Quantum Computing with Trapped Atomic Ions APS March Meeting - Montréal: March 21, 2004 Brian King Dept. Physics and Astronomy, McMaster University http://physserv.mcmaster.ca/~kingb/King_B_h.html Innsbruck Oxford Ann Arbor Boulder Garching
Jan 13, 2016
Quantum Computing with Trapped Atomic Ions
APS March Meeting - Montréal: March 21, 2004Brian King
Dept. Physics and Astronomy, McMaster University
http://physserv.mcmaster.ca/~kingb/King_B_h.html
Innsbruck Oxford
Ann Arbor
Boulder
Garching
Outline:
• building “quantum computers”• overview of ion trap quantum information processor• ion trapping• initialization and detection• single-qubit gates (internal)• coupling internal and external qubits• directions for the future...
Building Quantum Computers:
Need:
1. qubits• two-level quantum systems• superpositions isolated from outside world• confined, characterizable, scalable
2. preparation• prepare computer in standard start state
3. read-out
4. logic gates• controllable interactions with outside world!• single- and two-qubits gate sufficient (not nec.!)
* the Devil is in the details...
Why atomic qubits?
• unparallelled persistence of quantum superposition
• control over quantum states - internal and external
• atomic clocks - accuracy, precision
• BEC, Fermi degeneracy (controllable), Mott insulator transition, quantum squeezing, quantum state engineering...
• atomic ions - demonstration of building blocks for scalable* quantum “computer” architecture
* the Devil is in the details...
Trapped-Ion QC (Cirac, Zoller('95))
• a collection (string) of trapped atomic ions:• qubits: (1) internal atomic levels
• quantum memory• tdecoh À tgate
• T2 > 10 min.• clocks
• accuracy, stability> 1/1015
• “data bus:” (2) common-mode motion
• transitory• tdecoh tgate
• 10 2 10 3
E|0|1
|1|0
2. map qubit i state to motion with lasers
How it works...
A quantum logic gate between 2 different ions:1. prepare qubits using single-qubit gates
3. 2-qubit gate between motion and ion j4. put information from motion “back into” ion i
i j
laser laserlaser laserlaser
Dynamical RF trapping:
• want to confine charged atoms E fields!• Eherenfest/Gauss can’t use static fields
+V
+V
0
0
F
0
0
+V
+V
F
+V
+V
0
0
F
0
0
+V
+V
F
+V
+V
0
0
F
0
0
+V
+V
F
+V
+V
0
0
F
use oscillating fields!
• in 3-D:
• e.g. z:
assume:
Dynamical RF trapping:
• average over 1 RF period:
• full solution: Mathieu equation (same results...)
Quantum Motion:
• same results:• quantum harmonic oscillator• wavepackets “breathe” at T
Linear Ion Traps for QC:ra
dial
axial
V0,U0
• axial confinement - static!
(z) = (mz2/2q) (z2/2)
• z2=2qU0/m ~ 1 (geom.)
-1.0
-0.5
0.0
0.5
1.0
po
siti
on
1086420 time
"secular" motion
"micromotion"
• micromotion small, at different freq.
• radial confinement -dynamic!
(r) = (m/2q) (r2 z
2/2) (r2)• r
2 = q2V02/(2mRF4r4) ~ 1 (geom.)
• r < RF
V0,U0
Innsbruck MPQ/GarchingOxford
Ion Traps - initial micromachining:
2”
1 cm 0.2 mm
• DC: U0 ≈ 10 V• RF: V0 ≈ 750 V ≈ 230 MHz
HO ≈ 10 MHz
• pressure < 2×1011 torr• single ion lifetime: > 10
h.(cryogenic up to 100
days...)
centre of mass (COM)x
“stretch”x
Ion Motion in Trap:
• single ion:• like a mass on a spring
• multiple, cold ions:• “normal modes” - the string moves as one...
N ions:N modes per direction
Dirty little secrets - motional heating:
• after cooling to the ground state of motion, the ion heats back up!
• timescale for motional manipulation ~ 10 s• |0 |1 in ~ 100 s (1998...)
1. motion only sensitive to noise spectrum near mot
2. heating scales strongly with trap size ~ 10 4
• fluctuating patch potentials?• RF-assisted tunnelling?
3. heating seems related to atom source shield trap!
• 21st century: NIST < 1 /(4 ms)IBM: 1/(10 ms)Innsbruck: 1/(190 ms)
Q.A. Turchette et al. Phys. Rev. A 62, 053807, 2000.
• plus sympathetic cooling (multi-species...)
