Quantum Information Science with Trapped Ca+ Ions with Trapped ...

Quantum Information Science
with Trapped Ca + Ions

Quantum Information Science
with Trapped Ca + Ions
Rainer Blatt
Institute of Experimental Physics, University of Innsbruck,
Institute of Quantum Optics and Quantum Information,
Austrian Academy of Sciences
• Trapped Ca + for quantum information processing
• Tffli Toffoli gate with trapped ions
• Mølmer-Sørensen high fidelity gate operation
• Quantum computation with logical qubits
• Testing quantum mechanics with trapped ions
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The requirements for quantum computation
D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001)
I. Scalable physical system, well characterized qubits
II.
III.
Ability to initialize the state of the qubits
Long relevant coherence times, much longer than gate operation time
IV.
“Universal” set of quantum gates
V. Measurement capability specific to implementation
VI.
VII.
Ability to interconvert stationary and flying qubits
Ability to faithfully transmit flying qubits between specified locations
The seven commandments for QC !!

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

Qubits with trapped ions
Storing and keeping quantum information requires long-lived atomic states:
• optical transition frequencies
(forbidden transitions,
intercombination lines)
S – D transitions in alkaline earths:
Ca + , Sr + , Ba + , Ra + , (Yb + , Hg + ) etc.
• microwave transitions
(hyperfine transitions,
Zeeman transitions)
alkaline earths:
9
Be + , 25 Mg + , 43 Ca + , 87 Sr + ,
137
Ba + , 111 Cd + , 171 Yb +
P 1/2
D 5/2
S 1/2
P 3/2
TLS
S 1/2
TLS
S 1/2 Boulder 9 Be + ; Michigan 111 Cd + ;
Innsbruck 40 Ca + Innsbruck 43 Ca + , Oxford 43 Ca + ;
Maryland 171 Yb + ;

Level scheme of Ca +
P 3/2
854 nm
qubit on narrow S - D
quadrupole transition
1
s
P 1/2
393 nm
397 nm
866 nm
D 5/2
D 3/2
729 nm
40 Ca +
S 1/2

Innsbruck linear ion trap (2000)
1.0 mm
6 mm

Detection of 6 individual ions

Spectroscopy of the S 1/2 –D 5/2 transition
P 1/2 D 5/2
2 level system:
Fluorescence
detection
2-level-system:
S 1/2
Zeeman structure in
non-zero magnetic field:
D 5/2
-3/2 -1/2 1/2 3/2
-5/2
3/2
5/2
sideband
+ vibrational
quantum
cooling
degrees of
state
S 1/2
1/2 freedom processing
-1/2

Quantized Ion Motion
2-level-atom
harmonic trap
coupled system
excitation: various resonances
spectroscopy: carrier and sidebands
n = 0
D 5/2
n = 1
n = -1
S 1/2
n = 0 1 2
Laser detuning

Coherent state manipulation
D,0
D ,1
carrier
(C)
S,1
S,0
carrier and sideband
Rabi oscillations
with Rabi frequencies
blue sideband
(BSB)
Lamb-Dicke parameter

Quantum information processing with trapped ions
►algorithms:
sequence of single qubit and
two-qubit gate operations
►gate operations:
sequences of laser pulses
(carrier and/or sideband pulses)
carrier sideband
d
►analysis: l i
measure density matrix of
state or process (tomography)
measure entanglement
g
via parity oscillations

Scalable push-button generation of GHZ states
Fidelity
Fidelity

Eight – Ion W state
Fidelity: 0.76
6561 settings,
~ 10 h measurement time,
but reconstruction time:
~ several days on a
computer cluster
genuine
e
8-particle
entanglement !
The
Quantum Byte
H. Häffner et al., Nature 438, 643 (2005)

Quantum Process Tomography
control bit
target bit
target bit
control bit
characterizes gate operation completely

– matrix for observed CNOT gate operation

– matrix for observed CNOT gate operation

Scalable quantum computation requires error correction

Toffoli gate: controlled-controlled NOT
Toffoli gate (Tommaso Toffoli, 1980):
…… is a universal reversible logic gate, i.e. any reversible circuit can
be constructed from Toffoli gates.
also known as the controlled-controlled-NOT or CCNOT-gate operation
useful, e.g. for error correction

Toffoli gate: pulse sequence
use 2-phonon excitation
Th. Monz et al.,
Phys. Rev. Lett. 102, 040501 (2009)
B B B B B B B B
B
B
effective
for n=1,2
result:
n=1 iff SS
C B B B C
CNOT
iff n=1
undo encoding,
reverse phases

Toffoli gate: experimental truth table
density matrix
State Fidelity
Th. Monz et al., Phys. Rev. Lett. 102, 040501 (2009)

Toffoli gate: process tomography
- matrix for ideal TOFFOLI gate operation
III
IIXIIY
IIYIIZIXI
ZZZ
ZZZ
III

