Quantum Computing with Trapped Atomic Ions

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Quantum Computing with Trapped Atomic Ions
Garching
Boulder

• APS March Meeting - Montréal March 21, 2004
• Brian King
• Dept. Physics and Astronomy, McMaster University
• http//physserv.mcmaster.ca/kingb/King_B_h.html

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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...

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Building Quantum Computers

• Need
• qubits
• two-level quantum systems
• superpositions ? isolated from outside world
• confined, characterizable, scalable
• preparation
• prepare computer in standard start state
• read-out
• logic gates
• controllable interactions with outside world!
• single- and two-qubits gate sufficient (not nec.!)

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Why atomic qubits?

• unparallelled persistence of quantum
  superposition

• atomic clocks - accuracy, precision

• control over quantum states - internal and
  external

• 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...
the Devil is in the details...
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Trapped-Ion QC (Cirac, Zoller('95))

• a collection (string) of trapped atomic ions
• qubits (1) internal atomic levels

• quantum memory
• tdecoh À tgate
• T2 gt 10 min.
• clocks
• accuracy, stability
• gt 1/1015

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How it works...

• A quantum logic gate between 2 different ions

• prepare qubits using single-qubit gates

• map qubit i state to motion with lasers

• 2-qubit gate between motion and ion j

• put information from motion back into ion i

i
j
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Dynamical RF trapping

• want to confine charged atoms ? E fields!

• Eherenfest/Gauss ? cant use static fields

? use oscillating fields!
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Dynamical RF trapping

• full solution Mathieu equation (same results...)

• same results

• quantum harmonic oscillator

• wavepackets breathe at ?T

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Linear Ion Traps for QC

• axial confinement - static!

U0
U0
V0,W
V0,W

• F(z) (mwz2/2q) (z2/2)
• wz22aqU0/m a 1 (geom.)

• radial confinement -dynamic!

• F(r) (m/2q) (wr2 - wz2/2) (r2)
• wr2 q2V02/(2mWRFb4r4) b 1 (geom.)
• wr lt WRF

• micromotion small, at different freq.

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Ion Traps - initial micromachining

• DC U0 10 V
• RF V0 750 V
• ? 230 MHz
• ? wHO 10 MHz
• pressure lt 210?11 torr
• single ion lifetime gt 10 h.
• (cryogenic ? up to 100 days...)

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Ion Motion in Trap

• single ion
• like a mass on a spring

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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...)

• motion only sensitive to noise spectrum near ?mot

• fluctuating patch potentials?

• RF-assisted tunnelling?

• heating scales strongly with trap size 10 ? 4

• heating seems related to atom source ? shield
  trap!

Q.A. Turchette et al. Phys. Rev. A 62, 053807,
2000.

• 21st century NIST lt 1 /(4 ms)
• IBM 1/(10 ms)
• Innsbruck 1/(190 ms)

• plus sympathetic cooling (multi-species...)

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Internal-State Qubits

• long-lived electronic states

Ca, Sr, Ba, Hg
199Hg Qmeas 1.61014 _at_ 282 nm
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Internal-State Qubits

• ground-state hyperfine levels

Be (313 nm), Mg (280 nm), Cd (215 nm)
P3/2
g/2p 19 MHz t 8 ns
P1/2
313 nm
?1?
9Be Qmeas 3.41011 _at_ 303 MHz 173Yb Qmeas
1.51013
1.25 GHz
S1/2
?0?
Be
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State preparation

• electronic
• optical qubit - kT ? free!
• hyperfine qubit optical pumping
• vibrational Doppler sideband laser cooling

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Single-qubit logic gate
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Coupling qubit levels

• oscillating field induces dipole moment
• HI ? m E0 ei(kz - wLt)

• can change electronic level
• (resonance?)

• if ion vibrates, interaction strength modulated
• HI ? m E0 ei(kz0 cos(wzt)- wLt)

• Quantum HI ? ½mE0 (S S-) ei(kz0 (a a)-
  wLt)
• W (S S-) ei(h (a a)- wLt)
• can change motion!
• (k z0?nvib z0 / l ?nvib) (... and
  resonance...)

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CZ Realized

• motion-dependent spin transitions (conditional
  logic)

1ñm
?ñº 1ñe
0ñm
1ñm
ñº 0ñe
1ñm
0ñm
auxñ
0ñm
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CZ Realized - a two-ion logic gate!
F. Schmidt-Kaler, et al., Nature 422, 408 (2003)

• two 40Ca ions - CZ scheme
• but no aux? needed...

theoretical
measured F 70
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CZ Realized - a two-ion logic gate!

• doesnt use aux? - uses clever NMR trick!

