Quantum computing …

Applications in informatics and physics

P. Shor, 1994: factorization of large numbers is polynomial on a quantum computer, exponential on a classical computer

L. Grover, 1997: data base search N1/2 quantum queries, N classical

simulation of Schrödinger equations or any unitary evolution

quantum cryptography / repeaters / quantum links

improved atomic clocks

understanding the fundamentals of quantum mechanics / Gedanken-Experimente

Experiments with entangled matter

The prototype

Quantum gate proposal(s)

21121

Further gate proposals: • Cirac & Zoller• Mølmer & Sørensen, Milburn• Jonathan & Plenio & Knight• Geometric phases

0111

1101

1010

0000

control bitcontrol bit target bittarget bit

controlled NOTcontrolled NOT

D5/2

729 nm

|1>

|0>

internal qubit

Qubits in a single 40Ca+ ion

S1/2

motional qubit

|0>|1>1

n=0

2

|S,n> |D,n> : carrier transition ()

|S,n> |D,n±1> : sideband transition ()

"computational subspace"

|S,0>

|D,0>|D,1>

|S,1>

COHERENT LASER MANIPULATION (Rabi oscillations)

First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 4714 (1995).

2 ions + motion = 3 qubits

With several ions, the motional qubits are sharedWith several ions, the motional qubits are shared

vibrational modes computational subspace: 2 ions, 1 mode

|S,S,0>

|D,S,0>|D,S,1>

|S,S,1>

|S,D,0>|S,D,1>

|D,D,0>|D,D,1>

laser on ion 2

laser on ion 1laser on ion 2

laser on ion 1

Details of C-Z CNOT gate operation (Phase gate)

Experimental techniques

conditions vs. achievements

Experimental techniques

conditions vs. achievements

D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001)

● Qubits store superposition information, ion string, but scalability?scalable physical system

● Ability to initialize the state of the qubits ground state cooling

● Long coherence times, hard work much longer than gate operation time

● Universal set of quantum gates: Coherent pulses on Single bit and two bit gates carrier and sidebands,

addressing

● Qubit-specific measurement capability Shelving, imaging

Some requirements ...

See Toni's lecture today

Innsbruck linear ion trap

|1>

|0>

|1>

|0>

Two 2-level systems

5mm

MHz5radial

MHz27.0 axial

+HV +HV

RF

RF

GND

GND

P3/2

854 nm

393 nm

S1/2

P1/2

D3/2

397 nm

866 nm

s1D5/2

729 nm

Level scheme of 40Ca+

S1/2

Zeeman structure in non-zero magnetic field: :

(+ motional degrees of freedom ...)

S1/2

D5/2

5/23/2

-3/2-5/2

- /21

- /21/21

/21

2-level-system

1/2 5/2

Zeeman structure of the S1/2 – D5/2 transition

P3/2

S1/2

P1/2

D3/2

s1D5/2

729 nm

Manipulation by laser pulses on 729 nm transition(~ 1 ms coherence time)

|1>

|0>

qubit

Level scheme of 40Ca+

S1/2

Superpositions of S1/2(m=1/2) and D5/2(m=5/2) form qubits

P3/2

S1/2

P1/2

D3/2

397 nm

866 nm D5/2 |1>

|0>

qubit

Level scheme of 40Ca+

S1/2

State detection by photon scattering on S1/2 to P1/2 transition at 397 nm (> 99% in ~ 3 ms)

Detector

P3/2

854 nm

S1/2

P1/2

D3/2

D5/2

729 nm

|1>

|0>

qubit

Level scheme of 40Ca+

S1/2

Motional state preparation by sideband

cooling on 729 nm transition (> 99.9%)

S1/2

D5/2

|n> = |0> |1> |2>

coupled system & transitions

g

e

2-level-atom harmonic trap

spectroscopy: carrier and sidebands

Laser detuning

n = 1n = -1

n = 0

0n

12

Motional sidebands

Rabi frequencies

1 n

n

Carrier:

