Quantum Computer with Trapped Ions

Quantum Information Processing with Trapped Ions

Overview: Experimental implementation of quantum information processing with trapped ions

1. Implementation concepts of QIP with trapped ions 2. Quantum gate operations and entanglement with trapped ions 3. Advanced QIP procedures, scaling the ion-trap quantum computer

Quantum Information Science with Trapped Ions 1 Handouts for lectures

… are available from the organizers as pdf-files (ask Prof. Knoop, web-pages)

Part 1: Implementation concepts of QIP with trapped ions

Part 2: Quantum gate operations, entanglement etc.

Part 3: Advanced QIP procedures, scaling… plus 3 review articles, covering basics and advanced features … are available from organizers (as pdf-files)

Quantum dynamics of single trapped ions D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev. Mod. Phys. 75, 281 (2003)

Quantum computing with trapped ions H. Häffner, C. Roos, R. Blatt, Physics Reports 469, 155 (2008)

Entangled states of trapped atomic ions Rainer Blatt and David Wineland, Nature 453, 1008 (2008) Quantum Information Processing Experimental Implementation with Trapped Ions

Rainer Blatt Institute of Experimental Physics, University of Innsbruck, Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences

1. Implementation concepts and basic techniques

generics of QIP, the ion trap quantum computer, spectroscopy, measurements, basic interactions and gate operations

1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses Why Quantum Information Processing ?

applications in physics and mathematics

 factorization of large numbers (P. Shor, 1994) can be achieved much faster on a quantum computer than with a classical computer factorization of number with L digits: classical computer: ~exp(L1/3), quantum computer: ~ L2  fast database search (L. Grover, 1997) search data base with N entries: classical computer: O(N), quantum computer: O(N1/2)  simulation of Schrödinger equations  spectroscopy: quantum computer as atomic „state synthesizer“ D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)  quantum physics with „information guided eye“ Computation is a physical process

input physical object set

computation physical process shift

output physical object read

Classical Computer Quantum Computer - bits, registers - qubits, quantum registers - gates - quantum gates - classical processes (switches), - coherent processes dissipative - preparation and manipulation of entangled states The requirements for quantum information processing

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

I. Scalable physical system, well characterized qubits II. Ability to initialize the state of the qubits III. Long relevant coherence times, much longer than gate operation time IV. “Universal” set of quantum gates V. Qubit-specific measurement capability

VI. Ability to interconvert stationary and flying qubits VII. Ability to faithfully transmit flying qubits between specified locations

The seven commandments for QIP !! Quantum bits and quantum registers

 classical bit: physical object in state 0 or 1  register: bit rows 0 1 1 . . .  quantum bit (qubit): superposition of two orthogonal quantum states

 quantum register: L 2-level atoms, 2L quantum states 2L states correspond to numbers 0,....., 2L – 1

 most general state of the register is the superposition

(binary) (decimal) Universal Quantum Gates

Operations with single qubit: (1-bit rotations)

together universal !

Operations with two qubits: (2-bit rotations)

CNOT – gate operation (controlled-NOT) analogous to XOR

control bit target bit How a quantum computer works

quantum processor

input computation: sequence of quantum gates output Circuit quantum computer model: notation

control

qubit

target

input computation: sequence of quantum gates output 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses Which technology ?

Quantum 000 Processor 100 001 011 010 110 011 011

Cavity QED NMR

superconductors quantum dots

trapped ions © A. Ekert Realization concepts

Quantum Information Science and Technology Roadmaps: US – http://qist.lanl.gov/ EU – http://qist.ect.it/Reports/  Ion traps  Neutral atoms in traps (opt. traps, opt. lattices, microtraps)  Neutral atoms and cavity QED  NMR (in liquids)  Superconducting qubits (charge-, flux-qubits)  Solid state concepts (spin systems, quantum dots, etc.)  Optical qubits and LOQC (linear optics quantum computation) Quantum Systems  Electrons on L-He surfaces  Spectral hole burning that can be controlled and  …and more manipulated the DV criteria translated to an implementer…