Toffoli gate: process tomography
- matrix for the real TOFFOLI gate operation
Mean
Process Fidelity
Th. Monz et al., Phys. Rev. Lett. 102, 040501 (2009)

Quantum procedures and fidelities
► single qubit operations > 99 %
► 2-qubit CNOT gate ~ 93 %
► Bell states
► W and GHZ states
93-95 %
85-90 %
► Quantum teleportation 83 %
► Entanglement swapping
~ 80 %
► Toffoli gate operation 71 %
!
BUT: for fault-tolerant operation needed > 99 %

Mølmer - Sørensen gate - operation
bichromatic laser excitation
close to upper and lower
sidebands induces collective
state changes (spin flips)
K. Mølmer, A. Sørensen,
Phys. Rev. Lett. 82, 1971 (1999)
C. A. Sackett et al.,
Nature 404, 256 (2000)

measure the parity :
Measuring entanglement
C. A. Sackett et al., Nature 404, 256 (2000)
Ion 1
Ion 2
/2
CNOT /2
/2
oscillates t with 2
54% visibility
Fidelity =
0.5(P SS +P DD +visibility) =
71(3) %
F. Schmidt-Kaler et al., Nature 422, 408 (2003)

Deterministic Bell states using the Mølmer-Sørensen gate
entangled
J. Benhelm, G. Kirchmair,
C. Roos
Theory: C. Roos,
New Journ. of Physics 10,
013002 (2008)
measure entanglement
via parity oscillations
gate duration
average fidelity

Mølmer-Sørensen gate: thermal states
Gate operation after ground Doppler state cooling cooling
Fidelity :
F=98 99.3(1) 98.0(1) %
Gate operation ≈ independent of motional state !
G. Kirchmair et al., New. J. Phys. 11, 023002 (2009)

Collective entangling gates + individual light shifts
N ions
Basic set of operations:
Mølmer-Sørensen gate
individual light shift gates
collective spin flips
+ + + + +
● favorable ion addressing by light shifts (~ 2 )
● no interferometric stability between beams required
Arbitrary unitary operations can be achieved !
...but how

Optimal control for arbitrary quantum gates
Quantum optimal control:
V. Nebendahl et al.,
Phys. Rev. A 79,
012312 (2009)
Find
such that
Gradient ascent algorithm: N. Khaneja et al., , J. Magn. Res. 172, , 296 (2005)
Modification of search algorithm:
Example: quantum Toffoli gate
● no simultaneous application of several Hamiltonians
● sequence of pulses with variable length
1
2
=
3

Optimal control : Quantum Error Correction
Quantum Error Correction: 3 qubits encode logical qubit (protection against spin flips)
spin flip errors
spin flip errors
encoding
reset ancillas
error syndrome
detection
correction
step
Implementation : 34 laser pulses (11 entangling pulses)
V. Nebendahl et al.,
Phys. Rev. A 79,
012312 (2009)

Scalability is facilitated by error avoidance
Decoherence-free Bell states:
C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)
decoherence-time:
0.5 x 1.05(15) s

prepare qubits
in S states
t
D 1/2 3/2
-1/2
Robust entanglement
5/2
H. Häffner et al., Appl. Phys. B 81, 151 (2005)
Hiding states in S, S‘ states
avoids decoherence from
-5/2 -3/2 spontaneous emission
D 5/2
S 1/2
S
-1/2
1/2
S‘

Quantum gates within a decoherence free subspace
Idea: use long-lived Bell states as logical qubits
K. Kim et al., Innsbruck, 2006
Gate operations by
- addressing individual ions
- simultaneous addressing of innermost ions
Trade off more ions (2x) for much extended coherence times (100-1000x)

Universal quantum computation with logical qubits
Idea: use long-lived Bell states as logical qubits
Th. Monz, K. Kim,
Ph. Schindler (2008)
Gate operations by
- addressing individual ions and MS-gates
- simultaneous addressing of innermost ions
1 st experiment:
Analysis by state and process tomography

Gate operations with logical qubits
Z
Z
K. Kim et. al.
Phys. Rev. A77, 050303 (2008)
X
C. Roos,
New Journ. Phys. 10 (2008)
Z
single qubit phase shift by
AC Stark shifts
F. Schmidt-Kaler et. al.
Europhys. Lett. 65, 587 (2004)

CNOT gate operation with a logical qubit
spin echo

theory
Process tomography of CNOT with logical qubits

Process tomography of CNOT with logical qubits
experiment
mean gate fidelity

Quantum state manipulation with logical qubits
results and limitations:
● mean gate fidelity: 89(4)%
● limited by addressing errors, spurious laser noise, position stability,
equal illumination of neighbouring ions
advantages:
● insensitive to laser linewidth,
● insensitive to AC-Stark shifts
(work in progress)
● lifetime limited coherence time
(work in progress)

Testing quantum mechanics with trapped ions
• Do observables have predetermined values,
fixed by hidden variables
J. S. Bell
“No physical theory
of local hidden
variables can
reproduce all of
the predictions of
quantum mechanics” S. Kochen E. Specker
“Non‐contextual t lhidden
variable theory cannot
reproduce the predictions of
quantum mechanics”

