Quantum Information Processing with Trapped Ions. Experimental implementation of quantum information processing with trapped ions

Quantum Information Processing with Trapped Ions Overview: Experimental implementation of quantum information processing with trapped ions 1. Implementation concepts of QIP with trapped ions 2. Quantum gate operations and entanglement with trapped ions 3. Advanced QIP procedures, scaling the ion-trap quantum computer

Quantum Information Science with Trapped Ions 1

Handouts for lectures are available from the organizers as pdf-files (ask Prof. Knoop, web-pages) Part 1: Implementation concepts of QIP with trapped ions Part 2: Quantum gate operations, entanglement etc. Part 3: Advanced QIP procedures, scaling plus 3 review articles, covering basics and advanced features are available from organizers (as pdf-files) Quantum dynamics of single trapped ions D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev. Mod. Phys. 75, 281 (2003) Quantum computing with trapped ions H. Häffner, C. Roos, R. Blatt, Physics Reports 469, 155 (2008) Entangled states of trapped atomic ions Rainer Blatt and David Wineland, Nature 453, 1008 (2008)

Quantum Information Processing Experimental Implementation with Trapped Ions Rainer Blatt Institute of Experimental Physics, University of Innsbruck, Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences 1. Implementation concepts and basic techniques generics of QIP, the ion trap quantum computer, spectroscopy, measurements, basic interactions and gate operations

Why Quantum Information Processing? applications in physics and mathematics factorization of large numbers (P. Shor, 1994) can be achieved much faster on a quantum computer than with a classical computer factorization of number with L digits: classical computer: ~exp(l 1/3 ), quantum computer: ~ L 2 fast database search (L. Grover, 1997) search data base with N entries: classical computer: O(N), quantum computer: O(N 1/2 ) simulation of Schrödinger equations spectroscopy: quantum computer as atomic state synthesizer D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996) quantum physics with information guided eye

Computation is a physical process input physical object set computation physical process shift output physical object read Classical Computer - bits, registers - gates - classical processes (switches), dissipative Quantum Computer - qubits, quantum registers - quantum gates - coherent processes - preparation and manipulation of entangled states

The requirements for quantum information processing D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001) I. Scalable physical system, well characterized qubits II. Ability to initialize the state of the qubits III. Long relevant coherence times, much longer than gate operation time IV. Universal set of quantum gates V. Qubit-specific measurement capability VI. VII. Ability to interconvert stationary and flying qubits Ability to faithfully transmit flying qubits between specified locations The seven commandments for QIP!!

Quantum bits and quantum registers classical bit: physical object in state 0 or 1 register: bit rows 0 1 1... quantum bit (qubit): superposition of two orthogonal quantum states quantum register: L 2-level atoms, 2 L quantum states 2 L states correspond to numbers 0,..., 2 L 1 most general state of the register is the superposition (binary) (decimal)

Universal Quantum Gates Operations with single qubit: (1-bit rotations) together universal! Operations with two qubits: (2-bit rotations) CNOT gate operation (controlled-not) analogous to XOR control bit target bit

How a quantum computer works quantum processor input computation: sequence of quantum gates output

Circuit quantum computer model: notation control qubit qubit target input computation: sequence of quantum gates output

Which technology? 000 001 010 011 Quantum Processor 100 011 110 011 Cavity QED NMR superconductors quantum dots A. Ekert trapped ions

Ion traps Neutral atoms in traps (opt. traps, opt. lattices, microtraps) Neutral atoms and cavity QED NMR (in liquids) Superconducting qubits (charge-, flux-qubits) Solid state concepts (spin systems, quantum dots, etc.) Optical qubits and LOQC (linear optics quantum computation) Electrons on L-He surfaces Spectral hole burning and more Realization concepts Quantum Information Science and Technology Roadmaps: US http://qist.lanl.gov/ EU http://qist.ect.it/reports/ Quantum Systems that can be controlled and manipulated

the DV criteria translated to an implementer I. Find two-level systems, II. III. IV. that can be individually controlled that are stable and don t decay while you work on them that interact to allow for CNOT gate operation V. that can be efficiently measured or ~100% VI. VII. Find a way to interconnect remote qubits Make sure, your interconnection is good