Red SB:

Blue SB:

k <0|x2|0>1/2 «

Excitation spectrum of the S1/2 – D5/2 transition

ax = 1.0 MHzrad = 5.0 MHz

(only one Zeeman

component)

Excitation spectrum of two ions

Sideband cooling of two ions

Laser pulses for coherent manipulation

coh >> gate

AOM = acousto-optical modulator, based on Bragg diffraction

"Ampl" includes switching on/off

AOM = acousto-optical modulator, based on Bragg diffraction

"Ampl" includes switching on/off

Ampl

Ampl

cw laser

RF

AOM I

t

I

t

to trap

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

Tim e (µs)

Po

pula

tion

of D

sta

te

see also experiments at NISTRoos et al., PRL 83, 4713 (1999)

S1/2

D5/2

0

1

1

2

Quantum state engineering

Blue sideband

Blue sideband/2 /2

D-s

tate

po

pu

lati

on

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Pulse length (s)

Rabi-flops on blue sideband

Ramsey Interference

0 500

0.2

0.4

0.6

0.8

1

D-s

tate

po

pu

lati

on

100 150 200 250 300

Pulse length (s)

Qubit rotations

|S,0>

|D,0>|D,1>

|S,1>

|S,0>

|D,0>|D,1>

|S,1>

Addressing of ions in a string

Well-focussed laser beam

● beam steering with electro-optical deflector

● addressing waist ~ 2.5 - 3.0 mm

● < 1/400 intensity on neighbouring ion

Individual ion detectionon CCD camera

5µm

quantum state populations pSS,pSD,pDS,pDD

|SS>|DS>

|DD> |SD>

Two-ion histogram (1000 experiments)

region 1 region 2

|SS>

|SD>

|DS>

|DD>

Quantum state discrimination

Cirac-Zoller Quantum CNOT Gatewith two trapped ions

Cirac-Zoller Quantum CNOT Gatewith two trapped ions

Detection

ion 1

motion

ion 2

,S D

,S D

0 0

control qubit

target qubit

SWAP

1 2

Cirac-Zoller two-ion controlled-NOT gate

SWAP-1

Preparation |S> = bright|D> = dark

CNOT

"bus" qubit

Result : schematic

S S S S

S D S D

D D DS

D D D Scontrolcontrol targettarget

SS SS DS DD

SD SD DD DS

Result : full time evolution

Pre

pa

rati

on

De

tec

tio

n

every point = 100 single measurements, line = calculation (no fit)

Details of time evolution

Prep

aration

SW

AP

SW

AP

-1CNOT betweenmotion and ion 2

Detectio

n

SS SS

input

output

expideal >|2

Measured fidelity (truth table)

F. Schmidt-Kaler et al., Nature 422, 408 (2003)

|SS> + ei|DD>

{ |SS>+|DD>, |SD>+|DS>}

controlcontrol targettarget

(|S>+|D>)|S>

Experimental sequence:

Ion 1Ion 1

Ion 2Ion 2

CNOT /2

/2

/2

Deterministic entanglement

"Super-Ramsey experiment"

Detection: Parity check ...

CNOTCNOT

|S>|S>

local /2 rotation local /2 rotation

local (/2,) rotationlocal (/2,) rotation

outputpreparation gate detect

Gate coherence

Pro

ject

ion

CNOT

Fidelity = 0.5 ( PSS + PDD + contrast) = 71(3)%

Oscillation with 2 entanglement !

Parity and fidelity

Ion 1Ion 1

Ion 2Ion 2

CNOT /2

/2

/2

Phase P

arit

y: P

SS+

PD

D-P

DS-P

SD

54% contrast

|SS>+|DD> ↔ |SD>+|DS>

"super-Ramsey experiment"

Examples of experimental

problems & solutions

Examples of experimental

problems & solutions

computational subspace

,0S,1S

,0D,1D out of CS !