Kochen‐specker theorem
“Non‐contextual hidden variable theory cannot
reproduce the predictions of quantum mechanics”
Non‐Contextuality
Any measurement A should give a value independent
of other compatible measurements B :
(AB) = (A) ∙ (B)
Experimental tests:
Huang, Y.F., Phys. Rev. Lett. 90, 250401 (2003)
Hasegawa, Y., Phys. Rev. Lett. 97, 230401 (2006)
Specker, R., Dialectica 14, 239‐246 (1960)
Bell, J.S., Rev. Mod. Phys. 38, 447‐452 (1966)
Kochen, S., & Specker, E.P., J. Math. Mech. (1967)

The Peres – Mermin square
Measure observables
A ij
with v(A ij
) = ±1,
where the observables
in each row and column
are mutually compatible
A. Peres, Phys. Lett. A 151, 107–108 (1990)
N. D. Mermin, Phys. Rev. Lett. 65, 3373–3376 (1990)
(1)
z
(2)
z
(1)
z
(2)
z R 1 =1
1
z z z z
(2) x
(1)
x
(1)
x
(2)
x R 2 =11
z
(1) x
(2)
x
(1) z
(2)
y
(1) y
(2)
R 3 =1
C 1 =1 C 2 = 1 C 3 = ‐1

Testing the Kochen‐Specker theorem
● KS = R 1 + R 2 + R 3 + C 1 + C 2 ‐ C 3
● Non‐contextual hidden variable theory
KS ≤ 4
● quantum mechanics
KS =6
(1) z
(2) z
(1)
(2)
z z
R 1 =1
for any input state
(2)
x
(1)
x
(1)
x
(2)
x
R 2 =1
z
(1) x
2
x
(1) z
(2)
y
(1) y
(2)
R 3 =1
C 1 =1 C 2 = 1 C 3 = ‐1
A. Cabello, Phys. Rev. Lett. 101, 210401 (2008)

Measuring operators like x x
(1) (2) (1) (2)
z z z z
(2) x
(1)
x
(1)
x
(2)
x
z
(1) x
(2)
x
(1) z
(2)
y
(1) y
(2)
We have to measure all observables with the same setup
whether measured in a row or a column map to single ion

Experimental procedure
Measurement scheme for rows and columns of PM square:
unitary for respective
element of PM square
actual measurement

Results
z
(1)
z
(2)
z
(1) z
(2) R 1 =0.92(1)
Input state:
|SS |DD
(2)
x
(1)
x
(1)
x
(2)
x R 2 =0.93(1) 0 z
(1) x
2
x
(1) z
(2)
y
(1) y
(2) R 3 =0.90(1)
C =0.90(1) C = 0.89(1)C = ‐0.91(1) 1 2 3 = 5.46(4)
KS > 4

State independence
States are prepared with 97(2) % fidelity (quantum state tomography)
KS avg = 5.38(4)
> 4
State independence

Simulating the Dirac equation
… with a single trapped ion

Simulating the Dirac equation
Lamata et al., PRL 98, 253005 (2007)
The Dirac equation
can be cast in a 1+1 dimensional form (1 spatial + 1 spin degree of freedom)
resonant bichromatic excitation
Stark shift
with the replacements

Quantum Information Science with Trapped Ions
Ca + for quantum information processing
Quantum process tomography, gate operations
Toffoli gate with trapped ions
Mølmer-Sørensen high fidelity gate operation
Quantum gate operations with logical qubits
Testing quantum mechanics with trapped ions
Simulating the Dirac equation with a trapped ion
Future:
further optimization of logic operations
error correction protocols with three and five qubits
implementation with 43 Ca + , logical qubits + scalability
miniaturize traps, interface quantum information

Future goals and developments
more qubits (~20 – 50)
better fidelities
faster gate operations
faster detection
cryogenic trap, micro-structured traps
development of 2-d trap arrays, onboard addressing, electronics etc.
entangling of large(r) systems: characterization
implementation of error correction, keep „qubit alive“
applications
- small scale QIP (e.g. repeaters)
- quantum metrology, enhanced S/N, tailored atoms and states
- quantum simulations
- quantum computation

The international Team 2008
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The international Team 2008
C. Roos
H. Häffner
W. Hänsel
P. Schmidt
M. Hennrich
R. Gerritsma
F. Dubin
K. Kim
A. Villar
M. Brownnutt
N. Daniidilis
T. Northup
J. Barreiro
G. Hetet
M. Riebe
J. Benhelm
C. Russo
H. Barros
F. Splatt
S. Narayanan
F. Zähringer
P. Schindler
B. Brandstädter
M. Kumph
A. Stute
M. Harlander
B. Hemmerling
M. Chwalla
Th. Monz
G. Kirchmair
S. Gerber
L. Slodička
M. Niedermair
L. Anderlan
V. Nebendahl
R. Lechner
D. Nigg
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