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Quantum Information Processing and Quantum Error Correction and Quantum Error Correction with Trapped Ca+ Ions

Quantum Information Processing and h d Quantum Information Processing and h d Quantum Error Correction with Trapped Ca+ Ions Quantum Error Correction with Trapped Ca+ Ions

Rainer Blatt

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

 Trapped Ca+ for quantum information processing  Coherence of multi‐particle entangled states  Quantum error correction with trapped ions  Undoing a measurement by quantum error correction  Quantum simulations with trapped ions

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

s 1  

854 nm

P3/2 P1/2

866 nm

D3/2

397 nm 866 nm

D5/2

393 nm

40C +

3/2

729 nm

40Ca+ S1/2

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

2-level-system: 2-level-system: 2-level-system:

P1/2 D5/2 P1/2 D5/2

2 level system: 2 level system: 2 level system:

S

Fluorescence detection

S

Fluorescence detection Zeeman structure in non zero magnetic field: 5/2

S1/2 S1/2

non-zero magnetic field:

D

5/2

• 3/2
• 1/2

1/2 3/2



• 5/2

3/2 + vibrational degrees of sideband cooling quantum state

S

1/2

freedom processing

S

1/2

• 1/2

1/2

Spectroscopy with quantized fluorescence (quantum jumps)

P D

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

S

8

detection efficiency: 99.85% S tensity

6 7 8

D-Zustand besetzt S-Zustand besetzt ungen

ents

D state occupied S state occupied

scence in

3 4 5

der Messu

measureme

D Fluores

1 2 3

Anzahl

# of m

D

time (s)

20 40 60 80 100 120

Zählrate pro 9 ms

counts per 9 ms

Toolbox: entangling Mølmer‐Sørensen gate operation

n n-1 n+1

entangling operation with bichromatic excitation

n 1 n+1 n 1 n+1

+ + +

n n+1 n-1 n-1 n-1

MS‐gate generates GHZ states Realizes a two‐body Hamiltonian where every ion interacts with every other, e.g., for 3 ions

Toolbox: collective local rotations

collective local operations with resonant excitation

D5/2

40Ca+

resonant manipulation

+ + + S1/2 Generate rotations about x/y axis Examples: Examples:



Toolbox: addressed single‐qubit rotations

D5/2

40Ca+

Single‐qubit rotations:

• ff‐resonant

manipulation

+ + +

Single‐qubit rotations: ‐ far detuned laser AC stark shift S1/2 ‐ AC‐stark shift Generate rotations about z axis Examples: p

Quantum gate operations: universal toolbox

+ + + + +

collective local operations, F = 99% single‐qubit z‐rotations, F = 99% Mølmer‐Sørensen entangling operations, F = 98%

Entangling gates with more than two ions Entangling gates with more than two ions

Two‐body interaction by off‐resonant spin‐motion coupling

Effective spin‐spin interaction

Many ions:

Creation of GHZ states

Mølmer‐Sørensen gate: Two ions Mølmer‐Sørensen gate: Two ions

A Sørensen K Mølmer

• A. Sørensen, K. Mølmer,

PRL 82, 1971 (1999) …

1

…

1 1

… …

1 1 1

…

Creating GHZ‐states with 4 ions Creating GHZ‐states with 4 ions

DDDD DDDD SDDD DSDD DDSD DDDS SDDS DSSD DSDS DDSS SSDD SDSD SSSD SSDS SDSS DSSS

n = 1 n = 0

SSSD SSDS SDSS DSSS |0,SSSS>

Creating GHZ‐states with 8 ions Creating GHZ‐states with 8 ions

DDDDDDDD SSSSSSSS

n ‐ qubit GHZ state generation with global MS gates

Fidelity (%) P it i l 1 99.5(7) Parity signal single ion Ramsey fringe 2 3 98.6(2) 3 4 97.0(3) 95 7(3) 4 6 95.7(3) 89 2(4) 6 8 89.2(4) 81.7(4)

