 PRL u &.-- &6 &.-.-

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F Liu, You, Wei, Sun, Nori, You, Wei, Sun, Nori, Liu, Allowed three-level transitions in natural atoms

V - type 7$- type or ladder

8$- type No %$– type because of the electric-dipole selection rule. Some differences between artificial and natural atoms

In natural atoms, it is not possible to obtain cyclic transitions by only using the electric-dipole interaction, due to its well-defined symmetry.

However, these transitions can be obtained in a flux qutrit circuit, due to the broken symmetry of the potential of the flux qubit, when the bias flux deviates from the optimal point.

The magnetic-field-induced transitions in the flux qutrit are similar to atomic electric-dipole-induced transitions.

Liu, You, Wei, Sun, Nori, PRL (2005) Different transitions in three-level systems

V - type 7 - type or ladder ( f &$9:;$+

8$- type %$– type can be obtained using flux qubits (f away from 1/2)

Liu, You, Wei, Sun, Nori, PRL (2005) Superconducting flux qutrits:

Symmetry of potential energy

Creating cyclic transitions (not allowed for normal atoms)

Turning on-and-off inter-level transitions

Using this for generating photons on demand Superconducting flux qutrits:

Symmetry of potential energy

Creating cyclic transitions (not allowed for normal atoms)

Turning on-and-off inter-level transitions

Using this for generating photons on demand r e s e a c r m u o o r s c i n

m o Rev. B (2007). cond-mat (2005). t a o

h s p a - t e i l b g u n i q s You, Liu, Sun, Nori, Phys. x d u l n

F a Flux qutrit as a micromaser and single-photon source

You, Liu, Sun, Nori, Phys. Rev. B (2007)

We propose a tunable on-chip photon-generator using a superconducting quantum circuit.

By taking advantage of externally controllable state transitions, a state population inversion can be achieved and preserved. When needed, the circuit can generate a single photon. s t n e t n o

s m o t a

h t i w

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r Photon losses Weak atom-field coupling Atom in a cavity Atom in Atom Light sources Detectors T = 300 K a p m o

T = 30 mK t i

c r i qubit in a cavity Voltmeters and ammeters C Josephson junction device Dissipation in environment in Dissipation Electrodynamic environment Cavity Current and voltage sources Current and voltage Strong JJ-environment coupling Strong JJ-environment Circuit QED: Superconducting Yale group Yale

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f c g g J - c t (2003) 064509 68, PRB Nori, and You r i 4 ( / 2 ) cos !&!!

i &..

b ( ) 2 cos( / )

JXJX C u a cavity field: H E n C V e E

Hamiltonian

1) quantized field through the SQUID loop 1) quantized field applied to the gate voltage 2) quantized field The charge qubit two ways of interacting with has - * *+ 0 , ! n /0 >? 12

o 0 i ( t

c is replaced by

a X r !

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c E z z C x d < Q l &$! 22 11

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t ! i f - u t

i &..!-- c g g J !&!! c E cos

b ( ) 2 cos( / ) r JXJX 4 ( / 2 ) cos i u H I ga g a EE &.. C Q and we have H E n C V e E Then the magnetic flux Then the magnetic Quantized field is through the SQUID loop field is through Quantized Hamiltonian

JJ qubit photon Micromaser generator

JJ qubit in its ground Atom is thermally Before state then excited via excited in oven

Interaction JJ qubit interacts Flying atoms interact with with field via with the cavity field microcavity

Excited atom leaves Excited JJ qubit decays the cavity, decays to its After and emits photons ground state providing photons in the cavity.

Liu, Wei, Nori, EPL (2004); PRA (2005); PRA (2005) Interaction between the JJ qubit and the cavity field

Liu, Wei, Nori, EPL 67, 941 (2004); PRA 71, 063820 (2005); PRA 72, 033818 (2005)

details Cavity QED on a chip

Qubit is in The qubit returns carrier The qubit is in a superposed red sideband to its ground state its ground state state excitation and a superposition of the vacuum and single-photon states is created. s t n e t n o

C via variable frequency magnetic fields via variable frequency via data buses Scalable circuits Generating GHZ states. Testing Bell inequalities. Quantum tomography Conclusions Controllable coupling between qubits: Flux qubits on a chip Cavity QED qubits Coupling Qubits can be coupled either directly or indirectly, using a data bus (i.e., an “intermediary”) Let us now very quickly (fasten your seat belts!) see a few experimental examples of qubits coupled directly Capacitively coupled charge qubits