2~Rabi

1~Rabi

2π

naive idea : -pulse on blue SB composite SWAP (from NMR)

computational subspace

,0S,1S

,0D,1D

4

Gate pulses (I) : SWAP

(works if initial state is not |S,1>)

Swap information from internal into motional qubit and back

1

2

3

,0 ,1D Son

4 ,1 ,2D Son

1

3

I. Chuang et al., Innsbruck (2002)

3-step composite SWAP operation

computational subspace

Phase factor -1 for all except |D,0 >

,0S,1S

Phase factor -1 for |S,1 >

Cirac & Zoller (1995) Composite phase gate

,0S,1S

,0D,1D

22

Gate pulses (II) : Phase gate

use auxiliarylevel

0,Aux1,Aux

M. H. Levitt, Prog. NMR Spectrosc., 1986I. L. Chuang, Innsbruck, 2002

Phase factor conditioned on state

1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R R

1

2

3

4

,0 ,1S D2on

Composite phase gate (2 rotation)

1 1 1 1( , ) 2, 2 ,0 2, 2 ,0R R R R R

4

3

2

1

2 also on

2,1, DS

Action on |S,1> - |D,2>

no populationoutside CS !

ion 1

motion

ion 2

,S DSWAP-1

,S D

0 0SWAP

Ion 1Ion 1

Ion 2Ion 2

pulse sequence

control bitcontrol bit

target bittarget bit

Cirac-Zoller two-ion controlled-NOT operation

blue0

blue

c0

blue

blue½

0

blue½

0

blue

c

CNOT

SS → SS

Details of time evolution

Ion 1Ion 1

Ion 2Ion 2

blue0

blue

c0

blue

blue½

0

blue½

0

blue

c

Time (s)

AC Stark shift & its compensation

AC Stark shift & its compensation

1/2 → -5/2

1/2 → -1/2

1/2 → 3/2

Why Bell states ?

entangled massive particles, distinguishable

resource for quantum cryptography / repeaters / quantum links

improved atomic clocks

understanding the fundamentals of quantum mechanics /

EPR paradox, Gedanken-Experimente

Generation of Bell states with three pulses

atom 1 atom 2 atom 2

Carrier pulses:

Blue sideband pulses

2/

result

z

x

y

/2 - pulse

Rotation of the Bloch sphere prior to state measurement

Principle of tomography (1 atom)

Measurement of spin components

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

F=0.91

Bell state generation & tomography

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

F=0.90

Bell state generation & tomography

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

F=0.88

Bell state generation & tomography

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

F=0.91

Bell state generation & tomography

Fidelity : F = 0.91

Peres-Horodecki criterion :

Violation of a CHSH inequality: S(0°,90°,45°,135°)(exp) = 2.53(6) > 2

E(exp) = 0.79 (4)

Entanglement of formation for a pair of qubits (Wooters ’98) :

Entanglement characterization

Cirac-Zoller quantum CNOT gate with two trapped ionsCirac-Zoller quantum CNOT gate with two trapped ions

F. Schmidt-Kaler, C. Becher, J. E., H. Häffner, C. Roos, W. Hänsel, G. Lancaster, S. Gulde, M. Riebe, T. Deuschle,

I.L. Chuang, R. Blatt

F. Schmidt-Kaler et al., Nature 422, 408 (2003)

The works and the workers

Bell States of Atoms with Ultralong Lifetimes and Their Tomographic State Analysis

Bell States of Atoms with Ultralong Lifetimes and Their Tomographic State Analysis

C. F. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)

Deutsch-Jozsa quantum algorithm with a single trapped ionDeutsch-Jozsa quantum algorithm with a single trapped ion

S. Gulde et al., Nature 421, 48-50 (2003).

Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor

Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor

H. Häffner et al., Phys. Rev. Lett. 90, 143602 (2003).

http://heart-c704.uibk.ac.at/papers.html