Квантовые цепи и кубиты Твердотельные наноструктуры и устройства

для квантовых вычислений Лекция 5

А.В. Устинов Karlsruhe Institute of Technology, Germany

Russian Quantum Center, Russia

Alexey Ustinov Solid-state qubits 2

JC C

eE2

2

=π2

0Φ= c

JIECharging energy Josephson energy

• Friedman et al., Nature 406, 43 (2000) • van der Wahl et al., Science 290, 773 (2000)

flux qubit

JC EE <<

1≥∆⋅∆ ϕnUncertainty relation for a superconductor:

• Makhlin et al., Nature 398, 305 (1999) • Nakamura et al., Nature 398, 786 (1999)

charge qubit

JC EE >>

810electrons

Flux and charge: Two extremes

F VG

Alexey Ustinov Solid-state qubits 3

for a chosen Jc , the ratio EJ /EC depends on the junction area A

103 106 109 1

2AEE

C

J ∝charge flux phase

NEC Chalmers

Yale

Yale ETHZ Saclay

UCSB, NIST Karlsruhe, Grenoble

Maryland

Delft, NTT NEC, Jena Karlsruhe

0.5 – 10 µs ~ 10 µs 100 ns

Josephson junction qubits: Energy span

10-2

0.5 – 1 µs

Alexey Ustinov Solid-state qubits 4

NIST Chalmers NEC

TU Delft

flux phase charge charge/flux

NEC Chalmers Yale

Delft, Jena MIT, Berkeley NTT, NEC

Saclay Yale PTB

NIST, UCSB Erlangen Maryland

Realizations of Superconducting Qubits

Nakamura, Pashkin, Tsai et al. Nature 398, 421, 425 (1999, 2003, 2003)

Chiorescu, van der Wal, Mooij, Orlando, S. Lloyd et al. Science 285, 290, 299 (1999, 2000, 2003)

Vion, Esteve, Devoret et al. Science 296 (2002)

Martinis, Simmonds, Lang, Nam, Aumentado, Urbina et al. Phys. Rev. Lett. 89, 93 (2002, 2004)

Alexey Ustinov Solid-state qubits 5

VG

charge qubit flux qubits

charge/flux qubit

Φ

VG

NEC Chalmers

Yale Jena

Delft Jena MIT

Berkeley NTT NEC

Saclay Yale PTB

Overview of superconducting qubits

Φ

phase qubit

NIST UCSB

Karlsruhe Maryland

John Martinis

Lecture 4

Alexey Ustinov Solid-state qubits 6

Two extremes: Charge qubits and flux qubits

Charge qubits manipulation charge noise

Flux qubits manipulation flux noise

Readout circuits switching readout dispersive readouts

Alexey Ustinov Solid-state qubits 7

CE

n=0 n=1

0 1charge of the box n =q/(2e)G

ener

gy

Nakamura et al., Nature 398, 786 (1999) Nakamura et al., PRL 87, 246601 (2001) Nakamura et al., PRL 88, 047901 (2002)

box

reservoir

pulse gate

probedc gate

1 µm

VG

© Y.Nakamura

Charge qubit: NEC experiments

Alexey Ustinov Solid-state qubits 8

Charge qubit: NEC experiments

Nakamura, Pashkin and Tsai, Nature 398, 786 (1999)

Alexey Ustinov Solid-state qubits 9

Nakamura et al., Nature 398, 786 (1999)

Measurement

Simulation

charge

ns4≈ϕT

Charge qubit: NEC experiments

Alexey Ustinov Solid-state qubits 10

A circuit analog for cavity QED (Yale)

© A.Wallraff A. Wallraff, D. I. Schuster, A. Blais, et al., Nature 431, 162 (2004)

Alexey Ustinov Solid-state qubits 11

realization of superconducting cavity QED circuit A. Wallraff et al. , Nature (London) 431, 162 (2004)

Charge qubit in a cavity (Yale)

© A.Wallraff

Robert Schoelkopf

Andreas Wallraff

Alexey Ustinov Solid-state qubits 12

QED experiment @ Yale: scheme, resonator and charge qubit

A. Wallraff, D. I. Schuster, A. Blais, et al., Nature 431, 162 (2004)

Alexey Ustinov Solid-state qubits 13

A. Wallraff, D. I. Schuster, A. Blais, et al., Nature 431, 162 (2004)

off

resonance

on resonance

large shift

Photon-qubit anti-crossing: vacuum Rabi

splitting

Dispersive qubit-field interaction

Alexey Ustinov Solid-state qubits 14

J.E. Mooij et al., Science 285, 1036 (1999) C.H. van der Wal et al., Science 290, 773 (2000)

degeneracy point at

Three-junction flux qubit

Φ = Φ0/2

quantum states:

10 βα +=Ψ

Φ

clockwise current

Φ

counterclockwise current

Ic Ic

~0.8 Ic

0 1

TU Delft

Superconducting 3-junction flux qubit

15

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

flux quantization:

nππϕϕϕ 220

321 =ΦΦ

+++

ϕ3

Φ

Ic ϕ1 ϕ2 Ic

α Ic

ΦΦ

−−−++=0

ext2121 2coscoscos πϕϕαϕϕ

JEU

effective 2D potential:

U/EJ

effective 1D cross-section:

Superconducting flux qubit as a two-level system (artificial atom)

