Recent Experiments in Quantum Information with Trapped Ions

1
Recent experiments
in QI with Trapped Ions
Recent Experiments
in Quantum Information with Trapped Ions
T. Coudreau
Introduction
Teleportation
T. Coudreau
thomas.coudreau@univ-paris-diderot
Cat states
High fidelity
gate
Conclusion
Laboratoire Matériaux et Phénomènes Quantiques
Université Paris Diderot – Paris 7 et CNRS, Paris, France
http://mpq.univ-paris-diderot.fr/spip.php?rubrique29
MAPPI 2008 – Les Houches

Three levels of understanding
Recent experiments
in QI with Trapped Ions
Level 1 : Manipulation of qubits via single and two qubit
gates;
Level 2 : Manipulation of ion qubits via quantum gates;
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Level 3 : Manipulation of internal (energy levels) and
external (phonon modes) variables with lasers and e.m.
fields.
All three levels are important:
quantum gates have been originally devised without a
specific physical system in mind;
not taking into account the physical system is deemed
in the end to failure as specific problems & solutions
arise;
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3
Outline
Recent experiments
in QI with Trapped Ions
T. Coudreau
Teleportation
Transfer of an unknown quantum state from a site A to
a site B.
Experiment performed in Innsbruck in 2004 (and ...)
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Cat states
Generation of a highly non classical state
Experiment performed in Boulder in 2005 (and ...)
Towards fault tolerant computation
High fidelity gate
Experiment performed in Innsbruck in 2008

Some references
Recent experiments
in QI with Trapped Ions
Principle
G. Bennet et al., Teleporting an unknown quantum state via dual
classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett.
70, 1895 (1993);
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Photonic realizations
“Conditional” : D. Bouwmeester et al., Experimental quantum
teleportation, Nature 390, 575 (1997);
“Unconditional” : A. Furusawa et al., Unconditional quantum
teleportation, Science 282, 706 (1998)
Ion realizations
M Riebe et al., Deterministic quantum teleportation with atoms,
Nature 429, 734 (2004);
M.D. Barrett et al., Deterministic quantum teleportation of atomic
qubits, Nature 429, 737 (2004).
4

Principle - level 1
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Goal
Transmit an unknown quantum state from a place A (Alice)
to a place B (Bob) : ∣ ∣ ∣φ
〉
= α|0〉 + β|1〉
Method
Alice and Bob share an entangled state of two particles
Alice mixes her particle with the one to teleport and
makes an appropriate measurement;
Depending on the outcomes of the measurements, Bob
makes an appropriate transformation on his particle
Prerequisites
No-cloning theorem : the unknown state is destroyed
Special relativity : classical information is sent
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Detailed principle - level 2
Recent experiments
in QI with Trapped Ions
Entangled state
Alice and Bob share an entangled state : ∣ ∣ ∣ψ
〉 =
|0〉 a |1〉 b +|1〉 a |0〉 b √2 ;
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Mixing the states
Alice performs change of basis from the Bell-state basis to a
measurable basis (CNOT + Hadamard gates);
Measurement
Alice measures the state of her ions via fluorescence
Bob’s operation
Bob changes his state depending on Alice’s results

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Detailed operation - level 3
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
The system
Optical qubits : |1〉 = |S1/2,m = −1/2〉,
|0〉 = |D5/2,m = −1/2〉
Ions are stored in a linear Paul trap (ω = 2π × 1.2 MHz)
Single qubit gates are performed by a resonant laser at
729 nm
Two qubit gates are operated via the motional modes;

8
Detailed operation - level 3
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Bell state measurement
Instead of a standard measurement (CNOT +
Hadamard gate), the experiment was performed using
CZ + phase gates;
Fluorescence is observed by shining laser light
resonant with the S → P transition (397 nm) : S states
fluoresce, D states do not;
In order for the measurement result to be directly
exploitable, one must use a photo multiplier tube : since
it can not distinguish between |SD〉 and |DS〉, one of
the ions is put in a “dark” state, i.e. D5/2, m = -5/2 ;