Meeting the DiVincenzo criteria with trapped ions criterion physical implementation technique scalable qubits initialization long coherence times universal quantum gates internal atomic transitions (2-level-systems) laser cooling, state preparation narrow transitions (optical, microwave) single qubit operations, two-qubit operations linear traps (trap arrays) optical pumping, laser pulses coherence time ~ ms - min Rabi oscillations Cirac-Zoller CNOT qubit measurement quantum jump detection individual ion fluorescence convert qubits to flying qubits coupling of ions with high finesse cavity CQED, bad cavity limit T E faithfully transmit flying qubits coupling of cavities via fiber (photonic channel) coupling pulse sequences (CZKM) T E

Quantum computation with trapped ions J. I. Cirac P. Zoller other gate proposals (and more): Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases Leibfried & Wineland

Quantum computation with trapped ions control bit target bit other gate proposals (and more): Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases Leibfried & Wineland

Quantum bits (qubits) with trapped ions Storing and keeping quantum information requires long-lived atomic states: optical transition frequencies (forbidden transitions, intercombination lines) S D transitions in alkaline earths: Ca +, Sr +, Ba +, Ra +, (Yb +, Hg + ) etc. P 1/2 D 5/2 microwave transitions (hyperfine transitions, Zeeman transitions) alkaline earths: 9 Be +, 25 Mg +, 43 Ca +, 87 Sr +, 137 Ba +, 111 Cd +, 171 Yb + P 3/2 S 1/2 TLS S 1/2 TLS Innsbruck 40 Ca + Boulder 9 Be + ; Michigan 111 Cd + ; Innsbruck 43 Ca +, Oxford 43 Ca + ; Maryland 171 Yb + ; many more

L Ions in linear trap Quantum Computer with Trapped Ions J. I. Cirac, P. Zoller; Phys. Rev. Lett. 74, 4091 (1995) quantum bits, quantum register - narrow optical transitions - groundstate Zeeman coherences state vector of quantum computer 2-qubit quantum gate laser pulses entangle pairs of ions control bit target bit needs individual addressing, efficient single qubit operations small decoherence of internal and motional states quantum computer as series of gate operations (sequence of laser pulses)

Ion trap experiment

Linear Ion Trap Linear ion trap I. Waki et al., Phys. Rev. Lett. 68, 2007 (1992) M.G. Raizen et al., Phys. Rev. A 45, 6493 (1992) - up to about 30 ions (string) - ions separated by a few µm collective quantized motion - anisotropic oscillator: ν z << ν x, ν y - ion motion coupled by Coulomb repulsion - eigenmodes (nearly) independent of the number of ions center of mass mode breathing mode

Ion strings scale favorably... Coulomb repulsion defines a length scale Mode frequencies are nearly independent of ion number A. Steane, Appl. Phys. B 64, 623 (1997) D. James, Appl. Phys. B 66, 181 (1998)

Ion strings as quantum registers 1996 2006 1-10 qubits 2007-2012 32 qubits 40 70

Level scheme of Ca + qubit on narrow S - D quadrupole transition P 3/2 854 nm P 1/2 393 nm 397 nm 866 nm D 5/2 D 3/2 729 nm S 1/2

Spectroscopy with quantized fluorescence (quantum jumps) P monitor S D spectroscopy absorption and emission cause fluorescence steps (digital quantum jump signal) S histogram of absorption events absorptions D laser detuning

State detection by quantized fluorescence 8 7 D state occupied S state occupied # of measurements 6 5 4 3 2 e g detection efficiency: 99.85% 1 0 0 20 40 60 80 100 120 counts per 9 ms

Laser ion interactions...1 An ion trapped in a harmonic potential with frequency ω interacting with the travelling wave of a single mode laser tuned close to a transition that forms an effective two-level system, is described by the Hamiltonian k, ω l, φ: wavenumber, frequency and phase of laser radiation m 0 : mass of the ion for details see: Pauli matrices D. Leibfried, C. Monroe, R. Blatt, D. Wineland Rev. Mod. Phys. 75, 281 (2003)

Laser ion interactions...2 With the Lamb-Dicke parameter we write in terms of creation and annihilation operators In the interaction picture defined by we obtain for the Hamiltonian with coupling states with vibration quantum numbers