I. Find two-level systems,

II. that can be individually controlled

III. that are stable and don’t decay while you work on them

IV. that interact to allow for CNOT gate operation

V. that can be efficiently measured or ~100%

VI. Find a way to interconnect remote qubits

VII. Make sure, your interconnection is good 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses Meeting the DiVincenzo criteria with trapped ions

criterion physical implementation technique scalable qubits internal atomic transitions linear traps  (2-level-systems) (trap arrays) initialization laser cooling, optical pumping,  state preparation laser pulses long coherence times narrow transitions coherence time  (optical, microwave) ~ ms - min universal quantum gates single qubit operations, Rabi oscillations  two-qubit operations Cirac-Zoller CNOT qubit measurement quantum jump detection individual ion  fluorescence convert qubits to coupling of ions with high CQED, bad cavity limit T flying qubits finesse cavity E

faithfully transmit coupling of cavities via fiber coupling pulse T flying qubits (photonic channel) sequences (CZKM) E Quantum computation with trapped ions

J. I. Cirac

P. Zoller

other gate proposals (and more): • Cirac & Zoller • Mølmer & Sørensen, Milburn • Jonathan & Plenio & Knight • Geometric phases • Leibfried & Wineland Quantum computation with trapped ions

control bit target bit

other gate proposals (and more): • Cirac & Zoller • Mølmer & Sørensen, Milburn • Jonathan & Plenio & Knight • Geometric phases • Leibfried & Wineland Quantum bits (qubits) with trapped ions

Storing and keeping quantum information requires long-lived atomic states:

 optical transition frequencies  microwave transitions (forbidden transitions, (hyperfine transitions, intercombination lines) Zeeman transitions) S – D transitions in alkaline earths: alkaline earths: Ca+, Sr+, Ba+, Ra+, (Yb+, Hg+) etc. 9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 111Cd+, 171Yb+ P3/2 P1/2 D5/2

TLS S1/2 TLS S1/2 Boulder 9Be+; Michigan 111Cd+; 40 + Innsbruck Ca Innsbruck 43Ca+, Oxford 43Ca+; Maryland 171Yb+; … many more Quantum Computer with Trapped Ions

J. I. Cirac, P. Zoller; Phys. Rev. Lett. 74, 4091 (1995) L Ions in linear trap  quantum bits, quantum register  state vector of quantum computer - narrow optical transitions - groundstate Zeeman coherences

 2-qubit quantum gate

laser pulses entangle pairs of ions control bit target bit  needs individual addressing, efficient single qubit operations  small decoherence of internal and motional states  quantum computer as series of gate operations (sequence of laser pulses) 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses Ion trap experiment Linear Ion Trap

 Linear ion trap I. Waki et al., Phys. Rev. Lett. 68, 2007 (1992) M.G. Raizen et al., Phys. Rev. A 45, 6493 (1992)

- up to about 30 ions (string) - ions separated by a few µm

 collective quantized motion

- anisotropic oscillator: νz << νx, νy - ion motion coupled by Coulomb repulsion - eigenmodes (nearly) independent of the number of ions center of mass mode breathing mode Ion strings scale favorably....

Coulomb repulsion defines a length scale

Mode frequencies are nearly independent of ion number

A. Steane, Appl. Phys. B 64, 623 (1997) D. James, Appl. Phys. B 66, 181 (1998) Ion strings as quantum registers

1996 – 2006

1 - 10 qubits

2007 - 2012

32 qubits

70 Level scheme of Ca+ qubit on narrow S - D quadrupole transition

P3/2 854 nm

P1/2 866 nm D5/2 393 nm 397 nm D3/2

729 nm

S1/2 Spectroscopy with quantized fluorescence (quantum jumps)

P D absorption and emission cause fluorescence steps monitor spectroscopy (digital quantum jump signal) S

histogram of absorption events S

absorptions

D laser detuning State detection by quantized fluorescence

8 D state occupied S state occupied 7 s t n 6 detection efficiency: e m e 5 r 99.85% u s

a 4 e m

3 f o

# e 2 g

0 0 20 40 60 80 100 120 counts per 9 ms Laser – ion interactions ...1

An ion trapped in a harmonic potential with frequency ω interacting with the travelling wave of a single mode laser tuned close to a transition that forms an effective two-level system, is described by the Hamiltonian

k, ωl, φ: wavenumber, frequency and phase of laser radiation

m0: mass of the ion

for details see: D. Leibfried, C. Monroe, R. Blatt, D. Wineland Pauli matrices Rev. Mod. Phys. 75, 281 (2003) Laser – ion interactions ...2

With the Lamb-Dicke parameter we write in terms of creation and annihilation operators

In the interaction picture defined by we obtain for the Hamiltonian

with coupling states with vibration quantum numbers Quantized ion motion: ladder structure

2-level-atom harmonic trap e

Ω Γ ⊗ ν n g

+ coupled system n 1,e n,e n −1,e

n +1, g n, g n −1, g Laser – ion interactions ...3 with laser tuned close to a resonance coupling to other resonances can be neglected ( ).