16

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

persistent current states ± Ip

ϕ3

Φ

Ic ϕ1 ϕ2 Ic

α Ic

+ Ip - Ip

magnetic flux bias Φ ∼ Φ0/2

( )xzH σεσ ∆+=21

Superconducting flux qubit

17

J. Clarke and F. K. Wilhelm, Nature 453, 1031 (2008)

−=∆

C

JCJ exp

EEbEEa α

degeneracy point Φ = Φ0/2

22 ∆+= εν

away from degeneracy point

ϕ3

Φ

Ic ϕ1 ϕ2 Ic

α Ic

−

ΦΦ

Φ= 5.020

0pIε

energy level separation

18

qubit

Implementation of flux qubit

Al wires Al/AlOX/Al Josephson junctions

Alexey Ustinov

Ic

Ic α Ic

Sample made at E.Il’ichev group @IPHT Jena

19

readout LC resonator

qubit

Readout of an individual qubit

Al/AlOX/Al Josephson junctions

Alexey Ustinov

M. Jerger, S. Poletto, P. Macha, U. Hübner, A. Lukashenko, E. Il'ichev, and A. V. Ustinov, EPL, 96 40012 (2011)

20

Dispersive readout via a resonator

21

Anticrossings

22

Multiplexed readout of 7 flux qubits

• each qubit is coupled to a resonator, all resonators are coupled to a transmission line

• the qubit state changes the resonator frequency

• all resonators are measured via the same transmission line

all qubits are control- led and measured using only 1 microwave line.

M. Jerger, S. Poletto, P. Macha, et al. EPL, 96 40012 (2011)

23

Frequency-Division Multiplexing

#1 #2 #3 #4 #5 #6 #7

M. Jerger, S. Poletto, P. Macha, U. Hübner, A. Lukashenko, E. Il'ichev, and A. V. Ustinov, EPL 96, 40012 (2011)

24

Spectroscopy of 7-qubit array

All 7 qubits work and can be read out simultaneously ! M. Jerger, S. Poletto, P. Macha, U. Hübner, A. Lukashenko, E. Il'ichev, and A. V. Ustinov, EPL, 96 40012 (2011)

Quantum metamaterial to be made of superconducting artificial atoms

25 Alexey Ustinov

50 Ω transmission line

superconductor

d << λ

dielectric substrate

26

Superconducting quantum metamaterial: array of flux qubits

Alexey Ustinov Solid-state qubits 27

tunable coupling

T. Hime, P.A. Reichardt, B.L.T. Plourde, T.L. Robertson, C.-E. Wu, A.V. Ustinov, and J. Clarke, Science 314, 1427 (2006).

qubit A

qubit B

level anti-crossing

Flux qubits with current-controlled coupling

John Clarke

Alexey Ustinov Solid-state qubits 28

Plourde et al. Phys. Rev. B 70, 140501 (2004)

Circulating current in dc SQUID vs. applied flux

Alexey Ustinov Solid-state qubits 29

Chip layout

tunable coupling

Alexey Ustinov Solid-state qubits 30

typical double degeneracy point

lines of constant flux in qubit A

lines of constant flux in qubit B

Two-Qubit Flux Map

Alexey Ustinov Solid-state qubits 31

6.5 7.5

1

0.5

0 C

alcu

late

d |T

f0|2 (

a.u.

)

ΦA-Φ0/2 (mΦ0)

|T20|2

|T10|2

Matrix elements |<f|σzA + σzB|0>|2

Two coupled flux qubits

Alexey Ustinov Solid-state qubits 32

Quantum non-demolition measurement of a flux qubit

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

A quantum non-demolition (QND) measurement minimizes the disturbance by interaction with a detector that preserves the eigenstates of the quantum system.

The mutual inductance M=14 pH represents the sum of a geometric inductance and of the kinetic inductance of the narrow lines shared by the qubit and SQUID loops.

Alexey Ustinov Solid-state qubits 33

QND measurement of a flux qubit

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

conditional measurement

P(h)

P(h|h)

Alexey Ustinov Solid-state qubits 34

Demonstration of controlled-NOT quantum gates on a pair of flux qubits

J.H. Plantenberg, P.C. de Groot, C.J.P.M. Harmans and J.E. Mooij, Nature 447, 836 (2007)

C control qubit

T target qubit

Computational basis:

A resonant microwave pulse induces rotations in this basis, and its microwave phase determines the rotation axis.

A microwave pulse inducing a rotation around the x axis of the transition:

gate matrix CNOT

Experimental setup for demonstrating CNOT gate

© J. Platenberg, Ph.D. thesis, TU Delft (2007)

Schematic representation of the different components of the coupled-qubit measurement setup, displaying equipment used for signal generation, detection, data acquisition and timing as well as the various filter and attenuation stages.

Alexey Ustinov Solid-state qubits 36

Operation of the coupled-qubits device

J.H. Plantenberg, P.C. de Groot, C.J.P.M. Harmans and J.E. Mooij, Nature 447, 836 (2007)

Alexey Ustinov Solid-state qubits 37

Newest developments in superconducting qubits

Find your way through the zoo Inventing new names:

Quantronium Transmon Fluxonium Superinductor Double-SQUID qubit π-shift flux qubit

Alexey Ustinov Solid-state qubits 38

State-of-the-art of superconducting qubits

Josephson `artificial atoms´ are becoming scalable and reliable qubits Gates and simple algorithms have been reported for charge, flux and phase qubits More qubit circuits will be coming soon Dispersive QND readout is the key Challenges for the near future:

Hybrid systems Entangling Josephson qubits with other

types of qubits (photons, atoms, spins)