Quantum computer: basics, gates, algorithms

Quantum computer: basics, gates, algorithms single qubit gate various two qubit gates baby-steps shown so far with ion quantum processors and how to reach a scalable device in future Ulm, Germany: 40 Ca +

Laser coupling 2-level-atom harmonic trap D S dressed system D energy ladder picture molecular Franck Condon picture... n 1, D n 1, S n, D n, S n +1, n +1, D S... S

Laser coupling 2-level-atom harmonic trap D S dressed system D energy ladder picture molecular Franck Condon picture... n 1, D n 1, S n, D n, S n +1, n +1, D S... S

Coherent qubit rotation Carrier flops electronic excitation Prob. for D D,0 laser pulse length in µs S,0 electronic excitation

Coherent qubit rotation Vibrational quanta Carrier flops D,0 D,1 S,0 S,1 internal electronic state laser pulse length in µs

Basics of a quantum computer INPUT Temporal sequence of quantum logic operations OUTPUT Single qubit gate time two-qubit gate

Why? applications in physics and informatics P. Shor, 1994: factorization of large numbers, L digits, is much more efficient on a quantum computer than with a classical computer: classical computer: ~exp(l 1/3 ), quantum computer: ~ L 2 L. Grover, 1997: search data base - quantum computer: ~ L simulation of Schrödinger equations or any unitary evolution spin interactions, quantum phase transitions quantum cryptography / repeaters / quantum links improved atomic clocks understanding the fundamentals of quantum mechanics / Gedanken-Experimente Experiments with entangled matter

The requirements for experimental qc Qubits store superposition information, scalable physical system Ability to initialize the state of the qubits ψ = α 0 + β 1 Universal set of quantum gates: Single bit and two bit gates Long coherence times, much longer than gate operation time Qubit-specific measurement capability D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001) Qubit Transformation

Experimental status Scalable device? Quantum Information Roadmaps http://qist.ect.it/ http://qist.lanl.gov/

Quantum gate proposal W. Paul J. I. Cirac P. Zoller single bit rotations and quantum gates small decoherence unity detection efficiency scalable control bit bit target bit bit

Quantum gate proposal Controlled NOT : ε ε ε ε ε 1 2 1 1 2 J. I. Cirac P. Zoller 0 0 0 1 0 0 0 1 single bit rotations and quantum gates small decoherence 1 0 1 1 unity detection efficiency 1 1 1 0 scalable control bit bit target bit bit

Cirac & Zoller gate with two ions

Controlled-NOT operation ε1 ε 2 S S S S S D S D D S D D D D D S ion 1 motion ion 2 S S,, D 0 0 D SWAP control control target target control qubit target qubit

Controlled-NOT operation ε1 ε 2 S S S S S D S D D S D D D D D S ion 1 motion ion 2 S, D 0 0 S, D 0>, 1> control qubit target qubit

Controlled-NOT operation ε1 ε 2 S S S S S D S D D S D D D D D S ion 1 motion ion 2 S, D S, D 0>, 1> SWAP -1 0 0 control qubit target qubit

SWAP and SWAP -1 starting with n=0> phonons, write into and read from the common vibrational mode π-pulse on blue SB control bit control bit D,0 D,1 D,0 D,1 π π S,0 S,1 S,0 S,1 SWAP SWAP -1

Conditional phase gate target bit Composite pulse phase gate I.Chuang, MIT Boston 2π 2π Rabi frequency: Blue SB: Ω η n +1 Effect: phase factor of -1 for all, except D,0 >

Composite phase gate (2π rotation) ( ) ( ) ( ) ( ) R( θφ, ) = R ππ, 2 R π 2,0 R ππ, 2 R π 2,0 + + + + 1 1 1 1 1 2π on S,0 D,1 3 2 4

Population of S,1> - D,2> remains unaffected ( ) ( ) ( ) ( ) R( θφ, ) = R π 2, π 2 R π,0 R π 2, π 2 R π,0 + + + + 1 1 1 1 2 4 3 1

Controlled-NOT operation ion 1 motion ion 2 S S, D, D SWAP SWAP -1 0 0 control bit target bit pulse sequence: Ion 1 laser frequency pulse duration optical phase Ion 2

Fidelity of Cirac-Zoller CNOT <Y exp Y ideal > 2 F. Schmidt-Kaler et al., Nature 422, 408 (2003) Fidelity : 73% M. Riebe et al., PRL 97, 220407 (2006) Fidelity : 92,6% input output

Bichromatic two-qubit gate DS1 DS0 DD1 DD0......... SD1 SD0 Milburn, arxiv:quantph/9908037. Milburn, Schneider, and James, Fortschr. Phys. 48, 801 (2000). Sörensen and Mölmer, PRL 82, 1971 (1999). Sörensen and Mölmer, PRA 62, 022311 (2000). SS1 SS0... Optical qubit theory (manual): Roos C.F., New J. Phys. 10 No 1, 2008, 013002 The common absorption of red and blue detuned light leads to a coherent evolution SS> to DD>. No excitation of DS> states. Requires only Lamb Dicke limit Bell state with F=83% Sackett et al., Nature 406, 256 (2000)

Mølmer-Sørensen interaction time evolution Probabilities 13000 measurements Detuning δ = 20 khz gate time 50 µs Pulse length τ [µs] p 0 +p 2 = 0.9965(4) J. Benhelm, G. Kirchmair, C. F. Roos, R. Blatt, Nature Physics 4, 463 (2008)

Fidelity of the created Bell state Parity flops 29400 measurements in 35 min Phase φ of analysis pulse A = 0.990(1) p 0 +p 2 = 0.9965(4) F = 99.3(1)% F 21 =80% J. Benhelm, G. Kirchmair, C. F. Roos, R. Blatt, Nature Physics 4, 463 (2008)

Mølmer-Sørensen interaction Entangle Disentangle Detuning δ = 20 khz gate time 50 µs pulse shaping 2 µs Equivalent to 17 gate operations

Experimentelle Höhepunkte von QIPC Quantengatter Verschränkte Zustände MICROTRAP SCALA AQUTE mit 2 bis 8 Ionen Teleportation Quanten Simulation Frequenzstandards Photon-Atom Schnittstellen Quantensensoren Quanten-Phasenübergänge

Wesentlicher Milestone für STREP-MICROTRAP Partner

Vision Laserpulse erzeugen verschränkte Zustände Segmentierte Mikrofalle erlaubt das Positionieren vieler Ionen Skalierbarer Quantenprozessor

Seite 30 DPG AMOP Düsseldorf 21.03.2007 Vision Laserpulse erzeugen verschränkte Zustände Segmentierte Mikrofalle erlaubt das Positionieren vieler Ionen Skalierbarer Quantenprozessor