NEC-RIKEN Entanglement; conditional logic gates Inductively coupled flux qubits

A. Izmalkov et al., PRL 93, 037003 (2004) Entangled flux qubit states Inductively coupled flux qubits

J.B. Majer et al., PRL 94, 090501 (2005). Delft group Inductively coupled flux qubits

J. Clarke’s group, Phys. Rev. B 72, 060506 (2005) Capacitively coupled phase qubits

Berkley et al., Science (2003)

McDermott et al., Science (2005) Entangled phase qubit states Let us now consider qubits coupled indirectly using an LC databus

ting LC circuit has been observed. : s t i u c r i c

e l b a Delft, Nature, 2004 NTT, PRL 96, 127006 (2006) l a c

S Level quantization of a superconduc LC-circuit-mediated interaction between qubits between interaction LC-circuit-mediated + 0 X ! !

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a t qubit and a data bus a

d Current-biased junction Wei, Liu, Nori, PRB 71, 134506 (2005)

: g n i l p between the between u o c e l b a h c t

i The bus-qubit coupling is proportional to w Liu, Wei, Nori, EPL 67, 941 (2004) Liu, Wei, Nori, EPL 67, switchable coupling S A through the qubitthrough loops. could also be realized by changing the magnetic fluxes magnetic fluxes by changing the be realized could also Single-mode cavity field 89, 197902 (2002) via a common inductance via a common s using LC oscillating modes). t i Phys. Rev. Lett. u not c r i directly c

e l

b You, Tsai, and Nori, a l a c boxes, can couple any selected charge qubitsboxes, can couple the by common inductance ( S the SQUIDsSwitching on/off to the Cooper-pair connected Couple qubits Couple s t n e t n o

C via data buses via variable frequency magnetic fields via variable frequency (no data bus) Conclusions Scalable circuits Generating GHZ states. Testing Bell inequalities. Quantum tomography Controllable coupling between qubits: Controllable coupling Flux qubits on a chip Cavity QED Coupling qubits Coupling qubits ddiirreeccttllyy and (first) without VFMF (Variable Frequency Magnetic Flux)

H0 = Hq1 + Hq2 + HI = Total Hamiltonian

H I & MII12

M )EJ2 )EJ1

*;+ I *9+ I ! e 2 ! e 1

E E J2 EJ2 J1 EJ1 2 2 Pml Ppl ()()()l l l HEEEEql&---..2 Jl) Jl 2 Jl cos ( , ( ) ( m cos P Jl cos*+ 2 fl +2 m 22MMml pl

l=1,2 Now: let’s consider a Variable-Frequency- Magnetic-Flux (VFMF) Controllable couplings via VFMFs

Applying a Variable-Frequency Magnetic Flux (VFMF)

H &-H0 H* t + B(t) *9+ *;+ !e !e EJ1 EJ2 I (2) )EJ1 )EJ2 I (1) E EJ1 M I(1)( t ) I (2) J2 I(1)()()t@- I (1) I" t

Liu, Wei, Tsai, and Nori, Phys. Rev. Lett. 96, 067003 (2006) back

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a a g t m i o n t N i p l O p u ling of flux qubits, o c

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C The couplings in these two circuitssimilarly work Grajcar, Liu, Nori, Zagoskin, Nori, Grajcar, Zagoskin, Liu, Switchable coupling proposals (without using data buses)

Feature Weak Optimal No additional Proposal fields point circuitry

Rigetti et al. (Yale) No Yes Yes

Liu et al. (RIKEN-Michigan) OK No Yes Bertet et al. (Delft) OK Yes No Niskanen et al. (RIKEN-NEC) Grajcar et al. (RIKEN-Michigan)

Ashhab et al. (RIKEN-Michigan) OK Yes Yes

Depending on the experimental parameters, our proposals might be useful options in certain situations. details Now: let’s consider a Variable-Frequency- Magnetic-Flux (VFMF) and also a data bus s t n e t n o

C via variable frequency magnetic fields via variable frequency via data buses Scalable circuits Generating GHZ states. Testing Bell inequalities. Quantum tomography Conclusions Controllable coupling between qubits: Controllable coupling Flux qubits on a chip Cavity QED Coupling qubits A data bus using TDMF to couple several qubits

L I )E )EJ1 ! J2 (1) (2) e ! !dc dc + (2) - - (1) I !e I1 E I !e 2 I (1) I !(2)(t) J 1 ! (t) 2 !(t)

E E E C J1 EJ1 J2 J2

A data bus could couple several tens of qubits.

