Download Quantum Computation with Neutral Atoms

Quantum Computation
with Neutral Atoms
Marianna Safronova
Department of Physics and Astronomy
Why quantum information?
Information is physical!
Any processing of information
is always performed by physical means
Bits of information obey laws of classical physics.
1
Why quantum information?
Information is physical!
Any processing of information
is always performed by physical means
Bits of information obey laws of classical physics.
Why Quantum Computers?
Computer technology is
making devices smaller
and smaller…
…reaching a point where classical
physics is no longer a suitable model for
the laws of physics.
2
Bits & Qubits
Fundamental building
blocks of classical
computers:
Fundamental building
blocks of quantum
computers:
Quantum bits
or
QUBITS
BITS
STATE:
Basis states: 0 and 1
Definitely
Superposition:
0 or 1
ψ =α 0 +β 1
Bits & Qubits
Fundamental building
blocks of classical
computers:
BITS
STATE:
Fundamental building
blocks of quantum
computers:
Quantum bits
or
QUBITS
Basis states: 0 and 1
Definitely
0 or 1
3
Qubit: any suitable two-level
quantum system
Bits & Qubits:
Superposition
primary differences
ψ =α 0 +β 1
4
Bits & Qubits:
primary differences
Measurement
Classical bit: we can find out if it is in state 0 or 1 and the
measurement will not change the state of the bit.
Qubit:
Quantum calculation:
number of parallel processes
due to superposition
FR
QO
Bits & Qubits:
Superposition
Measurement
primary differences
ψ =α 0 +β 1
Classical bit: we can find out if it is in state 0 or 1 and the
measurement will not change the state of the bit.
Qubit: we cannot just measure α and β and thus determine
its state! We get either 0 or 1 with corresponding
probabilities |α|2 and |β|2.
2
2
α + β =1
The measurement changes the state of the qubit!
5
Hilbert space is a big place!
- Carlton Caves
Multiple qubits
Hilbert space is a big place!
- Carlton Caves
Multiple qubits
Two bits with states 0 and 1 form four definite states 00,
01, 10, and 11.
Two qubits: can be in superposition of four
computational basis set states.
ψ = α 00 + β 01 + γ 10 + δ 11
2 qubits
4 amplitudes
3 qubits
8 amplitudes
10 qubits
1024 amplitudes
20 qubits
1 048 576 amplitudes
30 qubits
1 073 741 824 amplitudes
500 qubits More amplitudes than our estimate of
number of atoms in the Universe!!!
6
Entanglement
Results of the measurement
ψ =
00 + 11
2
ψ ≠α ⊗β
First
qubit
Second qubit
0
0
1
1
Entangled
states
Quantum logic gates
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Logic gates
Quantum NOT gate
(X gate)
Classical NOT gate
NOT A
A
A
NOT A
0
1
1
0
α 0 +β 1
α 1 +β 0
X
Matrix form representation
0 1 
X =

1 0 
α   β 
X = 
 β  α 
The only non-trivial
single bit gate
More single qubit gates
Any unitary matrix U will produce a quantum gate!
1 0 
Z =

0 −1
α 0 +β 1
Hadamard gate:
α 0 +β 1
H
α 0 −β 1
Z
H=
α
1 1 1 


2 1 −1
0 +1
2
+β
0 −1
2
8
Two-qubit gates
Quantum CNOT gate
A
B
A
B'
AB
AB'
00
00
01
01
10
11
11
10
WE NEED TO BE ABLE TO MAKE
ONLY ONE TWO-QUBIT GATE!
Back to the real world:
What do we need to build a quantum computer?
Qubits which retain their properties.
Scalable array of qubits.
Initialization: ability to prepare one certain state
repeatedly on demand. Need continuous supply of 0 .
Universal set of quantum gates. A system in which
qubits can be made to evolve as desired.
Long relevant decoherence times.
Ability to efficiently read out the result.
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Real world strategy
“…If X is very hard it can be
substituted with more of Y.