Time evolution of pairwise coupled states is then determined by the set of coupled equations

With the detuning and the Rabi frequency we obtain Rabi oscillations on resonance Laser – ion interactions ... Lamb-Dicke regime

For small ion oscillation amplitudes (Lamb-Dicke regime) we expand and obtain for the coupling

carrier excitation

detuning

red sideband excitation blue sideband excitation Interactions in the ladder structure carrier

red sidebands blue sidebands Laser – ion interactions ... unitary operators

For later use we already note that the action of a laser on the respective transitions can be written in the following form:

Operates on internal degrees Carrier pulse of length of freedom

Red sideband pulse of length Operate on internal and external degrees of freedom Blue sideband pulse of length ENTANGLEMENT ! Level scheme of Ca+ qubit on narrow S - D quadrupole transition

P3/2 854 nm

P1/2 866 nm D5/2 393 nm 397 nm D3/2

729 nm

S1/2 Spectroscopy of the S1/2 – D5/2 transition

Zeeman structure in non-zero magnetic field:

5/2 2-level-system: 3/2 1/2 D -1/2 5/2 -3/2 -5/2

S 1/2 1/2 -1/2 ∆ + vibrational degrees of freedom Quantized ion motion

2-level-atom harmonic trap coupled system

excitation: various resonances spectroscopy: carrier and sidebands ∆n = 0

D5/2 ∆n = -1 ∆n = 1

ν S 1/2 n = 0 1 2 Laser detuning Absorption spectra: Experimental cycle

P3/2 P3/2 P3/2 854 nm P P P 1/2 1/2 866 nm 1/2 D5/2 D5/2 D5/2 397 nm 397 nm D3/2 D3/2 D3/2 866 nm 729 nm S1/2 S1/2 S1/2

1st step: 2nd step: 3rd step: probe with 729 nm detect fluorescence reset, Doppler cooling 5 µs – 3 ms 1 – 10 ms 1 – 10 ms

Average: Repeat experiment 100 - 1000 times, then step 729 nm laser to new detuning Absorption measurements: Pulse scheme

397nm 397nm σ+ P 854nm 3/2 866nm P 397nm 1/2 854nm 866nm D5/2 729nm D3/2 detection 729nm time S preparation detection 1/2 excitation

- typically 5ms for each sequence - repeated typically 100 times - triggered by 50Hz line frequency - flexibly controlled by Labview Excitation spectrum of the S – D transition

- 32 - 12 12 32 52 12 12 12 12 12 excitation probability

0.5

-25 -20 -15 -10 -5 0 5 ∆/MHz Laser detuning ∆ at 729 nm Excitation spectrum of the S – D transition

red blue axial radial radial axial sidebands sidebands 0.5 carrier

0 -1710 -920 0 920 1710 kHz

(spherical trap) Excitation spectrum of single ion in linear trap

ωax = 1.0 MHz

ωrad = 5.0 MHz 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses Ion trap experiment Addressing of individual ions

coherent

manipulation Paul trap 0.8 of qubits 0.7

0.6 electrooptic 0.5 deflector 0.4

0.3 Excitation

0.2

0.1

0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Deflector Voltage (V)

dichroic beamsplitter  inter-ion distance: ~ 4 µm Fluorescence  addressing waist: ~ 2.5 µm detection CCD < 0.1% intensity on neighbouring ions Detection of 6 individual ions 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses Coherent state manipulation

carrier (C)

blue sideband carrier and sideband (BSB) Rabi oscillations with Rabi frequencies

Lamb-Dicke parameter Coherent state manipulation: carrier

carrier

Carrier transitions leave the motion unchanged Coherent state manipulation: sideband

Sideband transitions entangle motion and internal excitation

sideband Qubit rotations in computational subspace

D,2 D,1 D,0 transitions are described by unitary matrices:

S,2 carrier: S,1 S,0

computational red sideband: subspace

keep all blue sideband: populations in subspace only !