2ñm
1ñm
?ñº 1ñe
0ñm
use (p,x) (p/?2,y) (p,x) (p/?2,y)
1ñm
ñº 0ñe
0ñm
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Scaling up

• problem
• as Nions ?
• ion string gets heavier ? gates get slower!
• more motional modes ? greater noise

• optical multiplexing

R. DeVoe, PRA 58, 910 (98) J.I. Cirac, et al. PRL
78, 3221 (97)
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Solutions (1) - optical

• MPQ, Garching (Ca) 4 2S1/24 2P1/2
• G.R. Guthöhrlein, et al., Nature 414 (01)

res. l/10

• U. Innsbruck (Ca) 4 2S1/23 2D5/2
• A.B. Mundt, et al., quant-ph/0202112

red shift
blue shift

• sweep PZT
• Þ Doppler shift
• Pex. gt 0.5 Þ coherent

• positioning
• node/antinode
• res. l/100

• differential coupling to motional sidebands

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Scaling up

• problem
• as Nions ?
• ion string gets heavier ? gates get slower!
• more motional modes ? greater noise

• quantum CCD

• quantum CCD
• Wineland, et al. J. Res. NIST 103, 259 (98)
• D. Kielpinski, et al. Nature 417, 709 (02)

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Solutions (2) - physical multiplexing
M. Rowe, et al., Quantum Information and
Computation 1, x (01).

• transporting ions between traps

(1) Ramsey interferometer

• no transport 96.8 0.3 contrast
• line triggered 96.6 0.5 contrast!
• 60 Hz fields...

spin echo 96 contrast
(2) separating ions
Dn200 quanta (2.9 MHz) for 10 ms sep.
time (separation electrode too wide!)
95 sep. eff. (5000 shots)
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Solutions (2) - physical multiplexing

• gold foil traps

• silicon traps

• easily micro-machined, smooth

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Ion Trap QC Wither thou?...

• single-qubit logic gates (40s) (gt98 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 algorithm
• Gulde et al. Nature 421, 48 (03).
• state preparation (fidelity gt 98)
• spin qubit t / tgate gt 1000
• motional data bus/qubit
• heating lt 1/(4, 10, 190 ms) (NIST, IBM, Innsbruck)

NIST Boulder, MPQ, IBM Almaden, U. Innsbruck,
Oxford, U. Michigan, McMaster
http//physserv.mcmaster.ca/kingb/King_B_h.html
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References

• Cirac Zoller New Frontiers in Quantum
  Information With Atoms and Ions, Physics Today
  57, 3, 38 (March '04).
• Steane Appl. Phys. B 64 , 623 ('97).
• Ghosh Ion Traps, (Clarendon Press, '97),
  ISBN 0198539959.
• Leibfried et al. Quantum dynamics of single
  trapped ions, Rev. Mod. Phys. 75, 281 ('03).
• Wineland, et al. Quantum information processing
  with trapped ions, Phil. Trans. Royal Soc.
  London A 361, 1349, ('03).
• Wineland, et al., Experimental Issues in
  Coherent Quantum-State Manipulation of Trapped
  Atomic Ions, J. Research NIST 103, 259 ('98).
• 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

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Nobel Sidebar - Ramseys expt.

• superpositions - how do we characterize phase?

t
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2 is better than one!...
D. Leibfried, et al., Nature 422, 412 (2003)

• spin-dependent motional Berrys phase

• 2 lasers with dwL ? 0 create standing wave
• dipole force

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2 is better than one!...

• resonant oscillating force displacement
  operator in phase space

• a set by strength of force
• phase set by phase between motion and lasers

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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

walking standing wave has different strengths
for ?,?
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2 is better than one!...

• IF ions in different electronic states, move
  quantum motional state in closed loop in phase
  space
• ? motional Berrys phase ? phase shift

• ???????Y? ???????? Y ?
• ??????? Y ? ? eip/2??????? Y ?
• ??????? Y ? ? eip/2 ??????? Y ?
• ??????? Y ? ???????? Y ?
• e-ip (eip/2 ???) ( eip/2???)? Y ?
• controlled-Phase single-qubit rotations (F
  97)

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and some 2s are better than others
in the lab

• 2-qubit gates utilize the motion
• gt cough, cough, mumblelt
• higher motional n gives faster gates
• ?? shining laser on only one ion!

• Motional gates (Mølmer-Sørensen, Milburn, etc.)
  can be done illuminating all ions!
• - keep n high ? fast motional gates
• - with expt. gate, can have different
  illuminations
• single-qubit operations can be done with weak
  trap
• the accordion quantum computer!

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Coupling qubit levels

• laser-ion interaction messy details

• in interaction picture

• rotating-wave approximation

• expand exponential