The TDMF introduces a nonlinear coupling between the qubit, the LC circuit, and the TDMF.

Liu, Wei, Tsai, Nori, PRL (2006) details Comparison between SC qubits and trapped ions

Superconducting Trapped ions Qubits circuits

Quantized Vibration mode LC circuit bosonic mode

Classical Lasers fields Magnetic fluxes s t n e t n o

g n i l Phys. Rev. B Phys. p u o c e Wei, Liu, Nori, d l a c i :

a m e a d i n y n i

D a

M Then single-qubit rotationscan be approximately obtained, even though the qubit-qubit interaction fixed. is (as a perturbation term) Then the interaction between qubits can be effectively incorporatedinto the single qubit term not very strong (coupling energy < qubit energy) Let us assume that the coupling between qubits the coupling assume that Let us is y t i l a (2005) u q e n i

’ Phys. Rev. B l l e

g n i Wei, Liu, Nori, coupled Josephson qubits inequality Bell’s test to t We propose how to use s e

T y t i l

a with a pair of capacitively u (2005) q decoupling approach to overcome e n i

s ’ l Rev. B Phys. l e

g testing Bell’s inequality n i Wei, Liu, Nori, t s e

T should allow coupled Josephson qubits. elemental logical gates for quantum computation. the “fixed-interaction” difficulty for effectively implementing 2) The proposed single-qubit operations and local measurements 1) Propose an effective dynamical . qubits

s e t a t s

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g n i (June 2006 t a r e n e macroscopic coupled superconducting We propose an efficient approach to produce and approach to an efficient We propose three of the quantum entanglement control

G We show that their Greenberger-Horne-ZeilingerWe show that their (GHZ) appropriate conditional operations. entangled states can be deterministically generated by Wei, Liu, Nori, Phys. Rev. Lett. s t n e t n o

y pr (2005); 014547 72, h p a r Phys. Rev. B tomographic reconstruction of g o m o t (2004); 874 67,

m u t for a general class of solid state systems in which state systems class of solid for a general n a Europhysics Letters u

Q Liu, Wei, Nori, Nori, Wei, Liu, We propose a method for the a method We propose qubit states This general method has been appliedto study superconducting charge qubitreconstruct any state. the qubit can be states for the spin Hamiltonians using experimental data. reconstructed by We analyze the implementation of the projective operator implementation of the projective We analyze the or spin measurements, on qubitmeasurements, states. All the Hamiltonians are represented by spin operators, e.g., with spin operators, by are represented the Hamiltonians XY- XXZ-, or Heisenberg-, interactions. type exchange Quantum tomography Quantum states

Liu, Wei, and Nori, Europhys. Lett. 67, 874 (2004) Quantum tomography

x back

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Q Superconducting charge qubit

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Quantum tomography for superconducting charge qubits

Liu, Wei, Nori, Europhysics Letters 67, 874 (2004); Phys. Rev. B 72, 014547 (2005); preprint (2007). y h p a r g o m o t

m u t n a u

Q details eprint (2007). (2007). eprint

y h p a 72, 014547 (2005); pr (2005); 014547 72, r g o m Phys. Rev. B o t

u (2004); 874 67, t n a u

Q Europhysics Letters Liu, Wei, Nori, Nori, Wei, Liu, Experiments on quantum tomography in SQs

PRL (2006), Martinis group s t n e t n o

S on interactions between different qubits between different verified experimentally, years after our prediction verified experimentally, qubits including ion traps, on-chip micromasers,cyclic using SC qubits photon states generating transitions, • quantum tomography Introduced solid state • qubits how to dynamically decouple Studied with always- • couplings several methods of controllable Proposed • We QED. It has been proposed and studied circuit • Studied phase qubits charge-flux, and charge, flux, SC • Studied SC physics and between atomic analogies many