A. M. Childs, I. L. Chuang, Phys. Rev. A 63, 012306 (2001) SWAP and composite SWAP operation

D,1 D,0 find simultaneously perfect SWAP operation with 4π rotation S,1 on the D,1↔↔ SS ,2( ,1 D ,2 ) S,0 transition(s) π 2 π out of composite SWAP operation: perfect subspace if SWAP |D,1> (|S,1>) operation populated

I. L. Chuang et al., Innsbruck (2002) |S,0> - |D,1> SWAP and composite SWAP operations

single-step SWAP operation 3-step composite SWAP operation 3-step composite SWAP operation

I. Chuang et al., Innsbruck (2002)

1 1 3

on on A phase gate with 4 pulses (2π rotation), blue sideband

1 on

4 2 Population of |S,1> - |D,2> remains unaffected

2 4

3 1 A single ion composite phase gate: Experiment state preparation S,0 , then application of phase gate pulse sequence

π 2 π π 2 π φϕ =00 φ = π 2 φϕ = 00 φϕ =ππ 22 1

0.9

0.8

0.7

0.6

0.5

excitation 0.4 -

5/2 0.3 D

0.2

0.1

0 0 20 40 60 80 100 120 140 160 180 Time (µs) A single ion composite CNOT gate: Experiment state preparation S,0 , then application of CNOT gate pulse sequence

π π Phase gate − 2 2 1

0.9 97.8 (5) % 0.8

0.7

0.6

0.5 excitation -

0.4 5/2

D 0.3

0.2

0.1

0 0 50 100 150 200 250 300 Time (µs) A single ion composite CNOT gate: Experiment

π π |D,1> phase gate − prep. 2 2 1

0.8

0.6

0.4

0.2 state excitation probability -

5/2 0

D 0 50 100 150 200 250 300 Time (µs) 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing

1.2 Realization concepts

1.3 Ion trap quantum computer – the basics

1.4 Spectroscopy in ion traps

1.5 Addressing individual ions

1.6 Basic gate operations, composite pulses 2. Quantum gate operations and entanglement with ions

2.1 Cirac-Zoller CNOT gate operation with two ions 2.2 Entanglement and Bell state generation 2.3 State tomography 2.4 Process tomography of the CNOT gate 2.5 Tripartite entanglement (W-states, GHZ-states) 2.6 Multipartite entanglement 2.7 Teleportation 2.8 Entanglement Swapping 2.9 Toffoli gate operation Quantum computer with trapped ions

 2-qubit quantum gate

control bit target bit

F. Schmidt-Kaler et al., Nature 422, 408 (2003) Cirac – Zoller two-ion controlled-NOT operation

ε1 ε2 SS→ SS SD→ SD DDS → D DD→ DS control target ion 1 SD, control qubit SWAP motion 0 0 ion 2 SD, target qubit Cirac – Zoller two-ion controlled-NOT operation

ε1 ε2 SS→ SS SD→ SD DDS → D DD→ DS control target ion 1 SD, control qubit |0>, |1> motion 0 0 ion 2 SD, target qubit Cirac – Zoller two-ion controlled-NOT operation

ε1 ε2 SS→ SS SD→ SD DDS → D DD→ DS control target ion 1 SD, control qubit |0>, |1> SWAP-1 motion 0 0 ion 2 SD, target qubit Cirac – Zoller two-ion controlled-NOT operation ion 1 SD, control qubit SWAP SWAP-1 motion 0 0 ion 2 SD, target qubit

pulse sequence: laser frequency pulse length Ion 1 optical phase

Ion 2

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

Ion 1 Ion 2 5µm SS SS control qubit SS

target qubit SD

DS gate Time τ sequence DD

Individual ion detection measure on CCD camera states Cirac – Zoller CNOT- gate operation

SS - SS DS - DD

SD - SD DD - DS Experimental fidelity of Cirac-Zoller CNOT operation

truth table:

control bit target bit

F. Schmidt-Kaler et al., input output Nature 422, 408 (2003) Superposition as input to CNOT - gate operation

prepare gate output detect 2. Quantum gate operations and entanglement with ions

CNOT |S+D>|S> |SS + eiφ DD>

control bit target with local rotations ϕ create

|SS+DD>, |SS-DD> ϕ on both ions Bell states |SD+DS>, |SD-DS> ϕ on individual ion

measurement sequence:

π/2 CNOT π/2 • Super-Ramsey experiment ϕ Ion 1 0 • Parity check π/2 ϕ Ion 2 C. A. Sackett et al., Nature 404, 256 (2000) Measuring entanglement

C. A. Sackett et al., Nature 404, 256 (2000) entangling operation: correlates states parity Π witnesses entanglement: Π oscillates with 2ϕ !