Internal-State Qubits:
• long-lived electronic states:
Ca+, Sr+, Ba+,Hg+
S1/2
D3/2
D5/2
P1/2
P3/2
397 nm
866 nm,1092 nm
422 nm194 nm
729 nm674 nm282 nm
= 1 s = 345 ms = 90 ms
199Hg+: Qmeas = 1.6·1014
@ 282 nm
Ene
rgy
Internal-State Qubits:
• ground-state hyperfine levels:
= 19 MHz= 8 ns
0
1
Be+ (313 nm),Mg+ (280 nm),Cd+ (215 nm)
9Be+: Qmeas = 3.4·1011
@ 303 MHz173Yb+: Qmeas = 1.5·1013
P1/2
P3/2
Be+
> 10,000 yr
313 nm
1.25 GHzS1/2
Ene
rgy
State Detection:
1
det.0
• cycling transition - excited state decays back to |0
State preparation:
• electronic:
•optical qubit - kT free!
•hyperfine qubit: optical pumping
• vibrational: Doppler & sideband laser cooling
optical: laser• single-photon• requires L ¿ mot
01
2P1/2
• strong E-gradients (optical)• motional coupling
• RF frequency diff. coupling• controllable strength• RF phase stability
2-photon stimulated Raman transitions
Single-qubit logic gate:
1086420
543210t (sec)
Avg
# c
oun
ts3020100
|0
3020100
|0 + |1
3020100
|1
Classically: · E0 m im Jm(kz0) eimzt e-iLt
sidebands!
L – 00
z
Quantum: HI ½E0 (S+ + S-) ei(kz0 (a + a†)- Lt)
= (S+ + S-) ei( (a + a†)- Lt)
• can change motion!(k z0nvib ~ [z0 / ]nvib(... and resonance...)
Coupling qubit levels:
• oscillating field induces dipole moment
• HI · E0 ei(kz - Lt)
+
• can change electronic level(resonance?)
• if ion vibrates, interaction strength modulated
• HI · E0 ei(kz0 cos(zt)- Lt)
CZ Realized:
• motion-dependent spin transitions (conditional logic)
|1m|0m
|1m|0m ||e
|1m|0m
|aux
( phase shift)
( phase shift)
( phase shift)
( phase shift)
c t c’t’| | | | | | | | | | | | | | | |
Controlled-Phase Gate (‘95):
-30 -20 -10 0 10 20 300.0
0.5
1.0
/2-pulse detuning (kHz)
initially |0m|0 initially |1m|0
Pr[|0]
phase
( phase shift)
/2
Initial State Final StateP(m=1) P() P(m=1) P()
0.02 0.01 0.09 0.160.03 0.99 0.04 0.880.92 0.05 0.77 0.880.94 0.98 0.88 0.19
/2 C-Phase /2 Controlled-NOT:
||e
CZ Realized - a two-ion logic gate!
• two 40Ca+ ions - CZ scheme
• but no |aux needed...
F. Schmidt-Kaler, et al., Nature 422, 408 (2003)
1 0 000 1 000 0 010 0 10
theoretical: measured: F ~ 70%
CZ Realized - a two-ion logic gate!
• doesn’t use |aux - uses clever NMR trick!
|1m|0m
|1m|0m ||e
( phase shift)
||e
|2m
( phase shift)
( phase shift)
coupling strength ~n> !•2 for n>=1 but 2 for n>=2
use (,x) (/2,y) (,x) (/2,y)
Scaling up:
• problem:• as Nions :
• ion string gets heavier gates get slower!• more motional modes greater “noise”
fibre
R. DeVoe, PRA 58, 910 (98)J.I. Cirac, et al. PRL 78, 3221 (97)
1. optical multiplexing:
laser(stim. Raman)
to other cavity/qubits
cavity mode(spont. Raman)
Solutions (1) - optical:
• MPQ, Garching (Ca+): 4 2S1/24 2P1/2G.R. Guthöhrlein, et al., Nature 414 (01)
res. /10
• U. Innsbruck (Ca+): 4 2S1/23 2D5/2A.B. Mundt, et al., quant-ph/0202112
• sweep PZT Doppler shift• Pex. > 0.5 coherent
Excitation Laser Det. (MHz)-0.2 0.2
Exc
itat
ion
Pro
b.
blueshift
redshift
• positioning:node/antinode• res. /100
• differential coupling to motional sidebands
Scaling up:
segmented electrodes
accumulator
memory register
• problem:• as Nions :
• ion string gets heavier gates get slower!• more motional modes greater “noise”
2. “quantum CCD:”
“quantum CCD”• Wineland, et al. J. Res. NIST 103, 259 (98)• D. Kielpinski, et al. Nature 417, 709 (02)
Solutions (2) - physical multiplexing:
M. Rowe, et al., Quantum Information and Computation 1, x (‘01).
• transporting ions between traps:
no transport: 96.8 ± 0.3% contrastline triggered: 96.6 ± 0.5% contrast!
