xR2(=µt+h)1e30tx0a0)= 5
The trapped-ion qubit tool box
Contemporary Physics, 52, 531-550 (2011)
Roee Ozeri
Weizmann Institute of Science Rehovot, 76100, Israel [email protected] Physical Implementation of a quantum computer
David Divincenzo’s criteria:
1. Well defined qubits.
2. Initialization to a pure state
3. Universal set of quantum gates.
4. Qubit specific measurement.
5. Long coherence times (compared with gate & meas. time). Universal Gate set
- For N qubits, a general unitary transformation U acts on a 2N- dimension Hilbert space.
- A finite set of unitary gates that spans any such U.
- The Deutsch-Toffoli gate
U
- D. Deutsch, Proc. R. Soc. London, Ser. A, 425, 73, (1989). Universal Gate set
- For N-qubits, and unitary transformation U on a 2N-dimension Hilbert space.
- A finite set of unitary gates that spans any such U.
Single qubit gate 2-qubit C-not gate
U
- Rotations can be approximated to e by concatenating k gates, from a finite set {Vi}, where k < polylog(1/e).
- Barenco et. al. Phys. Rev. A, 52, 3457, (1995). Physical Implementation of a quantum computer David Divincenzo’s criteria:
How well???
Well enough to allow for a large scale computation: Fault tolerance Fault-tolerant Quantum Computation Noisy operations
One level quantum error-correction codes ECC
concatenation; threshold theorem
k-levels of fault-tolerant encoding if
• Fault tolerance threshold. • Heavy resource requirements when • Depends on code, noise model, arch. constraints etc. • Current estimates for
Tutorial overview
1. The ion-qubit: different ion-qubit choices, Ion traps. 2. Qubit initialization.
3. Qubit measurement.
4. Universal set of quantum gates: single qubit rotations; two-ion entanglement gates
5. Memory coherence times
How well??? Benchmarked to current threshold estimates
Disclaimer: non exhaustive; focuses on laser-driven gates Different ion-qubit choises
2 - One electron in the valence shell; “Alkali like” S1/2 ground state. Electronic levels Structure
2 n P3/2
2 Fine structure n P
2 n P1/2 with D 2 + n-1 D5/2 Ca 397 nm Sr+ 422 nm w/o D 2 Ba+ 493 nm + n-1 D Be 313 nm Yb+ 369 nm Mg+ 280 nm 2 n-1 D3/2 Hg+ 194 nm Zn+ 206 nm Cd+ 226 nm
2 n S1/2 2 S1/2 Zeeman qubit (Isotopes w/o nuclear spin)
e.g. Advantages Disadvantages - RF separation. - Energy depends 24 + Mg - Tunable. linearly on B. 64 + Zn - Infinite T1. - Transition photon caries no 114Cd+ momentum 40 -Momentum transfer with off- Ca+ resonance lasers: photon 88Sr+ scattering. 138Ba+ - Detection. 174Yb+ 202Hg+ m = 1/2
2 Turn on small B field n S1/2 2.8 MHz/G m = -1/2 2 e.g. S1/2 Hyperfine qubit 9Be+ 25Mg+ Hyperfine structure (order depends on the sign of Ahf) 67 + Zn Turn on small B field 111Cd+ 43 Ca+ Disadvantages 87 Advantages Sr+ - MW energy separation. - Few GHz energy separation. 137Ba+ - B-field independent qubit. -Transition photon caries no momentum 171 Yb+ - Infinite T1. - Off resonance photon scattering. - State selective fluorescence Detection. 199Hg+ - initialization to clock transition can be more tricky m = 0 F = I – 1/2
2 1 - 40 GHz S1/2
F = I + 1/2 m = 0 Optical qubit
2 D5/2 Innsbruck/Weizmann with D
Ca+ 729 nm Advantages Sr+ 674 nm - Single optical photon (momentum). Ba+ 1760 nm - B-field independent qubit (if nuclear spin ≠ 0). - Hardly any spontaneous decay during gates. + Yb 411 nm - State selective fluorescence Detection: excellent + Hg 282 nm discrimination.
Disadvantages - Finite (~ 1 sec) lifetime. - Coherence time is limited by laser linewidth.
2 S1/2 Trapping
- Trap ions: a minimum/maximum to f, the electric potential.
- Impossible in all directions; Laplace’s equation: 2 0 Trapping
Linear RF Paul trap
Positive ion
RF electrode
High dc potential control electrode
Low dc voltage control electrode Dynamic trapping (pondermotive forces)
Oscillating electric field:
Large and slow small and fast
Newton’s E.O.M: Dynamic trapping
Field expansion:
To 0th order:
Next order:
Average over one period: Dynamic trapping
+ For electric quadruple:
d + Sign of e Pseudo-potential:
Harmonic frequency: eVrf trap ~ 2 mrf d Trapping
Linear RF Paul trap
Potential : 2 i iV0 cos(rf t)X i i Solve E.O.M : MX F e d 2 X (Mathieu eq‘: i a 2 q cos( 2 ) X 0 ) d 2 i i i Stable solution: eV Drive frequency ~ 20-30 MHz • RF RF amplitude ~ 200-300 V • trap ~ 2 Secular frequency • mrf d Radial ~ 2-3 MHz – d – the distance between the ion and the Axial ~ 1 MHz – electrodes Weizmann trap.