π/2 CNOT π/2 ϕ Ion 1 0

π/2 ϕ Ion 2

F = 71(3)% Fidelity =

0.5 (PSS+PDD+visibility)

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

Probability of Probability of

Parity =

or or

Parity = +1 Parity = -1 Bell states

Ion 1 D,1 D,0 π / 2

S,1 S,0

π/2 pulse, BSB Bell states

Ion 2 D,1 D,0 π S,1 S,0

π/2 pulse, BSB

π pulse, carrier Bell states

Ion 2 D,1 D,0 π

S,1 S,0

π/2 pulse, BSB

π pulse, carrier

π pulse, BSB Preparation of Bell states and measurement

Ion 1

Ion 2

parity measurement

C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) 2. Quantum gate operations and entanglement with ions

Representation of ρ as a sum of orthogonal observables Aj :

ρ is completely determined by the expectation values :

For a two-qubit system :

Joint measurements of all spin components Maximum likelihood estimation

The reconstructed is not necessarily positive semidefinite obtain ρ from maximum likelihood estimation: Z. Hradil, Phys. Rev. A 55, R1561 (1997), K. Banaszek et al., Phys. Rev. A 61, 010304 (1999) choose ρ such that

is minimized

optimization of 15 parameters Experimental tomography procedure

two 40Ca+ ions trapped in linear trap

Individual qubit operations

Experimental cycle (~20 ms):

100-200 1. Laser cooling to the motional ground state experiments 2. Quantum state preparation 3. Application of tomography pulses 4. State detection

Experimental tomography procedure

1. Initialization in |SS0> 10 ms of Doppler and sideband cooling

2. Quantum state manipulation on the

S1/2 – D5/2 transition

3. Quantum state measurement by fluorescence detection

5µm

P1/2 D5/2

DopplerQuantum stateFluorescence Detection with coolingmanipulation detectionSideband CCD camera: cooling

S1/2

Qubit transition : Measurement of the density matrix

Measurement of prepare Bell state no rotation 200 repetitions measure

prepare Bell state

ion #1, σx rotation ion #2, identity 200 repetitions measure

Rotation of Bloch sphere prior to 9 different measurement: settings π/2 -pulse prepare Bell state ion #1, σ rotation y 200 repetitions ion #2, σy rotation measure Art by Information from a Tim Noble & Sue Webster single projection is not enough … Tomography pulses for state reconstruction

Setting Transformation applied to Measured expectation values # ion 1 ion 2 (1) (2) (1) (2) 1 - - σz σz σz σz

(1) (2) (1) (2) 2 R(π/2 ,3π/2) - σx σz σx σz

(1) (2) (1) (2) 3 R(π/2, π) - σy σz σy σz

(1) (2) (1) (2) 4 - R(π/2, 3π/2) σz σx σz σx

(1) (2) (1) (2) 5 - R(π/2, π) σz σy σz σy

(1) (2) (1) (2) 6 R(π/2, 3π/2) R(π/2, 3π/2) σx σx σx σx

(1) (2) (1) (2) 7 R(π/2, 3π/2) R(π/2, π) σx σy σx σy

(1) (2) (1) (2) 8 R(π/2, π) R(π/2, 3π/2) σy σx σy σx

(1) (2) (1) (2) 9 R(π/2, π) R(π/2, π) σy σy σy σy

Push-button preparation and tomography of Bell states

Fidelity: F = 0.91

Entanglement of formation:

E(rexp) = 0.79

Violation of Bell inequality:

S(rexp) = 2.53(6) > 2

C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) 1. Implementation concepts and basic techniqes

1.1 Generics of quantum information processing 1.2 Realization concepts 1.3 Ion trap quantum computer – the basics 1.4 Spectroscopy in ion traps 1.5 Addressing individual ions 1.6 Basic gate operations, composite pulses

2. Quantum gate operations and entanglement with ions

2.1 Cirac-Zoller CNOT gate operation with two ions 2.2 Entanglement and Bell state generation 2.3 Quantum state tomography