• T. Monz, P. Schindler, J. Barreiro et al., Phys. Rev. Lett. 106, 130506 (2011)

Phase  of analysing pulse ()

n ‐ qubit GHZ state generation with global MS gates

genuine N‐particle Entanglement P it i l 8 96 by Parity signal 10 40 12 18 12 18 14 17

W Dü I Ci W Dü I Ci

• W. Dür, I. Cirac
• J. Phys A 34,

6837, (2001)

• W. Dür, I. Cirac
• J. Phys A 34,

6837, (2001)

• T. Monz, P. Schindler, J. Barreiro et al., Phys. Rev. Lett. 106, 130506 (2011)

Fidelity decay of n ‐ qubit GHZ states (simple model)

• T. Monz, P. Schindler, J. Barreiro, M. Chwalla, M. Hennrich, Innsbruck (2009)

single‐ion Ramsey contrast s g e o a sey co as

Assumption: Assumption:

2 ions

p magnetic field fluctuations cause collective dephasing p magnetic field fluctuations cause collective dephasing

4 ions 8 i

• s

3 ions

• s

6 ions 8 ions Waiting time (µs)

Dephasing of a single qubit (decay of Ramsey contrast)

• M. Chwalla, T. Monz, P. Schindler, Innsbruck 2010

1/2 ‐1/2 ‐3/2 3/2 5/2 / 3/2 ‐5/2 ‐1/2 1/2



Fidelity decay in randomly fluctuating magnetic field

• T. Monz et al., Phys. Rev. Lett. 106, 130506 (2011)

Hamiltonian with Fidelity Fidelity with (GHZ state) and denotes an ensemble average over the random fluctions For stationary noise of the magnetic field, characterized by a decay we obtain:

B‐field autocorrelation and coherence decay

Coherence of large‐scale entanglement

• T. Monz et al., Phys. Rev. Lett. 106, 130506 (2011)
• ne qubit

relative error probability

correlated noise correlated noise

two qubits

‐> superdecoherence ‐> superdecoherence

two qubits three qubits four qubits six qubits

Coherence of large‐scale entanglement

• T. Monz et al., Phys. Rev. Lett. 106, 130506 (2011)

BUT BUT application of a MS gate operation to the initial state yields the symmetric GHZ state the symmetric GHZ state insensitive to global dephasing by magnetic field noise insensitive to global dephasing by magnetic field noise measured coherence time consistent with expectation from lifetime limit:

Coherence of large‐scale entanglement

• T. Monz et al., Phys. Rev. Lett. 106, 130506 (2011)

D5/2

5/2

S1/2 consistent with expectation from lifetime (D5/2) limit:

Error Protection: robust entanglement Error Protection: robust entanglement

5/2

prepare qubits i S t t

• H. Häffner et al., Appl. Phys. B 81, 151 (2005)
• 1/2

1/2 3/2 5/2

Hiding states in S, S‘ states avoids decoherence from spontaneous emission

D

in S states

D

5/2

• 5/2
• 3/2

spontaneous emission

S‘ S

1/2

• 1/2

1/2

S‘ S

Error Protection: Decoherence free subspace Error Protection: Decoherence free subspace

Idea: use long-lived Bell states as logical qubits

• Th. Monz et al., Phys. Rev. Lett. 103, 200503 (2009)

Idea: use long-lived Bell states as logical qubits Gate operations by Gate operations by

• addressing individual ions
• simultaneous addressing of innermost ions

Trade off more ions (2x) for much extended coherence times (100-1000x)

Error Protection: Decoherence free subspace Error Protection: Decoherence free subspace

Idea: use long-lived Bell states as logical qubits

• Th. Monz et al., Phys. Rev. Lett. 103, 200503 (2009)

Idea: use long-lived Bell states as logical qubits Gate operations by 1st experiment: Gate operations by

• addressing individual ions and MS-gates
• simultaneous addressing of innermost ions