0.6 mm 200 Vpp AC
+ 21 MHz
m + + 10 V DC
m
0.2 mm
3
. 0 + n = 1 – 2 MHz (axial) - Laser machined Alumina z n r = 2 – 3 MHz (radial) 1.2 mm
~ 2-5 mm Nitzan Akerman Scale up?
One (out of many) problem: isolating one mode of motion for gates The ion vision: Multiplexed trap array
interconnected multi-trap structure subtraps completely decoupled
routing of ions by controlling electrode voltages
Subtrap for different purpose: Gates, readout, etc.
D. J. Wineland, et al., J. Res. Nat. Inst. Stand. Technol. 103, 259 (1998); D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002). Other proposals: DeVoe, Phys. Rev. A 58, 910 (1998) . Cirac & Zoller, Nature 404, 579 ( 2000) . Multi-zone ion trap
NIST Gold on alumina • rf filter board construction RF quadrupole • realized in two layers Six trapping zones • segmented Both loading and • linear trapping experimental region zones One narrow • separation zone rf control Closest electrode • view along axis: ~140 mm from ion control rf Ion transport NIST Ions can be moved • separation zone 200 mm between traps. Electrode potentials – varied with time Ions can be separated • efficiently in sep. zone Small electrode’s – potential raised Motion (relatively) fast • 100 mm Shuttling (adiabatic): – several 10 ms Separating: few 100 ms – 6-zone alumina/gold trap (Murray Barrett, John Jost) NIST Surface-electrode traps
1 mm
Field lines:
Trapping center
Filter capacitor Microfabricated filter resistor Control lead RF electrodes Control electrodes
Elbows and tee- junctions possible:
J. Chiaverini et al. Quant. Inform. Comp. 5, 419 (2005). S. Seidelin et al. Phys. Rev. Lett. 96, 253003 (2006). Micro-fabricated traps
Shappert; Georgia Tech
Allcock; Oxford
Amini et al. New J. Phys. 12, 033031 (2010). Allcock et al. App. Phys. Lett. 102, 044103 (2012). Shappert et al. arXiv quant-ph\1304. 6636 (2013).
Amini; NIST Trapped ions on the tabletop
EMCCD
PMT
• Vacuum ~ 10-11 Torr • Room temperature (or a bit above) Flip Mount • RF created with coax/helical resonator • Atoms created by oven. • Ions created by photo ionization. • Approx. F1 imaging optics to EMCCD/PMT (res = 0.8 um). RF resonator
f/1 lens B-field Microwaves cooling ,detection and excitation beams Harmonic oscillator levels
Vibrational mode quantum number
1 0
- Doppler cooling. 1 - Resolved side band cooling. 0 Sideband Spectroscopy
2 4 D5/2 Scan the laser frequency • across the S →D transition Dn < 80 Hz 674 nm 2 5 S1/2
carrier
Red sidebands Blue sidebands
axial radial Resolved-sideband Cooling
P 1033 nm D
674 nm S
D
~4 MHz Red sideband S
n = 0 n = 1 n = 2 n = 3 Sideband cooling to the ground state
• T ≈ 2 mK • Uncertainty in ion position = ground state extent 6nm 2mho Anomalous heating
• Fluctuating charges • f-1.5 noise • Thermally activated • Due to monolayer of C on eletrodes: gone after Ar+ ion cleaning
Deslauriers et al. Phys. Rev. Lett. 97, 103007 (2006) Hite et al. Phys. Rev. Lett. 109, 103001 (2012) Qubit Initialization Zeeman qubit -Optical pumping into a dark state.
-CPT possible into any superposition..
2 P1/2
s
Error sources: - Polarization purity. e ~ 10-6 – 10-3
2S 1/2 Qubit Initialization Zeeman qubit
Process tomography of optical pumping Limited to 10-3 due to stress- induced birefringence in vacum chamber optical viewports Qubit Initialization Zeeman qubit: the D level option
2 P3/2
g = 0.4 Hz 2 P1/2 +3/2 2D -1/2 5/2
• Limited by off-resonance coupling of ↓ to D • Power broadening and in-coherent noise • e 10-4 2 D3/2
2 S1/2
Keselman, Glickman, Akerman, Kotler and Ozeri, New J. of Phys. 13, 073027, (2011) Qubit Initialization Hyperfine qubit
- Optical pumping into a dark state.
2 P1/2 Estimated:
C. Langer, Ph.D Thesis, University of Colorado, (2006). s p
F=0
2 F=1 S1/2
Qubit Initialization Optical qubit
- state initialization: same as previous. 2 P3/2
- state initialization:
- Rapid adiabatic passage. -2 e ~ 10 2D (Wunderlich et. al. Journal of Modern Optics 54, 1541 (2007)) 5/2
- p-pulse. e ~ 10-2
2 S1/2