Quantum Information Processing with Trapped Ions E. Knill C. Langer D. Leibfried R. Reichle S. Seidelin T. Schaetz D. J. Wineland NIST-Boulder Ion QC group Be+ R. Ozeri M. Barrett J. Britton B. R. Blakestad J. Chiaverini W. M. Itano D. Hume J. D. Jost

Overview Trapped ions Experimental System –The trap –Initialization, detection. –Coherent control of the ion-qubit –Deterministic entanglement Deterministic Teleportation Between Atomic Qubits. Quantum Error-Correction.

Linear RF Paul trap RF electrode High dc potential control electrode Low dc voltage control electrode Positive ion Drive freq ~ MHz RF amp ~ V Secular freq –Radial ~ 15 MHz –Axial ~ 4 MHz

Multi-zone ion trap control rf view along axis: ~1 cm rf filter board Gold on alumina construction RF quadrupole realized in two layers Six trapping zones Both loading and experimental zones One narrow separation zone Closest electrode ~140  m from ion segmented linear trapping region

Ion transport 100  m 6-zone alumina/gold trap (Murray Barrett, Tobias Schaetz et al.) 200  m separation zone Ions can be moved between traps. –Electrode potentials varied with time Ions can be separated efficiently in sep. zone –Small electrode’s potential raised Motion (relatively) fast –Shuttling: several 10  s –Separating: few 100  s

Electronic levels in 9 Be + 2 P 1/2 2 P 3/2 2 S 1/ GHz F = 1 F = 2 2P2P Fine structure Hyperfine structure Turn on small B field (P also has hfs, but it’s negligible) 197 GHz

Qubit levels in 9 Be + 2 P 1/2 2 P 3/2 2 S 1/2  1.25 GHz  Vibrational mode quantum number     m F = -2 m F = -1 m F = 0 m F = 1 m F = 2 F = 1 m F = 1 m F = 0 m F = -1 F = 2

Cooling and initialization 2 P 1/2 2 P 3/2 2 S 1/2  1.25 GHz  m F = -2 m F = -1 m F = 0 m F = 1 m F = 2 m F = 1 m F = 0 m F = -1 Raman side-band cooling + Optical pumping = Ion initialized in the  with better than 99% efficiency.

Qubit detection by resonance fluorescence 2 P 1/2 2 P 3/2 2 S 1/2  1.25 GHz    F = 2, m F = -2     F = 1, m F = -1   Detection (   ) 313 nm Detection efficiency >99%

Coherent control of qubits 2 P 1/2 2 P 3/2 2 S 1/2  1.25 GHz ~ 80 GHz  Raman 313 nm     kk kk Vib. mode quantum #

Coherent qubit rotations Any single qubit rotation can be composed of 1-3 pulses    i   i        Bloch sphere

Entanglement on demand Geometric phase gate: F   F  Polarization gradient “walking-standing” wave Long. motional modes: 1 st mode (COM) 2 nd mode (stretch) Set ion spacing such that stretch is not excited for  or . Can give opposite spin states a phase relative to same spin states. Brennen, et al. PRL 1999 Jaksch, et al. PRL 1999 Mandel, et al. Nature 2003

Phase gate A  or   or    x x p p Tune  from stretch mode, displace for  Start with  Perform  D  obtain,  with fidelity=0.97: (  i  )  (Didi Leibfried et al. Nature 2003) (Universal 2-qubit gate)

Good year for the ions! Creation of and spectroscopy with GHZ states. (NIST & Innsbruck) Enhanced state detection efficiency with QIP. Quantum Dense coding Deterministic teleportation between atomic qubits. (NIST & Innsbruck) Quantum error-correction with atomic qubits. Implementation of a semmi-classical Quantum Fourier transform. (D. Leibfried et al. Science 2004) (T. Schaetz et al. PRL 2004) (M. Barrett et al. Nature 2004) (T. Schaetz et al. PRL 2004) (M. Riebe et al. Nature 2004) (C. F. Roos et al. Science 2004) (J. Chiaverini et al. Nature 2004) (J. Chiaverini et al. Submitted)