Of course, in many cases both X and Y are beyond
the present experimental state of the art …”
David P. DiVincenzo
The physical implementation of quantum computation.
Experimental proposals
Liquid state NMR
Trapped ions
Cavity QED
Trapped atoms
Solid state schemes
And other ones …
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1. A scalable physical system with
well characterized qubits: memory
(a) Internal atomic state qubits:
ground hyperfine states of neutral trapped atoms
well characterized
Very long lived!
MF=-2,-1,0,1,2
F=2
5s1/2
1
87Rb:
6.8 GHz
F=1
0
Nuclear spin I=3/2
MF=-1,0,1
1. A scalable physical system with
well characterized qubits: memory
(b) Motional qubits : quantized levels in the trapping potential
also well characterized
http://www.colorado.edu/physics/2000/index.pl
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1. A scalable physical system with
well characterized qubits: memory
(a)Internal atomic state qubits
(b) Motional qubits
Advantages: very long decoherence times!
Internal states are well understood: atomic spectroscopy & atomic clocks.
1. A scalable physical system with
well characterized qubits
Optical lattices: loading of one atom per site
may be achieved using Mott insulator transition.
Scalability: the properties of optical lattice
system do not change in the
principal way when the size of the system
is increased.
Designer lattices may be created
(for example with every third site loaded).
Advantages: inherent scalability and parallelism.
Potential problems: individual addressing.
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2: Initialization
Internal state preparation: putting atoms in the ground hyperfine state
Very well understood (optical pumping technique is in use since 1950)
Very reliable (>0.9999 population may be achieved)
Motional states may be cooled to motional ground states (>95%)
Loading with one atom per site: Mott insulator transition and other
schemes.
Zero’s may be supplied during the computation (providing individual
or array addressing).
3: A universal set of quantum
gates
Hadamard gate:
CNOT
A
A
B
B'
AB
AB'
00
00
01
01
10
11
11
10
A
A'
H
A
0
1
A'
0 +1
2
0 −1
2
H = (X + Z)/ 2
π/8 gate:
T
0 
1
T =
iπ / 4 
0 e

Phase gate S:
1 0 
S=

0 i 
S =T2
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3: A universal set of quantum
gates
1. Single-qubit rotations: well understood and had been carried out
in atomic spectroscopy since 1940’s.
2. Two-qubit gates: none currently implemented
(conditional logic was demonstrated)
Proposed interactions for two-qubit gates:
(a) Electric-dipole interactions between atoms
(b) Ground-state elastic collisions
(c) Magnetic dipole interactions
Only one gate proposal does not involve moving atoms (Rydberg gate).
Advantages: possible parallel operations
Disadvantages: decoherence issues during gate operations
Two-qubit quantum gates
(a) Electric-dipole interactions between atoms
Brennen et al. PRL 82, 1060 (1999), PRA 61, 062309 (2000),
Pairs of atoms are brought to occupy the same site in far-off-resonance
optical lattice by varying polarization of the trapping laser.
Two “types” of atoms: trapped in σ+ and σ- polarized wells.
Near-resonant electric-dipole is induced by auxiliary laser (depending on the
atomic state).
Brennen, Deutch, and Willaims PRA 65, 022313 (2002)
Deterministic entanglement of pairs of atoms trapped in optical
lattice is achieved by coupling to excited state molecular hyperfine
potentials.
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Two-qubit quantum gates
(a) Electric-dipole interactions between atoms … cont.
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
Gate operations are mediated by excitation of Rydberg states
(b) Ground-state elastic collisions
Calarco et al. Phys. Rev. A 61, 022304 (2000)
Cold collisions between atoms conditional on internal states.
Cold collisions between atoms conditional on motional-state tunneling.
(c) Magnetic-dipole interactions between pairs of atoms
Rydberg gate scheme
Gate operations are mediated by excitation of Rydberg states
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
Why Rydberg gate?
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Rydberg gate scheme
Gate operations are mediated by excitation of Rydberg states
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
Do not need to
move atoms!
FAST!
Local blockade of Rydberg excitations
Excitations to Rydberg
states are suppressed
due to a dipole-dipole
interaction or van der
Waals interaction
http://www.physics.uconn.edu/~rcote/
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Rydberg gate scheme
FAST!
Rb
40p
R
Apply a series of laser pulses to
realize the following logic gate:
∆
1
0
00
10
01
11
5s
1
2
→ 00
→ 01
→ 10
→ − 11
Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)
Rb
Rydberg gate scheme
5s
40p
R
π [1]
∆
00
01
10
11
1
0
1
→
→
→
→
00
01
R0
R1
2π [ 2]
→
00
→ − 01
→
R0
→
R1
π [1]
→
00
→ − 01
→ − 10
→ − 11
2
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Decoherence
One of the decoherence sources: motional heating.
Results from atom “seeing” different lattice in ground
and Rydberg states.
Solution: choose the lattice photon frequency ω to
match frequency-dependent polarizability α(ω) of the
ground and Rydberg states.
Error correction: possible but error rate has to be
really small (< 10-4 ).
Other decoherence sources
Photoionization
Spontaneous emission
Transitions induced by black-body radiation
Laser beam intensity stability
Pulse timing stability
Individual addressing accuracy
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4. Long relevant decoherence times
Memory: long-lived states.
5s1/2
F=2
1
F=1
0
6.8 GHz
Fundamental decoherence mechanism for
optically trapped qubits: photon scattering.
Decoherence during gate operations:
a serious issue.
5: Reading out a result
“Quantum jump” method via cycling transitions.
Advantages: standard atomic physics technique, well understood and reliable.
Quantum computation with
NEUTRAL ATOMS: ADVANTAGES
Scalability
Possible massive parallelism
due to lattice geometry
Long decoherence times
(weak coupling to the
environment)
Availability of the controlled
interactions
Well-developed experimental
techniques for initialization,
state manipulation, and
readout
Accurate theoretical description of the system is possible.
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Quantum computation with
NEUTRAL ATOMS: PROBLEMS
Decoherence during the gate operations
(various sources)
Reliable lattice loading and individual addressing
QC architecture for lattice geometry:
Error-correcting codes and fault-tolerant computation,
how to run algorithms on neutral atom quantum
computer.
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Quantum information processing
with trapped ions
Courtesy of Timo Koerber
Institut für Experimentalphysik
Universität Innsbruck
1.
Basic experimental techniques
2.
Two-particle entanglement
3.
Multi-particle entanglement
4.
Implementation of a CNOT gate
5.
Teleportation
6.
Outlook
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
1
Experimental Setup
P1/2
D5/2
„quantum
bit“
S1/2
Important energy levels
• The important energy levels are shown on the next
slides; a fast transition is used to detect ion fluorescence
and for Doppler cooling, while the narrow D5/2
quadrupole transition has a lifetime of 1 second and is
used for coherent manipulation and represents out
quantum bit. Of course a specific set of Zeeman states is
used to actually implement our qubit. The presence of
other sublevels give us additional possibilities for doing
coherent operations.
2
Ca+: Important energy levels
S1/2 – D5/2 : quadrupole transition
τ = 7 ns
P1/2
τ=1s
D5/2
397 nm
729 nm
S1/2
Ca+: Important energy levels
τ = 7 ns
P1/2
D5/2
S1/2
„qubit“
„quoctet“ (sp?)
3
Qubits with trapped ions
Encoding of quantum information requires long-lived atomic states:
optical transitions
microwave transitions
Ca+, Sr+, Ba+, Ra+, Yb+, Hg+ etc.
9Be+, 25Mg+, 43Ca+, 87Sr+,
137Ba+, 111Cd+, 171Yb+
P3/2
P1/2
D5/2
qubit
S1/2
qubit
S1/2
String of Ca+ ions in Paul trap
row of qubits in a
linear Paul trap forms
a quantum register
4
String of Ca+ ions in linear Paul trap
row of qubits in a
linear Paul trap forms
a quantum register
ω z ≈ 0.7 − 2 MHz
ω x , y ≈ 1.5 − 4 MHz
50 µm
String of Ca+ ions in linear Paul trap
row of qubits in a
linear Paul trap forms
a quantum register
ω z ≈ 0.7 − 2 MHz
ω x , y ≈ 1.5 − 4 MHz
50 µm
5
Addressing of individual ions
0.8
0.7
Paul trap
0.6
Excitation
coherent
manipulation
of qubits
0.5
0.4
electrooptic
deflector
0.3
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
addressing waist: ~ 2.5 µm
< 0.1% intensity on neighbouring ions
Fluorescence
detection
CCD
Ion addressing
The ions can be addressed individually on the qubit
transition with an EO deflector which can quickly move the
focus of the 729 light from one ion to another, using the
same optical path as the fluorescence detection via the
CCD camera.
How well the addressing works is shown on the previous
slide: The graph shows the excitation of the indiviual ions
as the deflector is scanned across the crystal.
6
External degree of freedom: ion motion
Notes for next slides:
Now let's have a look at the qubit transition in the presence of the
motional degrees of freedom. If we focus on just one motional
mode , we just get a ladder of harmonic oscillator levels.
The joint (motion + electronic energy level) system shows a
double ladder structure. With the narrow laser we can selectively
excite the carrier transition, where the motional state remains
unchanged...
Or use the blue sideband and red sideband transitions, where we
can change the motional state.
We can walk down the double ladder by exciting the red
sideband and returning the ion dissipatively to the grounsstate.
With this we can prepare the ions in the motional ground state
with high probability, thereby initializing our quantum register.
External degree of freedom: ion motion
harmonic trap
…
7
External degree of freedom: ion motion
2-level-atom
harmonic trap
joint energy levels
…
External degree of freedom: ion motion
2-level-atom
harmonic trap
joint energy levels
…
Laser cooling to the motional ground state:
Cooling time: 5-10 ms
> 99% in motional ground state
8
Coherent manipulation
2-level-atom
harmonic trap
joint energy levels
…
Interaction with a resonant laser beam :
Ω : Rabi frequency
φ : phase of laser field
Laser beam switched on for duration τ :
θ : rotation angle
If we resonantly shine in light pulse at the carrier transition, the system evolves for a time tau with this Hamiltonian,
where the coupling strength Omega depends on the sqroot of the intensity, and phi is the phase of the laser
field with respect to the atomic polarization.
Coherent manipulation
Let's now begin to look at the coherent state manipulation. If we resonantly
shine the light pulse at the carrier transition, the system evolves for a time τ
with this Hamiltonian, where the coupling strength Ω depends on the
square root of the intensity, and φ is the phase of the laser field with
respect to the atomic polarization.
The effect of such a pulse is a rotation of the state vector on the Bloch
sphere, where the poles represent the two states and the equator
represents superposition states with different relative phases. The roation
axis is determined by the laser frequency and phase. The important
message is here that we can position the state vector anywhere on the
Bloch sphere, which is a way of saying that we can create arbitrary
superposition states.
The same game works for sideband pulses. With a π/2 pulse, for example,
we entangle the internal and the motional state! Since the motional state is
shared by all ions, we can use the motional state as a kind of bus to
mediate entanglement between different qubits in the ion chain.
9
Coherent excitation: Rabi oscillations
„Carrier“ pulses:
Bloch sphere
representation
D state population
Coherent excitation on the sideband
„Blue sideband“ pulses:
coupled system
Entanglement between internal and motional state !
