Quantum Computation with Neutral Atoms

Quantum Computation with Neutral Atoms Marianna Safronova Department of Physics and Astronomy Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. 1

Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. Why Quantum Computers? Computer technology is making devices smaller and smaller reaching a point where classical physics is no longer a suitable model for the laws of physics. 2

Bits & Qubits Fundamental building blocks of classical computers: BITS STATE: Definitely 0 or 1 Fundamental building blocks of quantum computers: Basis states: Quantum bits or QUBITS 0 and 1 Superposition: ψ = α 0 + β 1 Bits & Qubits Fundamental building blocks of classical computers: BITS Fundamental building blocks of quantum computers: Quantum bits or QUBITS STATE: Definitely 0 or 1 Basis states: 0 and 1 3

Qubit: any suitable two-level quantum system Bits & Qubits: primary differences Superposition ψ = α 0 + β 1 4

Bits & Qubits: primary differences Measurement Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. Qubit: Quantum calculation: number of parallel processes due to superposition QO FR Bits & Qubits: primary differences Superposition Measurement ψ = α 0 + β 1 Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. Qubit: we cannot just measure α and β and thus determine its state! We get either 0 or 1 with corresponding probabilities α 2 and β 2. 2 2 α + β = 1 The measurement changes the state of the qubit! 5

Multiple qubits Hilbert space is a big place! - Carlton Caves Multiple qubits Hilbert space is a big place! Two bits with states 0 and 1 form four definite states 00, 01, 10, and 11. Two qubits: can be in superposition of four computational basis set states. ψ = α 00 + β 01 + γ 10 + δ 11 - Carlton Caves 2 qubits 4 amplitudes 3 qubits 8 amplitudes 10 qubits 1024 amplitudes 20 qubits 1 048 576 amplitudes 30 qubits 1 073 741 824 amplitudes 500 qubits More amplitudes than our estimate of number of atoms in the Universe!!! 6

Entanglement ψ = 00 + 11 2 Results of the measurement First qubit 0 1 Second qubit 0 1 ψ α β Entangled states Quantum logic gates 7

Logic gates Classical NOT gate Quantum NOT gate (X gate) A NOT A α 0 + β 1 X α 1 + β 0 A N O T A Matrix form representation 0 1 1 0 0 1 X = 1 0 The only non-trivial single bit gate α β X = β α More single qubit gates Any unitary matrix U will produce a quantum gate! Z 1 0 = 0 1 α 0 + β 1 α 0 β 1 Z Hadamard gate: H 1 1 1 = 2 1 1 α 0 + β 1 H 0 + 1 0 1 α + β 2 2 8

Two-qubit gates Quantum CNOT gate A A B B' AB AB' 00 00 01 01 10 11 11 10 WE NEED TO BE ABLE TO MAKE ONLY ONE TWO-QUBIT GATE! Back to the real world: What do we need to build a quantum computer? Qubits which retain their properties. Scalable array of qubits. Initialization: ability to prepare one certain state repeatedly on demand. Need continuous supply of 0. Universal set of quantum gates. A system in which qubits can be made to evolve as desired. Long relevant decoherence times. Ability to efficiently read out the result. 9

Real world strategy If X is very hard it can be substituted with more of Y. Of course, in many cases both X and Y are beyond the present experimental state of the art David P. DiVincenzo The physical implementation of quantum computation. Experimental proposals Liquid state NMR Trapped ions Cavity QED Trapped atoms Solid state schemes And other ones 10

1. A scalable physical system with well characterized qubits: memory (a) Internal atomic state qubits: ground hyperfine states of neutral trapped atoms well characterized Very long lived! F=2 5s 1/2 6.8 GHz F=1 1 0 M F =-2,-1,0,1,2 87 Rb: Nuclear spin I=3/2 M F =-1,0,1 1. A scalable physical system with well characterized qubits: memory (b) Motional qubits : quantized levels in the trapping potential also well characterized http://www.colorado.edu/physics/2000/index.pl 11

1. A scalable physical system with well characterized qubits: memory (a)internal atomic state qubits (b) Motional qubits Advantages: very long decoherence times! Internal states are well understood: atomic spectroscopy & atomic clocks. 1. A scalable physical system with well characterized qubits Optical lattices: loading of one atom per site may be achieved using Mott insulator transition. Scalability: the properties of optical lattice system do not change in the principal way when the size of the system is increased. Designer lattices may be created (for example with every third site loaded). Advantages: inherent scalability and parallelism. Potential problems: individual addressing. 12

2: Initialization Internal state preparation: putting atoms in the ground hyperfine state Very well understood (optical pumping technique is in use since 1950) Very reliable (>0.9999 population may be achieved) Motional states may be cooled to motional ground states (>95%) Loading with one atom per site: Mott insulator transition and other schemes. Zero s may be supplied during the computation (providing individual or array addressing). 3: A universal set of quantum gates CNOT Hadamard gate: π/8 gate: A A A H A' T B AB AB' 00 00 01 01 10 11 B' A 0 1 A' 0 + 1 2 0 1 2 1 0 T = i /4 0 e π Phase gate S: 1 0 S = 0 i 11 10 H ( X Z ) / 2 S = T = + 2 13

3: A universal set of quantum gates 1. Single-qubit rotations: well understood and had been carried out in atomic spectroscopy since 1940 s. 2. Two-qubit gates: none currently implemented (conditional logic was demonstrated) Proposed interactions for two-qubit gates: (a) Electric-dipole interactions between atoms (b) Ground-state elastic collisions (c) Magnetic dipole interactions Only one gate proposal does not involve moving atoms (Rydberg gate). Advantages: possible parallel operations Disadvantages: decoherence issues during gate operations Two-qubit quantum gates (a) Electric-dipole interactions between atoms Brennen et al. PRL 82, 1060 (1999), PRA 61, 062309 (2000), Pairs of atoms are brought to occupy the same site in far-off-resonance optical lattice by varying polarization of the trapping laser. Two types of atoms: trapped in σ+ and σ- polarized wells. Near-resonant electric-dipole is induced by auxiliary laser (depending on the atomic state). Brennen, Deutch, and Willaims PRA 65, 022313 (2002) Deterministic entanglement of pairs of atoms trapped in optical lattice is achieved by coupling to excited state molecular hyperfine potentials. 14

Two-qubit quantum gates (a) Electric-dipole interactions between atoms cont. Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Gate operations are mediated by excitation of Rydberg states (b) Ground-state elastic collisions Calarco et al. Phys. Rev. A 61, 022304 (2000) Cold collisions between atoms conditional on internal states. Cold collisions between atoms conditional on motional-state tunneling. (c) Magnetic-dipole interactions between pairs of atoms Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Why Rydberg gate? 15

Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Do not need to move atoms! FAST! Local blockade of Rydberg excitations Excitations to Rydberg states are suppressed due to a dipole-dipole interaction or van der Waals interaction http://www.physics.uconn.edu/~rcote/ 16

Rydberg gate scheme Rb 40p FAST! R Apply a series of laser pulses to realize the following logic gate: 1 5s 0 1 2 Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) 00 00 10 01 01 10 11 11 Rydberg gate scheme Rb 5s R 1 0 40p 1 2 [ 1] 2 [ 2] [ 1] π π π 00 00 00 00 01 01 01 01 10 R0 R0 10 11 R1 R1 11 17

Decoherence One of the decoherence sources: motional heating. Results from atom seeing different lattice in ground and Rydberg states. Solution: choose the lattice photon frequency ω to match frequency-dependent polarizability α(ω) of the ground and Rydberg states. Error correction: possible but error rate has to be really small (< 10-4 ). Other decoherence sources Photoionization Spontaneous emission Transitions induced by black-body radiation Laser beam intensity stability Pulse timing stability Individual addressing accuracy 18

4. Long relevant decoherence times F=2 Memory: long-lived states. 5s 1/2 6.8 GHz F=1 0 1 Fundamental decoherence mechanism for optically trapped qubits: photon scattering. Decoherence during gate operations: a serious issue. 5: Reading out a result Quantum jump method via cycling transitions. Advantages: standard atomic physics technique, well understood and reliable. Quantum computation with NEUTRAL ATOMS: ADVANTAGES Scalability Possible massive parallelism due to lattice geometry Long decoherence times (weak coupling to the environment) Availability of the controlled interactions Well-developed experimental techniques for initialization, state manipulation, and readout Accurate theoretical description of the system is possible. 19

Quantum computation with NEUTRAL ATOMS: PROBLEMS Decoherence during the gate operations (various sources) Reliable lattice loading and individual addressing QC architecture for lattice geometry: Error-correcting codes and fault-tolerant computation, how to run algorithms on neutral atom quantum computer. 20

Quantum information processing with trapped ions Courtesy of Timo Koerber Institut für Experimentalphysik Universität Innsbruck 1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook 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 1

Experimental Setup P 1/2 D 5/2 quantum bit S 1/2 Important energy levels The important energy levels are shown on the next slides; a fast transition is used to detect ion fluorescence and for Doppler cooling, while the narrow D5/2 quadrupole transition has a lifetime of 1 second and is used for coherent manipulation and represents out quantum bit. Of course a specific set of Zeeman states is used to actually implement our qubit. The presence of other sublevels give us additional possibilities for doing coherent operations. 2

Ca+: Important energy levels τ = 7 ns S 1/2 D 5/2 : quadrupole transition P 1/2 τ = 1 s D 5/2 397 nm 729 nm S 1/2 Ca+: Important energy levels τ = 7 ns P 1/2 D 5/2 S 1/2 qubit quoctet (sp?) 3

Qubits with trapped ions Encoding of quantum information requires long-lived atomic states: optical transitions Ca +, Sr +, Ba +, Ra +, Yb +, Hg + etc. microwave transitions 9 Be +, 25 Mg +, 43 Ca +, 87 Sr +, 137 Ba +, 111 Cd +, 171 Yb + P 1/2 D 5/2 P 3/2 S 1/2 qubit S 1/2 qubit String of Ca+ ions in Paul trap row of qubits in a linear Paul trap forms a quantum register 4

String of Ca+ ions in linear Paul trap row of qubits in a linear Paul trap forms a quantum register 0.7 2 MHz 1.5 4 MHz ω z ω x, y 50 µm String of Ca+ ions in linear Paul trap row of qubits in a linear Paul trap forms a quantum register 0.7 2 MHz 1.5 4 MHz ω z ω x, y 50 µm 5

Addressing of individual ions 0.8 electrooptic deflector coherent manipulation of qubits Paul trap Excitation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-10 -8-6 -4-2 0 2 4 6 8 10 Deflector Voltage (V) dichroic beamsplitter Fluorescence detection CCD inter ion distance: ~ 4 µm addressing waist: ~ 2.5 µm < 0.1% intensity on neighbouring ions Ion addressing The ions can be addressed individually on the qubit transition with an EO deflector which can quickly move the focus of the 729 light from one ion to another, using the same optical path as the fluorescence detection via the CCD camera. How well the addressing works is shown on the previous slide: The graph shows the excitation of the indiviual ions as the deflector is scanned across the crystal. 6

Notes for next slides: External degree of freedom: ion motion Now let's have a look at the qubit transition in the presence of the motional degrees of freedom. If we focus on just one motional mode, we just get a ladder of harmonic oscillator levels. The joint (motion + electronic energy level) system shows a double ladder structure. With the narrow laser we can selectively excite the carrier transition, where the motional state remains unchanged... Or use the blue sideband and red sideband transitions, where we can change the motional state. We can walk down the double ladder by exciting the red sideband and returning the ion dissipatively to the grounsstate. With this we can prepare the ions in the motional ground state with high probability, thereby initializing our quantum register. External degree of freedom: ion motion harmonic trap 7

External degree of freedom: ion motion 2-level-atom harmonic trap joint energy levels External degree of freedom: ion motion 2-level-atom harmonic trap joint energy levels Laser cooling to the motional ground state: Cooling time: 5-10 ms > 99% in motional ground state 8

Coherent manipulation 2-level-atom harmonic trap joint energy levels Interaction with a resonant laser beam : Ω : Rabi frequency φ : phase of laser field Laser beam switched on for duration τ : θ : rotation angle If we resonantly shine in light pulse at the carrier transition, the system evolves for a time tau with this Hamiltonian, where the coupling strength Omega depends on the sqroot of the intensity, and phi is the phase of the laser field with respect to the atomic polarization. Coherent manipulation Let's now begin to look at the coherent state manipulation. If we resonantly shine the light pulse at the carrier transition, the system evolves for a time τ with this Hamiltonian, where the coupling strength Ω depends on the square root of the intensity, and φ is the phase of the laser field with respect to the atomic polarization. The effect of such a pulse is a rotation of the state vector on the Bloch sphere, where the poles represent the two states and the equator represents superposition states with different relative phases. The roation axis is determined by the laser frequency and phase. The important message is here that we can position the state vector anywhere on the Bloch sphere, which is a way of saying that we can create arbitrary superposition states. The same game works for sideband pulses. With a π/2 pulse, for example, we entangle the internal and the motional state! Since the motional state is shared by all ions, we can use the motional state as a kind of bus to mediate entanglement between different qubits in the ion chain. 9

Coherent excitation: Rabi oscillations Carrier pulses: Bloch sphere representation D state population Coherent excitation on the sideband Blue sideband pulses: coupled system Entanglement between internal and motional state! D state population 10

Experimental procedure P 1/2 D 5/2 τ =1s Doppler Quantum state Fluorescence cooling manipulation detection Sideband cooling S 1/2 40 Ca + S 1/2 One ion : Fluorescence histogram 8 D 7 5/2 state 6 5 4 3 2 1 S 1/2 state 0 0 20 40 60 80 100 120 counts per 2 ms 1. Initialization in a pure quantum state: laser cooling,optical pumping 2. Quantum state manipulation on S 1/2 D 5/2 qubit transition 3. Quantum state measurement by fluorescence detection 50 experiments / s Repeat experiments 100-200 times Experimental procedure P 1/2 D 5/2 τ =1s Doppler Quantum state Fluorescence cooling manipulation detection Sideband cooling S 1/2 40 Ca + S 1/2 1. Initialization in a pure quantum state: Laser sideband cooling 2. Quantum state manipulation on S 1/2 D 5/2 transition 3. Quantum state measurement by fluorescence detection Multiple ions: Spatially resolved detection with CCD camera: 50 experiments / s Repeat experiments 100-200 times 11

1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook Creation of Bell state Pulse sequence: 12

Generation Creation of of Bell Bell states Pulse sequence: Ion 1: π/2, blue sideband Creation of Bell states Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier 13

Creation of Bell states Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier Ion 2: π, blue sideband Analysis of Bell states Fluorescence detection with CCD camera: Coherent superposition or incoherent mixture? What is the relative phase of the superposition? Ψ + Measurement of the density matrix: SS SDDS DD SS DDDS SD 14

Reconstruction of a density matrix Representation of ρ as a sum of orthogonal observables A i : ρ is completely detemined by the expectation values <A i > : Finally: maximum likelihood estimation (Hradil 97, Banaszek 99) For a two-ion system : Joint measurements of all spin components Preparation and tomography of Bell states Fidelity: F = 0.91 SS SDDS SS SDDS DD SS SD DD DS DD SS SD DD DS Entanglement of formation: E(ρ exp ) = 0.79 SS SDDS SS SDDS Violation of Bell inequality: DD SS SD DD DS DD SS SD DD DS S(ρ exp ) = 2.52(6) > 2 C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) 15

Different decoherence porperties sensitive to: laser frequency magnetic field exc. state lifetime 1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook 16

Generation of W-states Pulse sequence: Ion 2,3: π, carrier Ion 1: θ 1, blue sideband 1. 2. 3. Ion 2: θ 2, blue sideband Ion 3: θ 3, blue sideband Density matrix of W state Fidelity: 85 % experimental result theoretical expectation 17

Four-ion W-states DDDD DDDS 14.4.2005 SSSS DDDD SSSS Five-ion W-states DDDDD DDDDS 15.4.2005 SSSSS DDDDD SSSSS 18

Detection of six individual ions 5µm 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 6 5 4 3 2 1 all ions in S> ion 1 in S> ion 6 in S> ion 4 in S> ion 5 in S> ions 1 and 5 in S> ions 1,2,3, and 5 in S> ions 1,3 and 4 in S> Ion detection on a CCD camera (detection time:4ms) Six-ion W-state F=73% preliminary result Is there 6-particle entanglement present? 6-particle W-state can be distilled from the state (O. Gühne) 6-particle entanglement present, unresolved issues with error bars 22.4.2005 729 settings, measurement time >30 min. 19