Analysis by state and process tomography

Process tomography of CNOT gate with logical qubit Process tomography of CNOT gate with logical qubit

experiment mean gate fidelity

• Th. Monz et al., Phys. Rev. Lett. 103, 200503 (2009)

Quantum gate operations: universal toolbox

+ + + + +

collective local operations, F = 99% single‐qubit z‐rotations, F = 99% Mølmer‐Sørensen entangling operations, F = 98%

Collective entangling gates + individual light shifts

Basic set of operations: N ions

Mølmer-Sørensen gate collective spin flips individual light shift gates

+ + + + +

• favorable ion addressing by light shifts (~2)

no interferometric stability between beams required

• no interferometric stability between beams required

 Arbitrary unitary operations can be achieved !  Arbitrary unitary operations can be achieved ! ...but what is an optimal way to do this ?

Optimal control for arbitrary quantum gates

Quantum optimal control:

• V. Nebendahl et al.,
• Phys. Rev. A 79,

012312 (2009) Find such that 012312 (2009) Find such that Gradient ascent algorithm:

• N. Khaneja et al., J. Magn. Res. 172, 296 (2005)

g j , g , ( ) Modification of search algorithm:

• no simultaneous application of several Hamiltonians
• no simultaneous application of several Hamiltonians
• sequence of pulses with variable length

Example: quantum Toffoli gate

=

1

Example: quantum Toffoli gate

=

2 3   

Scalable quantum computation requires error correction

Optimal control : Quantum Error Correction

Quantum Error Correction: 3 qubits encode logical qubit (protection against spin flips)

spin flip errors spin flip errors spin flip errors spin flip errors error syndrome correction encoding reset ancillas y detection step

• V. Nebendahl et al.,

Implementation : 34 laser pulses (11 entangling pulses)

• Phys. Rev. A 79,

012312 (2009)

Quantum Error Correction, reduced

encoding error decoding corr. reset

 i l h i d f fi bi  requires only three instead of five qubits  repetition requires re‐encoding

Quantum error correction, our implementation

►correct for phase errors instead of bit flips -> add Hadamards reset ► add a dummy operation D, D-1 that simplifies encoding ► operation U does not matter since ancillas are reset ► operation U does not matter since ancillas are reset (but gives an additional degree of freedom for optimisation)

Quantum error correction, our encoding

reset

• ptimization procedure using a modified GRAPE algorithm
• ptimization procedure using a modified GRAPE algorithm
• V. Nebendahl et al., Phys. Rev. A 79, 012312 (2009)

Quantum error correction: reset procedure

reset the ancillas: reset the ancillas: shelve population of ancillas l

D

• ptical pumping

heating: 0.014 phonons/reset

• P. Schindler et al., Science 332 , 1059‐1061 (2011)

Experimental repetitive quantum error correction quantum error correction

d encode error repeat correct correct

Repetitive Quantum Error Correction: full procedure

• P. Schindler et al.,

Innsbruck (2010)

Repetitive Quantum Error Correction: Experiment

process tomography after each step F = 97% identity process F = 97% identity process F = 90% w/ errors F = 90% w/o errors F = 90% w/ errors F = 90% w/o errors F = 80% w/ errors F = 80% w/o errors F = 80% w/ errors F = 80% w/o errors step 1 step 2 step 3F = 70% w/ errors F = 72% w/o errors F = 70% w/ errors F = 72% w/o errors

pulse sequence

• P. Schindler et al., Science 332 , 1059‐1061 (2011)

Repetitive Quantum Error Correction: Results Repetitive Quantum Error Correction: Results

• P. Schindler et al.,

Science 332 , 1059‐1061 (2011)

Quantum Simulations

Goal

Simulate the physics of a quantum system of interest by another system that is easier to control and to measure

Experimental requirements

another system that is easier to control and to measure.