D state population
10
Experimental procedure
P1/2
1. Initialization in a pure quantum state:
laser cooling,optical pumping
D5/2
τ =1s
Quantum state
Doppler
Fluorescence
Sideband
manipulation
cooling
detection
cooling
40
Ca+
SS1/2
1/2
2. Quantum state manipulation on
S1/2 – D5/2 qubit transition
3. Quantum state measurement
by fluorescence detection
One ion : Fluorescence histogram
8
7
D5/2 state
S1/2 state
6
50 experiments / s
5
4
Repeat experiments
100-200 times
3
2
1
0
0
20
40
60
80
100
120
counts per 2 ms
Experimental procedure
P1/2
D5/2
τ =1s
Quantum state
Doppler
Fluorescence
Sideband
manipulation
cooling
detection
cooling
40
Ca+
SS1/2
1/2
1. Initialization in a pure quantum state:
Laser sideband cooling
2. Quantum state manipulation on
S1/2 – D5/2 transition
3. Quantum state measurement
by fluorescence detection
Multiple ions:
Spatially resolved
detection with
CCD camera:
50 experiments / s
Repeat experiments
100-200 times
11
1.
Basic experimental techniques
2.
Two-particle entanglement
3.
Multi-particle entanglement
4.
Implementation of a CNOT gate
5.
Teleportation
6.
Outlook
…
Pulse sequence:
…
…
…
Creation of Bell state
12
…
Creation ofofBell
Generation
Bellstates
states
Pulse sequence:
…
…
…
Ion 1: π/2 , blue sideband
…
Creation of Bell states
Pulse sequence:
Ion 1: π/2 , blue sideband
carrier
…
,
…
…
Ion 2: π
13
…
Creation of Bell states
Pulse sequence:
Ion 2: π
,
Ion 2: π
, blue sideband
…
carrier
…
…
Ion 1: π/2 , blue sideband
Analysis of Bell states
Fluorescence
detection with
CCD camera:
Coherent superposition or incoherent mixture ?
What is the relative phase of the superposition ?
Ψ+
Measurement of the density matrix:
SS
SD
DS
DD
SDSS
DDDS
14
Reconstruction of a density matrix
Representation of ρ as a sum of orthogonal observables Ai :
ρ is completely detemined by the expectation values <Ai> :
Finally: maximum likelihood estimation (Hradil ’97, Banaszek ’99)
For a two-ion system :
Joint measurements of all spin components
Preparation and tomography of Bell states
Fidelity:
F = 0.91
SS
SD
DS
DD
DD DS
SD SS
SS
SD
DS
DD
DD DS
SD SS
Entanglement
of formation:
E(ρexp) = 0.79
SS
SD
DS
DD
DD DS
SD SS
SS
SD
DS
DD
Violation of
Bell inequality:
DD DS
SD SS
S(ρexp) = 2.52(6)
> 2
C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)
15
Different decoherence porperties
sensitive to:
laser frequency
magnetic field exc. state lifetime
1.
Basic experimental techniques
2.
Two-particle entanglement
3.
Multi-particle entanglement
4.
Implementation of a CNOT gate
5.
Teleportation
6.
Outlook
16
Generation of W-states
…
Pulse sequence:
Ion 2,3: π
,
carrier
1.
2.
3.
Ion 2: θ2 , blue sideband
…
…
Ion 1: θ1 , blue sideband
…
Ion 3: θ3 , blue sideband
Density matrix of W – state
Fidelity: 85 %
experimental result
theoretical expectation
17
Four-ion W-states
DDDD
DDDS
SSSS
SSSS
14.4.2005
DDDD
Five-ion W-states
DDDDD
DDDDS
SSSSS
15.4.2005
SSSSS
DDDDD
18
Detection of six individual ions
5µm
1
all ions in |S>
2
3
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
5
10
15
20
25
1
ion 1 in |S>
2
3
Ion detection
on a CCD camera
(detection time:4ms)
1
ion 6 in |S>
2
3
1
ion 4 in |S>
2
3
1
ion 5 in |S>
2
3
1
ions 1 and 5 in |S>
2
3
1
ions 1,2,3, and 5 in |S>
2
3
1
ions 1,3 and 4 in |S>
2
3
6
5
4
3
2
1
Six-ion W-state
F=73%
preliminary
result
Is there 6-particle
entanglement present?
• 6-particle W-state
can be distilled from the
state (O. Gühne)
• 6-particle entanglement
present, unresolved
issues with error bars
22.4.2005
729 settings, measurement time >30 min.
19