Quantized ion motion: ladder structure e 2-level-atom harmonic trap g Ω Γ ν n coupled system n 1,e n,e n +1,e n 1, g n, g n +1, g

Laser ion interactions...3 with laser tuned close to a resonance coupling to other resonances can be neglected ( ). Time evolution of pairwise coupled states is then determined by the set of coupled equations With the detuning and the Rabi frequency we obtain Rabi oscillations on resonance

Laser ion interactions... Lamb-Dicke regime For small ion oscillation amplitudes (Lamb-Dicke regime) we expand and obtain for the coupling carrier excitation detuning red sideband excitation blue sideband excitation

Interactions in the ladder structure carrier red sidebands blue sidebands

Laser ion interactions... unitary operators For later use we already note that the action of a laser on the respective transitions can be written in the following form: Carrier pulse of length Operates on internal degrees of freedom Red sideband pulse of length Blue sideband pulse of length Operate on internal and external degrees of freedom ENTANGLEMENT!

Level scheme of Ca + qubit on narrow S - D quadrupole transition P 3/2 854 nm P 1/2 393 nm 397 nm 866 nm D 5/2 D 3/2 729 nm S 1/2

Spectroscopy of the S 1/2 D 5/2 transition Zeeman structure in non-zero magnetic field: D 5/2-5/2-3/2-1/2 1/2 3/2 5/2 2-level-system: S 1/2 1/2-1/2 + vibrational degrees of freedom

Quantized ion motion 2-level-atom harmonic trap coupled system excitation: various resonances spectroscopy: carrier and sidebands n = 0 D 5/2 n = -1 n = 1 S 1/2 n = 0 1 2 ν Laser detuning

Absorption spectra: Experimental cycle P 3/2 P 1/2 P 3/2 P 1/2 866 nm P 3/2 P 1/2 854 nm S 1/2 D 5/2 D 3/2 729 nm 397 nm S 1/2 D 5/2 D 3/2 397 nm S 1/2 866 nm D 5/2 D 3/2 1st step: probe with 729 nm 5 µs 3 ms 2nd step: detect fluorescence 1 10 ms 3rd step: reset, Doppler cooling 1 10 ms Average: Repeat experiment 100-1000 times, then step 729 nm laser to new detuning

Absorption measurements: Pulse scheme 397nm 397nm σ+ 866nm 854nm 729nm detection preparation time detection excitation 397nm 854nm 866nm 729nm S 1/2 P 3/2 P 1/2 D 5/2 D 3/2 - typically 5ms for each sequence - repeated typically 100 times - triggered by 50Hz line frequency - flexibly controlled by Labview

Excitation spectrum of the S D transition excitation probability 0.5-32 - 12 12 32 52 12 12 12 12 12-25 -20-15 -10-5 0 5 Laser detuning at 729 nm /MHz

Excitation spectrum of the S D transition red axial radial sidebands 0.5 carrier blue radial axial sidebands 0-1710 -920 0 920 1710 khz (spherical trap)

Excitation spectrum of single ion in linear trap ω ax = 1.0 MHz ω rad = 5.0 MHz

Ion trap experiment

Addressing of individual ions electrooptic deflector coherent manipulation of qubits Paul trap Excitation 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 dichroic beamsplitter inter-ion distance: ~ 4 µm Fluorescence detection CCD 0-10 -8-6 -4-2 0 2 4 6 8 10 Deflector Voltage (V) addressing waist: ~ 2.5 µm < 0.1% intensity on neighbouring ions

Detection of 6 individual ions

Coherent state manipulation carrier (C) carrier and sideband Rabi oscillations with Rabi frequencies blue sideband (BSB) Lamb-Dicke parameter

Coherent state manipulation: carrier carrier Carrier transitions leave the motion unchanged

Coherent state manipulation: sideband Sideband transitions entangle motion and internal excitation sideband

Qubit rotations in computational subspace D,0 D,1 D,2 transitions are described by unitary matrices: S,0 S,1 S,2 carrier: computational subspace red sideband: keep all populations in subspace only! blue sideband: A. M. Childs, I. L. Chuang, Phys. Rev. A 63, 012306 (2001)

SWAP and composite SWAP operation D,0 S,0 π perfect SWAP operation D,1 S,1 π 2 out of subspace if D,1> ( S,1>) populated find simultaneously perfect SWAP operation with 4π rotation ( ) on the D,1 S,2 S,1 D,2 transition(s) composite SWAP operation: I. L. Chuang et al., Innsbruck (2002)

S,0> - D,1> SWAP and composite SWAP operations 1 3 2 single-step SWAP operation 3-step composite SWAP operation

3-step composite SWAP operation I. Chuang et al., Innsbruck (2002) 1 1 3 3 2 2 on on

A phase gate with 4 pulses (2π rotation), blue sideband 1 on 3 2 4

Population of S,1> - D,2> remains unaffected 2 4 3 1

A single ion composite phase gate: Experiment state preparation S,0, then application of phase gate pulse sequence 1 π 2 π π 2 π ϕ = 0 ϕ = 0 ϕ = π 2 φ = 0 φ = π 2 φ = 0 φ = π 2 0.9 0.8 D 5/2 - excitation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 Time (µs)

A single ion composite CNOT gate: Experiment state preparation S,0, then application of CNOT gate pulse sequence 1 π 2 Phase gate π 2 0.9 0.8 97.8 (5) % D 5/2 - excitation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 Time (µs)

A single ion composite CNOT gate: Experiment D 5/2 -state excitation probability 1 0.8 0.6 0.4 0.2 D,1> π prep. 2 Time (µs) phase gate π 2 0 0 50 100 150 200 250 300

2. Quantum gate operations and entanglement with ions 2.1 Cirac-Zoller CNOT gate operation with two ions 2.2 Entanglement and Bell state generation 2.3 State tomography 2.4 Process tomography of the CNOT gate 2.5 Tripartite entanglement (W-states, GHZ-states) 2.6 Multipartite entanglement 2.7 Teleportation 2.8 Entanglement Swapping 2.9 Toffoli gate operation

L Ions in linear trap Quantum computer with trapped ions J. I. Cirac, P. Zoller; Phys. Rev. Lett. 74, 4091 (1995) quantum bits, quantum register - narrow optical transitions - groundstate Zeeman coherences 2-qubit quantum gate state vector of quantum computer laser pulses entangle pairs of ions control bit target bit needs individual addressing, efficient single qubit operations small decoherence of internal and motional states quantum computer as series of gate operations (sequence of laser pulses)

The Cirac-Zoller CNOT gate operation with 2 ions allows the realization of a universal quantum computer! control bit target bit F. Schmidt-Kaler et al., Nature 422, 408 (2003)

Cirac Zoller two-ion 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 SWAP control target control qubit target qubit

Cirac Zoller two-ion 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 target control qubit target qubit

Cirac Zoller two-ion 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> control target SWAP -1 0 0 control qubit target qubit

Cirac Zoller two-ion controlled-not operation ion 1 motion ion 2 S S, D, D SWAP SWAP -1 0 0 control qubit target qubit pulse sequence: Ion 1 laser frequency pulse length optical phase Ion 2 F. Schmidt-Kaler et al., Nature 422, 408 (2003)

Individual ion detection Ion 1 Ion 2 SS SS control qubit 5µm SS target qubit SD DS gate sequence Time τ DD measure states Individual ion detection on CCD camera

Cirac Zoller CNOT- gate operation SS - SS DS - DD SD - SD DD - DS

Experimental fidelity of Cirac-Zoller CNOT operation truth table: control bit target bit input output F. Schmidt-Kaler et al., Nature 422, 408 (2003)

Superposition as input to CNOT - gate operation prepare gate output detect

Entanglement of two ions using the CNOT-gate operation S+D> S> CNOT SS + e iφ DD> control bit target with local rotations ϕ create Bell states SS+DD>, SS-DD> SD+DS>, SD-DS> ϕ on both ions ϕ on individual ion measurement sequence: Ion 1 Ion 2 π/2 0 CNOT π/2 ϕ π/2 ϕ Super-Ramsey experiment Parity check C. A. Sackett et al., Nature 404, 256 (2000)

Measuring entanglement C. A. Sackett et al., Nature 404, 256 (2000) entangling operation: parity Π witnesses entanglement: correlates states Π oscillates with 2ϕ! Ion 1 π/2 0 CNOT π/2 ϕ Ion 2 π/2 ϕ F = 71(3)% Fidelity = 0.5 (P SS +P DD +visibility) F. Schmidt-Kaler et al., Nature 422, 408 (2003)

Parity Measurement Probability of Probability of Parity = or or Parity = +1 Parity = -1

Bell states Ion 1 D,0 D,1 π / 2 S,0 S,1 π/2 pulse, BSB

Bell states Ion 2 D,0 S,0 π D,1 S,1 π/2 pulse, BSB π pulse, carrier

Bell states Ion 2 D,0 D,1 π S,0 S,1 π/2 pulse, BSB π pulse, carrier π pulse, BSB

Preparation of Bell states and measurement Ion 1 τ Ion 2 parity measurement C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)

Reconstruction of a density matrix ρ Representation of ρ as a sum of orthogonal observables A j : ρ is completely determined by the expectation values <A j > : For a two-qubit system : Joint measurements of all spin components

Maximum likelihood estimation The reconstructed is not necessarily positive semidefinite obtain ρ from maximum likelihood estimation: choose ρ such that Z. Hradil, Phys. Rev. A 55, R1561 (1997), K. Banaszek et al., Phys. Rev. A 61, 010304 (1999) is minimized optimization of 15 parameters

Experimental tomography procedure two 40 Ca + ions trapped in linear trap Individual qubit operations Experimental cycle (~20 ms): 1. Laser cooling to the motional ground state 2. Quantum state preparation 3. Application of tomography pulses 4. State detection 100-200 experiments

Experimental tomography procedure 1. Initialization in SS0> 10 ms of Doppler and sideband cooling 2. Quantum state manipulation on the S 1/2 D 5/2 transition 3. Quantum state measurement by fluorescence detection 5µm P 1/2 D 5/2 Doppler Quantum state Fluorescence cooling manipulation detection Sideband cooling Detection with CCD camera: S 1/2 Qubit transition :

Measurement of the density matrix Measurement of prepare Bell state no rotation measure prepare Bell state ion #1, σ x rotation ion #2, identity measure 200 repetitions 200 repetitions Rotation of Bloch sphere prior to measurement: 9 different settings π/2 -pulse prepare Bell state ion #1, σ y rotation ion #2, σ y rotation measure 200 repetitions

Information from a single projection is not enough Art by Tim Noble & Sue Webster

Tomography pulses for state reconstruction Setting Transformation applied to Measured expectation values # ion 1 ion 2 1 - - σ (1) z σ (2) z σ (1) z σ (2) z 2 R(π/2,3π/2) - σ (1) x σ (2) z σ (1) x σ (2) z 3 R(π/2, π) - σ (1) y σ (2) z σ (1) y σ (2) z 4 - R(π/2, 3π/2) σ (1) z σ (2) x σ (1) z σ (2) x 5 - R(π/2, π) σ (1) z σ (2) y σ (1) z σ (2) y 6 R(π/2, 3π/2) R(π/2, 3π/2) σ (1) x σ (2) x σ (1) x σ (2) x 7 R(π/2, 3π/2) R(π/2, π) σ (1) x σ (2) y σ (1) x σ (2) y 8 R(π/2, π) R(π/2, 3π/2) σ (1) y σ (2) x σ (1) y σ (2) x 9 R(π/2, π) R(π/2, π) σ (1) y σ (2) y σ (1) y σ (2) y

Push-button preparation and tomography of Bell states Fidelity: F = 0.91 Entanglement of formation: E(r exp ) = 0.79 Violation of Bell inequality: S(r exp ) = 2.53(6) > 2 C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)

1. Implementation concepts and basic techniqes 1.1 Generics of quantum information processing 1.2 Realization concepts 1.3 Ion trap quantum computer the basics 1.4 Spectroscopy in ion traps 1.5 Addressing individual ions 1.6 Basic gate operations, composite pulses 2. Quantum gate operations and entanglement with ions 2.1 Cirac-Zoller CNOT gate operation with two ions 2.2 Entanglement and Bell state generation 2.3 Quantum state tomography