Experimental requirements

• engineering of interactions
• measurement of relevant observables

Trapped ions

• Small quantum system (qubits + continuous variables)
• Excellent quantum control
• Excellent measurement capabilities

Analog quantum simulator Analog quantum simulator

Goal

Simulate the physics of a quantum system of interest by another system that is easier to control and to measure. Engineer a Hamiltonian Hsim exactly matching the H il i H

Approach

system Hamiltonian Hsys

pp Examples:

Ult ld t i

• Ultracold atoms in
• ptical lattices

(MIT, Harvard, MPQ, NIST,…)

• Ion crystals

(JQI MPQ IQOQI ) (JQI, MPQ, IQOQI …)

• I. Buluta, F. Nori, Science 326, 108 (2009)

Digital Quantum simulator Digital Quantum simulator

Goal

Simulate the physics of a quantum system of interest by another system that is easier to control and to measure.

Approach

Use a quantum computer as a quantum simulator Decompose dynamics induced by system Hamiltonian q p q into sequence of quantum gates

Example:

Simulating the Dirac equation

… with a single trapped ion

Simulating the Dirac equation

• L. Lamata, J. León, T. Schätz, E. Solano, PRL 98, 253005 (2007)

The Dirac equation can be cast in a 1+1 dimensional form (1 spatial + 1 spinor degree of freedom) can be cast in a 1+1 dimensional form (1 spatial + 1 spinor degree of freedom) resonant bichromatic excitation Stark shift with the replacements

Dynamics of a relativistic wave packet

For the 1d Dirac equation we trace over the spin degrees of freedom and the remaining spinor represents the matter and anti‐matter components Zitt b “ „Zitterbewegung“ due to interference between positive and negative energy parts of spinor

Experiment: Simulating the Dirac equation

Apply the Hamiltonian and measure the expectation value of the position

massless particle massless particle Zitterbewegung with increasing mass term

• R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature 463, 68 (2010)

Dirac particle in external potentials

• J. Casanova, J. Garcia‐Ripoll, R. Gerritsma, C. Roos, E. Solano, PR A 82, 020101(R) (2010)

For : Klein tunneling Klein tunneling Realization of a linear potential with trapped ions: A second ion driven by bichromatic light field and prepared in an eigenstate of  A second ion driven by bichromatic light field and prepared in an eigenstate of x

Experimental simulation of Klein tunneling Experimental simulation of Klein tunneling

steep

Linear potential

shallow

No potential

steep shallow ime Ti

Free propagation Reflection Reflection and tunneling positive and negative energy parts

• R. Gerritsma et al, Phys. Rev. Lett. 106, 060503 (2011)

Trapped ion quantum simulation: digital approach Trapped ion quantum simulation: digital approach

Decompose dynamics induced by system Hamiltonian p y y y into sequence of quantum gates

System: qubit register Quantum gate toolbox:

• Single qubit‐gates

g q g

• Entangling two‐qubit gates
• Engineering of many‐body interactions

E i i f di i i

• Engineering of dissipation

Trapped ion quantum simulation: digital approach Trapped ion quantum simulation: digital approach

Decompose dynamics induced by system Hamiltonian p y y y into sequence of quantum gates

System: qubit register Quantum gate toolbox:

• Single qubit‐gates

g q g

• Entangling two‐qubit gates
• Engineering of many‐body interactions

E i i f di i i

• Engineering of dissipation

Universal Quantum Simulator: The Algorithm Universal Quantum Simulator: The Algorithm

model of some local system to be simulated for a time t

0. have a universal set on ‘encoding’ degrees of freedom 1 b ild h l l l i l 1. build each local evolution operator separately, for small time steps, using operation set 2 approximate global evolution operator 2. approximate global evolution operator using the Trotter approximation

… …

“Efficient for local quantum systems“

• S. Lloyd,

“Efficient for local quantum systems“

• S. Lloyd,

1 Trotter (digital) time step

y , Science 273, 1073 (1996) y , Science 273, 1073 (1996)

Proof‐of‐principle demonstration Proof‐of‐principle demonstration

2‐spin Ising system 69%

………

Mølmer‐Sørensen gate 85% AC‐Stark gate dynamics to simulate: 91%

………

y

bability

………

pro

91%

Some of the spin systems simulated Some of the spin systems simulated

Ising Ising XY XY XYZ XYZ 2‐spin simulations Ising type 1 Ising type 1 Ising type 2 Ising type 2 n‐body n‐body 3‐spin simulations >3‐spin simulations 4 spins 4 spins 6 spins 6 spins

• B. Lanyon et al.,

Science 334, 6052 (2011)

New two‐spin simulations New two‐spin simulations

• B. Lanyon et al., Science 334, 6052 (2011)

Simulating many‐body interactions Simulating many‐body interactions

• B. Lanyon et al.,

Science 334, 6052 (2011)

Simulating many‐body interactions Simulating many‐body interactions

Simulated phase evolution with 6 spins t l t b d d ith entanglement bounds measured with the procedure according to

• H. Hofmann, Phys. Rev. Lett. 94, 160504 (2005)
• B. Lanyon et al., Science 334, 6052 (2011)

Outlook and future work Outlook and future work

J=2B

• Time dependent dynamics allows preparation of complex eigenstates exploration of
• Time dependent dynamics allows preparation of complex eigenstates, exploration of

ground state properties and quantum phase changes

• Frequencies tell you about spectrum:
• Frequencies tell you about spectrum:

Fourier transform the data and get the energy gaps in the simulated Hamiltonian

• Energy eigenvalues could be extracted by embedding into phase estimation algorithm

Energy eigenvalues could be extracted by embedding into phase estimation algorithm

• Current limiting source of error:

thought to be laser intensity fluctuations limiting possible simulation size and complexity g y g p p y

• Inclusion of error correction and error protection

QIP and Quantum Error Correction with Trapped Ca+ Ions QIP and Quantum Error Correction with Trapped Ca+ Ions

 Gate ops and coherence of multi‐partite entanglement  Q t i f ti t lb  Quantum information toolbox  Quantum error correction with trapped ions  Undoing a quantum measurement  Quantum Simulation of the Dirac equation and Klein‐tunneling  Universal quantum simulator with ion traps (Trotter)

Future: Future:

 further optimization of logic operations further optimization of logic operations  error correction protocols with five qubits, different encoding  implementation of logical qubits scalability issues  implementation of logical qubits, scalability issues  miniaturize traps, interface quantum information

… …

Future goals and developments

 more qubits (~20 – 50)  b tt fid liti  better fidelities  faster gate operations cryogenic trap, micro‐structured traps  faster detection  development of 2‐d trap arrays, onboard addressing, electronics etc.  entangling of large(r) systems: characterization ?  implementation of error correction, keep „qubit qubit alive alive“ “  applications ‐ small scale QIP (e.g. repeaters) ‐ quantum metrology, enhanced S/N, tailored atoms and states ‐ quantum simulations (Majorana equation, spin Hamiltonians, incl. QEC) ‐ quantum computation (period finding, quantum Fourier transform, factoring)

The international Team 2011 The international Team 2011

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The international Team 2011 The international Team 2011

• C. Roos

J Barreiro

• T. Monz
• P. Schindler

D Nigg N Röck

• F. Zähringer
• C. Hempel

M Niedermair

• M. Hennrich
• M. Brownnutt
• T. Northup
• J. Barreiro
• G. Hetet
• R. Gerritsma

B Lanyon

• D. Nigg
• M. Harlander
• A. Stute

B Brandstätter

• N. Röck
• M. Rambach
• J. Ghetta

K Schüppert

• M. Niedermair
• M. Kumph
• R. Lechner

B Ames

• M. Chwalla
• B. Lanyon
• B. Brandstätter
• B. Casabone
• L. Slodička
• K. Schüppert
• B. Ames
• M. Brandl
• P. Jurcevic

Theory collaboration: P. Zoller, M. Müller

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