9
Time table
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion

10
The “real” experimental implementation
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion

11
Experimental fidelity
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion

Some references
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
Photonic realizations
Single photons in a cavity : Observing the Progressive
Decoherence of the “Meter” in a Quantum
Measurement, M. Brune et al., Phys. Rev. Lett. 77,
4887 (1996)
Free e.m. fields : Generating Optical Schrödinger
Kittens for Quantum Information Processing, A.
Ourjoumtsev et al., Science 312, 83 (2006)
High fidelity
gate
Conclusion
Ion realizations
Creation of a six-atom ‘Schrödinger cat’ state, D.
Leibfried et al., Nature 438, 639 (2005)
Scalable multiparticle entanglement of trapped ions, H.
Haffner et al., Nature, 438, 643 (2005)
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Principle - level 1
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
Goal
Create a superposition of
“two maximally different
quantum states”;
Create a multiparticle
entangled state.
∣
∣ψ 〉 =
1 √
2
(|↑↑ ... ↑〉 + |↓↓ ... ↓〉)
High fidelity
gate
Conclusion
Test of the cat / entangled state
Measure the fidelity, i.e. overlap with the cat state;
Measure the coherence between the two “feet” of the
cat.

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Principle - level 2
Some useful definitions and relations
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Dicke operators : ⃗ J =
N∑
⃗σ j
j=1
Dicke states : N + 1 eigenstates of J 2 and J z with
eigenvalues J = N/2 and M (−J ≤ M ≤ M)
N e and N g are given by N e = J + M and N g = J − M.
|↑↑ ... ↑〉 = |J,J〉 and |↓↓ ... ↓〉 = |J,−J〉;
∣
∣ψ 〉 = √ 1 (|J,J〉 + |J,−J〉);
2
For N even, |J,J〉 + |J,−J〉 is even in J y while
|J,J〉 − |J,−J〉 is odd;
|J,−J〉 = 1 2 (|J,J〉 + |J,−J〉) − 1 2
(|J,J〉 + |J,−J〉)

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Principle - level 2
Entangling gate
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
The application of J 2 y
will entangle the ions
(
exp i π )
(
2 J2 y |J,−J〉 = exp i π )[ 1
2 J2 y (|J,J〉 + |J,−J〉)−
2
]
1
2 (|J,J〉 + |J,−J〉)
= 1 2 (|J,J〉 + |J,−J〉) − i (|J,J〉 + |J,−J〉)
2
= 1 − i
2 |J,J〉 + 1 + i
2 |J,−J〉
= 1 − i
2
|↑ ... ↑〉 +
1 + i
2
|↓ ... ↓〉
The state produced at the end of the interaction is, apart
from a phase shift, the desired cat state.

Principle - level 3
Entangling gate
Recent experiments
in QI with Trapped Ions
Yet another gate
A simple relation:
(
exp i π 2 J2 y
)
= exp
(
−i π ) (
2 J x exp i π ) (
2 J2 z exp i π )
2 J x
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
State dependent phase shift
Under the influence of a force, the vibrational state can
follow a closed curve.
Phase shift proportional to the area enclosed in the
curve : to have a phase shift proportional to J 2 z , one
must have a force proportional to J z ;
Since J z = (N e − N g )/2, the force must have a state
dependent sign.
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Principle - level 3
Setting up the force
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
The qubit is encoded in hyperfine ground states of Be +
ions;
The force is produced by shining Raman beams on the
ions;
Illuminating the ions : the ions spacing is chosen so
that each ion receives the same light (in practice, error
≤ 4%);
State dependent force
The detuning is chosen so
that the two Raman
amplitudes have opposite
signs.