Quantum teleportation Resources required: 2 cbits + entangled pair Transmit classical information Bell state measurement Entangle pair Distribute entanglement Apply conditional operation Arbitrary state to be teleported Bennet et al., 1993

Properties of Q. Teleportation Effectively transmit a qubit –Use a classical channel Actually transmit only 2 cbits –To classically define qubit: infinite # of cbits No information contained in 2 cbits Information in the correlations Entangled pair can be distributed anytime Initial qubit contains no info afterward

QT in the lab Prepare ions in state  and motional ground state Create entangled state on outer ions             Alice prepares state to be teleported              Alice performs Bell basis decoding using phase gate on ions 1 and 2 Alice measures ion 1 Alice measures ion 2 Bob performs conditional rotation dep. on meas. Bob recovers      on ion 3 and checks the state (Murray Barrett et al., Nature 04) Entire protocol requires ~2.5 msec (also demonstrated at Innsbruck with ions) Photons: Bouwmeester et al., Nature (1997) Furusawa, et al., Science (1998)

Teleportation results Average fidelity 78(2)% Best possible without entanglement: 2/3 A range of states was teleported:  i  i       

Classical error correction Decoding or parity check allows reconstruction With a noisy line, “B” is hard to distinguish from “C” A solution is to encode these letters in longer words B becomes Bravo, C becomes Charlie 0  0001  111 Digitally: Send each encoded bit An error occurs Decode using majority rule Repetition code

Classical error checking Measurement result Error action Correction operation Flip bit 1 None 000 or 111 No qubits flipped 1 st bit flipped 2 nd bit flipped 3 rd bit flipped 100 or or or 110 Flip bit 2 Flip bit 3 Probability that more than 1 bit flips: So rep. code provides an improvement when

Quantum error correction Problems in converting from classical –Can’t look at the quantum info. –No cloning of an unknown quantum state –Errors are continuous (not just a bit flip) Solution –Use entanglement –Make meas. that tell nothing about state (QND).

Three bit repetition code Encode state in three qubits via entanglement |  Now an error E (rotation around x axis) occurs in one of the qubits Apply E  I  I to our state

Three bit repetition code Now measure the ancilla qubits Decode Ancilla qubits

Error correction with rep. code Measurement result Syndrome Correction operation  X  I  I I  I  I I  X  I I  I  X |  |  |  |  No qubits flipped 1 st qubit flipped 2 nd qubit flipped 3 rd qubit flipped If we get | , we apply  X  I  I to Correction is independent of , , and 

Error correction protocol G includes a three-ion entangling gate that gives all states but  and  a phase of  Error rotation  e applied to all qubits Ancilla qubits are measured after decoding R is either  X,  Y, or I dep. on measurement ee ee ee R GG -1

Results Uncorrected infid. ~  e 2, corrected infid. ~  e 4 Qubits genuinely protected for  e ~ 1 rad. (J. Chiaverini et al. Nature 2005)

Summary oInitialization and detection efficiency > 99% oMemory coherence time > 10 sec. oTrapped ion qubit can be coherently manipulated, Fidelity > 99%. oTwo or more qubits can be deterministically entangled, fidelity > 97%. oEntanglement can be distributed across different traps (~mm). o sympathetic cooling with 24 Mg + ions demonstrated. oGoing for more ions!

From left to right: Joe Britton, Jim Bergquist, John Chiaverini, Windell Oskay, Marie Jensen, John Bollinger, Vladislav Gerginov, Taro Hasegawa, Carol Tanner, Wayne Itano, Jim Beall, David Wineland, Dietrich Leibfried, Chris Langer, Tobias Schaetz, John Jost, Roee Ozeri, Till Rosenband, Piet Schmidt, Brad Blakestad NIST Ion Storage Group, March, ‘04