• 60 Hz fields...
360 m
400 m
(1) Ramsey interferometer:
“spin echo”96% contrast
95% sep. eff. (5000 shots)
n=200 quanta (2.9 MHz) for 10 ms sep. time(separation electrode too wide!)
(2) separating ions:
Solutions (2) - physical multiplexing:
• “gold foil” traps:
• silicon traps:
alumina silicon
• easily micro-machined, smooth
Ion Trap QC: Wither thou?...
• single-qubit logic gates (´40’s) (>98% fidelity)• single-ion 2-qubit logic gate (´95) (80% fidelity)
C. Monroe et al. Phys. Rev. Lett. 75, 4714 (‘95).
• 2-ion 2-qubit logic gates 2 (80% / 97% fidelity)Gulde et al. Nature 422, 408 (‘03).
Leibfried et al. Nature 422, 412 (‘03).
• Deutsch-Jozsa algorithmGulde et al. Nature 421, 48 (‘03).
• state preparation (fidelity > 98%)• spin qubit: t / tgate > 1000*• motional data bus/qubit
• heating < 1/(4, 10, 190 ms) (NIST, IBM, Innsbruck)
http://physserv.mcmaster.ca/~kingb/King_B_h.htmlNIST Boulder, MPQ, IBM Almaden, U. Innsbruck, Oxford, U. Michigan, McMaster
References:
1. Cirac & Zoller: “New Frontiers in Quantum Information With Atoms and Ions,” Physics Today 57, #3, 38 (March '04).
2. Steane: Appl. Phys. B 64 , 623 ('97).
3. Ghosh: Ion Traps, (Clarendon Press, '97), ISBN: 0198539959.
4. Leibfried et al.: “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281 ('03).
5. Wineland, et al.: “Quantum information processing with trapped ions,” Phil. Trans. Royal Soc. London A 361, 1349, ('03).
6. Wineland, et al., “Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions”, J. Research NIST 103, 259 ('98).
7. Monroe, et al.: “Experimental Primer on the Trapped Ion Quantum Computer,” Forschr. Physik 46, 363 ('98).
http://jilawww.colorado.edu/pubs/recent_theses/
D. Kielpinski, “Entanglement and Decoherence in a Trapped-Ion Quantum Register”
B.E. King, “Quantum State Engineering and Information Processing withTrapped Ions”
Nobel Sidebar - Ramsey’s expt.:
• superpositions - how do we characterize phase?
t
T/2:create
superposition
T/2, phase :try to undo
superposition!
tR:phase evolves(Schrodinger)
*
• interferometer
-30 -20 -10 0 10 20 300.0
0.5
1.0N
t
2 is better than one!...
• spin-dependent motional Berry’s phase
D. Leibfried, et al., Nature 422, 412 (2003)
• 2 lasers with L create “standing wave”• dipole force
P1/2
S1/2
• 2 lasers with L z create “walking standing wave” which can resonantly drive ion motion
2 is better than one!...
• resonant oscillating force = displacement operator in phase space
x
p
D()
D()•|| set by strength of force•phase set by phase between motion and lasers
D() D() = ei Im(*)D()
geometric Berry’s phase!
2 is better than one!...
• “stretch mode:”• need different force on each ion to drive• can only excite if ions in different electronic levels!
• move ions in closed loop in phase space
0
1
P1/2
differentcouplingstrengths
S1/2
“walking standing wave” has different strengths for ,
z
pz
| ei |
2 is better than one!...
• IF ions in different electronic states, move quantum motional state in closed loop in phase space
motional “Berry’s phase” phase shift
ei/2 ei/2
= ei (ei/2 ) ( ei/2) •controlled-Phase + single-qubit rotations (F ~ 97%)
and some 2’s are “better” than others
• 2-qubit gates utilize the motion• > cough, cough, mumble…<
• higher motional gives faster gates
shining laser on only one ion!• Motional gates (Mølmer-Sørensen, Milburn, etc.) can
be done illuminating all ions!
- keep high fast motional gates
- with expt. gate, can have different illuminations
• single-qubit operations can be done with weak trap• the “accordion quantum computer!”
…in the lab…
Coupling qubit levels:
• laser-ion interaction: messy details:
• in interaction picture:
• rotating-wave approximation:
• expand exponential: