Short Course in Quantum Information Lecture 8 Physical Implementations

Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture : Intro Lecture 2: Formal Structure of Quantum Mechanics Lecture 3: Entanglement Lecture 4: Qubits and Quantum Circuits Lecture 5: Algorithms Lecture 6: Decoherence and Error Correction Lecture 7: Quantum Cryptography Lecture 8: Physical Implementations

Quantum Information Processing Classical Input! in QUANTUM WORLD! out Hilbert space inside. Dimension 2 n. Classical Output

The DiVincenzo Criteria Special Issue: Fortschritte der Physik, 49 (2). I. Scalable physical system, well characterized qubits. - (Could be qudits ) Quantum many-body system. II. Ability to initialize the state of the qubits. - Usually a pure state on n-qubits!n =,,K,. III. Long relevant coherence times. - Much longer than gate operations (fault-tolerance). IV. Universal set of quantum gates. - Quantum control on the 2 n dimensional Hilbert Space. V. Qubit-specific measurement capability. - Readout.

Example: Rydberg atom http://gomez.physics.lsa.umich.edu/~phil/qcomp.html

Quantum computing in a single atom Characteristic scales are set by atomic units Length Momentum Action Energy Bohr Hilbert-space dimension up to n 3 degrees of freedom

Quantum computing in a single atom Characteristic scales are set by atomic units Length Momentum Action Energy Bohr Poor scaling in this unary quantum computer 5 times the diameter of the Sun

Hilbert space and physical resources The primary resource for quantum computation is Hilbert-space dimension. Hilbert-space dimension is a physical quantity that costs physical resources. Single degree of freedom, e.g. motion of an oscillator Action

Hilbert space and physical resources Primary resource is Hilbert-space dimension. Hilbert-space dimension costs physical resources. Many degrees of freedom x 3, p 3 (A) x, p x 2, p 2 (B) = = 2 = 3 = 4 = 5 = 6 = 7 = x, p

Important Lessons The dimension of Hilbert is a resource. Physics determines the structure of Hilbert space. Systems with multiple physical degrees of freedom give Hilbert space a tensor-product structure. Control of a many-body system is a necessary condition to have an exponentially large Hilbert space without using an exponential physical resource. R. Blume-Kohout, C. M. Caves, and I. H. Deutsch, Found. Phys. 32, 64(22).

QIP = Many-body Control n-body Hilbert Space H = h! h 2!L! h n (dimension 2 n ) subsystem = body (qubit 2 n ) Fundamental Theorem of QIP An arbitrary unitary map on H can be constructed from a tensor product of: A finite set of single-body unitaries. { () u } i Any chosen entangling two-body unitary. u (2) ij! u () () i " u j

Generating Single Qubit Rotations Rabi oscillations -- Two-level quantum dynamics:! = E " E h Given:! () = c + c! (t) = c e "ie t / h + c e "ie t / h Bloch Sphere. Natural oscillation frequency!

Generating Single Qubit Rotations Rabi oscillations -- Two-level quantum dynamics:! = E " E h Apply coherent ac-field that couples! c and, E c cos (! c t + "). Field acts to torque the Bloch vector Resonance:! c =! Bloch Sphere. In Rotating Frame.!! c Rotation on Bloch sphere! ~ d E c!

Generating Single Qubit Rotations Rabi oscillations -- Two-level quantum dynamics:! = E e " E g h Apply coherent ac-field that couples! c e and g, E c cos (! c t + "). Field acts to torque the Bloch vector Off-Resonance:! = " c # " Bloch Sphere. In Rotating Frame.! Rotation on Bloch sphere! =! 2 + " 2

General Qubit Rotations Hamiltonian in rotating frame: ˆ H =! h" 2 ˆ Z + h# 2 ( cos$ X ˆ + sin$ Y ˆ ) Rotation matrix: ( ) = cos " 2 U ˆ (t) = exp!ih ˆ t /h Rabi solution:! () = P (t) = ˆ U (t) 2 = I ˆ! isin " & 2! # Z ˆ + $ $ $ (cos% X ˆ + sin% Y ˆ ( ) ' #! cos( " t) & % ( $ 2 ' " 2 " 2 + ) 2 )! = " t! On resonance and $=: P (t)!/2 2! #t/! "= "=# "=2# ˆ U (t) = cos! 2 ˆ I " isin! 2 ˆ X U ˆ $ (t =! /") = #i ' & % ) ( U ˆ $ (t = 2! /") = # ' & % ) (

Two-Qubit Entangling Gates Canonical Example: CNOT Example: CPhase! $ # & # & # & # " & % " % $ ' $ ' $ ' $ #! ' & Example: SWAP x x! $ # & # & # & # " & % SQRT of SWAP " % $ +i!i ' $ 2 2 '!i +i $ ' 2 2 $ # ' &

Designing CPHASE Gates Two-Qubit Separable Hamiltonian: ˆ H AB = ˆ H A + H ˆ B! U ˆ AB = U ˆ A " U ˆ B Suppose we have an interaction between qubits such that Hamiltonian is diagonal in the logical basis. ˆ H AB =! E # E # # # " $ & & E & E & % ˆ U AB = # % % % % $ e!i" e!i" e!i" e!i" & ( ( ( ( '! ij = E ij t /h Equivalent to CPhase by single-qubit rotations for when: #h! +! " (! "! ) = # $ t = E + E " (E " E ) Requires interaction between qubits (inseparable)! ij "! i +! j

The Tao of Quantum Computing Coupling qubits. Control fields. Coherence Coupling to environment. Coupling to neglected degrees of freedom. Decoherence

http://qist.lanl.gov/

A Variety of Platforms Atomic-Molecular-Optical Ion traps. Neutral atom traps. Linear optics. Solid-State Semiconducting quantum dots: Excitons-electronics. Spintronics. Superconducting: Cooper-pair boxes. Mesoscopic circuits. Cross-Cutting Systems NMR - Canonical example: Molecules in solvents. Cavity QED - Canonical example: Single atom in a Fabry-Perot.

Carriers of Quantum Information Electron Spin / Nuclear Spins - Nuclear spins in molecular NMR. - Spintronics. - Atoms ground state electronic manifold. Electronic motion - Metastable excited states of atoms. - Excitons in quantum dots. - Macroscopic coherent current in a superconductor. Photonics - Polarization state of photon. - Frequency/time encoding. - Spatial mode.

Ion traps Ernshaw s Theorem!"E = #! 2 V = Secular motion in oscillating trap V (x,z)

Paul Trap Typical numbers Dimensions: r ~ µm!cm Voltage: U ~! 5V, U ~! 5V rf-frequency:! rf ~ khz "MHz Secular frequencies:! rf ~ khz "MHz

Normal modes of motion center of mass stretch mode! cm =!! st = 3! Courtesy of R. Blatt

Internal states Encoding quantum information Ca + S / 2 D 5 / 2 D 3 / 2 One-photon optical Be + P 3 / 2 P / 2 Two-photon Raman S / 2 Hyperfine sublevels Motional states 2

Readout P 3 / 2 P 3 / 2 P / 2 D 5 / 2 D 3 / 2! " S / 2 S / 2 Cycling transition : Many photons scattered. Ion lights up when in

Quantized Motion Resolved Sidebands - Quantized Doppler Effect. Both internal and external degrees of freedom quantized.! = " int # $ ext! int = { e or g }! ext = { n } e 2 e e! osc g 2 g g! eg! drive =! eg carrier transition

Experimental Results Rabi Oscillations Carrier Transition Blue sideband

Ions in a linear Paul trap. Use quantized normal modes as a quantum bus. Resolved sidebands couple internal-external qubits.

Two-qubit gate: Cirac-Zoller I Atomic qubit { g, e } Motional qubit {, } Employ auxiliary level aux. Apply a 2! rotation on the blue-sideband of g! aux e e aux aux 2! g g Achieves C-Phase between internal and external qubit g! g e! e g! " g e! e

Two-qubit gate: Cirac-Zoller II Motional qubit as quantum bus : Oscillation mode shared. Swap control-qubit with motional qubit e c e c U =!ix!!-pulse on red-sideband g c g c g c g t! g c g t g c e t! g c e t e c g t! "i g c g t e c e c! "i g c e c

Two-qubit gate: Cirac-Zoller II Motional qubit as quantum bus : Oscillation mode shared. C-Phase between motional qubit and target e t e t aux t aux t g c g t! g c g t g c e t! g c e t 2! U =!I g t g t "i e c g t! +i e c g t "i e c e t! "i e c e t

Two-qubit gate: Cirac-Zoller II Motional qubit as quantum bus : Oscillation mode shared. Swap motional qubit back with control e c e c U =!i" x!!-pulse on red-sideband g c g t! g c g t g c g c g c e t! g c e t +i e c g t! e c g t "i e c e t! " e c e t

Experiments on Cirac-Zoller Gate 995 Monroe et al.: Entangling motional and internal qubit.!/2 %!/2 Q-circuit = Ramsey Interferometer 23 Schmidt-Kaler et al.: Multiqubit entanglement CNOT gate

Early Issues with Ion Trap QC Anomalous Heating - Fluctuating patch potentials. - Poor scaling with trap size. - Want large oscillation freq. Dense normal mode spacing

A more flexible trap architecture

Scaling Up: Multiplex Segmented Trap D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 47, 79 (22).

Qubits in Quantum Dots Science 39, 284 (25).

Qubits in Josephson Junctions Science 299, 89 (23).

Qubits in Neutral Atoms Ramsey Fringes $= $=! $=2! Nature 425, 937 (23).

Qubits in Photons Nature 49, 46 (2) Nature 426, 264 (23)

Conclusions Scalable quantum computing requires coherent control of a many-body system. Arbitrary control through single-body (qubit) unitaries operations plus two-body entangling operations. Must overcome the intrinsic conflict: Require strong coupling between qubits and to controlling fields. Variety of platforms from single photons to macroscopic persistent current. Experiments are proceeding to the point where scaling up to many qubits requires an engineering solution.