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Principle : summary
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Level 1 Level 2 Level 3
Initialize state
Put system in
|↓ ... ↓〉|0〉
Optical pumping
+ laser cooling
expi π 2 J Raman π/2
x gate
pulse on carrier
Raman pulse at
Entangling gate
exp ( )
i π expi π
2 J2 y gate 2 J2 z gate frequency ω 2 −
ω 1 = ν + δ for duration
τ g
expi π 2 J x gate
Raman π/2
pulse on carrier

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State read-out by fluorescence
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
The first measurement consists in measuring the
fluorescence;
ions in |g〉 will emit an average of 10 photons while
those in |e〉 emit none;
The corresponding histogram must have two peaks, the
first around 0 photons (|e ...e〉), the second one around
6 × 10 photons (|g ...g〉).
Cat states
High fidelity
gate
Conclusion

20
Coherence measurement
Recent experiments
in QI with Trapped Ions
It is necessary to test the coherence between |e ...e〉
and |g ...g〉;
This can be done by an interference measurement :
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
|g ...g〉 expi π 2 J2 y
−→
GHZ expiφJ z
−→
expi π 2 J2 y
−→
read-out

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High fidelity operations
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Motivation
Quantum error correction imposes “strict” bounds on
the tolerable errors : 10 −4 – 10 −2
The operations described previously failed to achieve
this bound e.g. preparation of a 6 ions entangled state
F ≈ 50%
Cat states
High fidelity
gate
Conclusion
Recent results
J. Benhelm et al., Towards fault-tolerant quantum
computing with trapped ions, Nature Physics (apr.
2008)

Towards fault-tolerant quantum computing with
trapped ions
System
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Recent experiments
in QI with Trapped Ions
T. Coudreau
two 40 Ca ions in a linear Paul trap
Mølmer - Sørensen entangling gate (H ∝ J 2 y )
Adiabatic switching during a well chosen time, τ gate
gives a propagator
Introduction
Teleportation
Cat states
High fidelity
gate
It is an entangling gate with
U gate = exp(−i(π/8)J 2 y )
Conclusion
|SS〉 τ gate
−→ |SS〉 + i |DD〉 τ gate
−→ |DD〉 τ gate
−→ |DD〉 + i |SS〉 τ gate
−→ |SS〉

Towards fault-tolerant quantum computing with
trapped ions
Experimental results (a)
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
|SS〉 → |SS〉 + i |DD〉 → |DD〉
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Towards fault-tolerant quantum computing with
trapped ions
Experimental results (b)
24
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion

Towards fault-tolerant quantum computing with
trapped ions
Experimental results (c)
25
Recent experiments
in QI with Trapped Ions
T. Coudreau
State populations
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
The total time corresponds to the application of 17
gates
Even after 21 gates, the fidelity is still on the order of
80%

Towards fault-tolerant quantum computing with
trapped ions
Some causes of errors
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Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
State initialization
2 × 10 −3
State discrimination
1.5 × 10 −3
Pulse shaping
10 −4
Coupling strength variation (laser intensity
variation, beam pointing, ion < 4 × 10 −4
motion . . . )
Non resonant excitation
< 4 × 10 −4
Laser frequency uncertainty < 2 × 10 −3 (*)
Total
7 ×10 −3
(*) not directly measured

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Conclusion
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
di Vincenzo’s criteria have been “met”
A large number of gates have been demonstrated
A great variety of “useful” applications have been
demonstrated
Teleportation
Multiparticle entangled states
Large fidelity gate
Three–qubit gate
Long coherence time
Entanglement enhanced measurements
Quantum error correction
Quantum search (Grover’s algorithm)
...

Outlook
Recent experiments
in QI with Trapped Ions
T. Coudreau
High fidelity operations
Large constraints implied by quantum error correction
schemes;
Improve all stages of the gate: state preparation, operation,
read-out;
Introduction
Teleportation
Cat states
High fidelity
gate
Conclusion
Scalable traps
Large number of qubits and interactions needed;
Devise an efficient & coherence preserving trap & shuttling
method
Other schemes
Quantum simulations / Topological quantum computing.
Ions as atomic ensembles : Aarhus, Paris
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29
On-going work in Paris
Recent experiments
in QI with Trapped Ions
T. Coudreau
Introduction
Teleportation
Cat states
Towards large ensembles
Micro-traps
up to 4×10 4 Sr + ions
Light atom interface
See S. Removille’s poster
on thursday.
High fidelity
gate
Conclusion
Developed in
collaboration with Thales
Research and
Technology
No trapped ions yet !