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Short Course in Lecture 8

Physical Implementations

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Course Info • All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/DeutschClasses.html

• Syllabus Lecture 1: Intro Lecture 2: Formal Structure of Lecture 3: Entanglement Lecture 4: and Quantum Circuits Lecture 5: Algorithms Lecture 6: Decoherence and Error Correction Lecture 7: Lecture 8: Physical Implementations

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Quantum Information Processing

Classical Input

! in QUANTUM WORLD

! out

Hilbert space inside. Classical Output Dimension 2n.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information The DiVincenzo Criteria Special Issue: Fortschritte der Physik, 49 (2000).

I. Scalable physical system, well characterized qubits. - (Could be “qudits”) Quantum many-body system.

II. Ability to initialize the state of the qubits. n 0 ! = 0,0, ,0 - Usually a pure state on n-qubits K .

III. Long relevant times. - Much longer than gate operations (fault-tolerance).

IV. “Universal” set of quantum gates. - Quantum control on the 2n dimensional Hilbert Space.

V. -specific measurement capability. - Readout. I. H. Deutsch, University of New Mexico Short Course in Quantum Information Example: Rydberg atom http://gomez.physics.lsa.umich.edu/~phil/qcomp.html

I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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

I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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

I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Hilbert space and physical resources

Primary resource is Hilbert-space dimension Hilbert-space dimension. costs physical resources. Many degrees of freedom x3, p3 (A) 1 0 0 1 1 0 1

1 0 1 1 1 1 0 1 0 0 x2, p2 0 0 1 0 1 1 0 0 1 1 x1, p1 (B) 0 = 000 1 = 001 2 = 010 3 = 011 4 = 100 5 = 101 6 = 110 7 = 111 x, p

I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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, 1641(2002).

I. H. Deutsch, University of New Mexico Short Course in Quantum Information QIP = Many-body Control

n-body Hilbert Space H = h ! h ! ! h (dimension 2n) 1 2 L n “subsystem” = “body” (qubit 2n)

Fundamental Theorem of QIP An arbitrary unitary map on H can be constructed from a tensor product of: u(1) • A finite set of single-body unitaries . { i } (2) (1) (1) • Any chosen entangling two-body unitary. uij ! ui " u j

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Generating Single Qubit Rotations

Rabi oscillations -- Two-level : 0 Given: ! (0) = c0 0 + c1 1 E 0 " E1 !01 = h ! (t) = c e"iE1t / h 1 + c e"iE0t / h 0 1 1 0 0

Bloch Sphere. Natural oscillation frequency

!01 1

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Generating Single Qubit Rotations

Rabi oscillations -- Two-level quantum dynamics: 0 E " E Apply coherent ac-field that couples ! = 0 1 ! 1 01 c 0 and , E c cos ( ! c t + " ) . h 1 Field acts to “torque” the Bloch vector 0 : !c = !01 ! Bloch Sphere. Rotation on Bloch sphere In Rotating Frame. !c ! ~ 0 d 1 E c !01 1 I. H. Deutsch, University of New Mexico Short Course in Quantum Information Generating Single Qubit Rotations

Rabi oscillations -- Two-level quantum dynamics: 0 E " E Apply coherent ac-field that couples e g ! g !01 = c e and , E c cos ( ! c t + " ) . h 1 Field acts to “torque” the Bloch vector 0 Off-Resonance: ! = "c # "01

Bloch Sphere. Rotation on Bloch sphere In Rotating Frame. ˜ 2 2 !˜ ! = ! + " 1

I. H. Deutsch, University of New Mexico Short Course in Quantum Information

General Qubit Rotations

" # Hamiltonian in rotating frame: Hˆ = ! h Zˆ + h (cos$ Xˆ + sin$ Yˆ ) 2 2 Rotation matrix: " " & # $ ( Uˆ (t) = exp !iHˆ t / = cos Iˆ ! isin ! Zˆ + (cos% Xˆ + sin% Yˆ ) ˜ ( h) 2 2 ' ˜ ˜ ) ! = "t $ $ Rabi solution: 2 2 # 1! cos("˜ t)& " ! (0) = 1 ˆ P0(t) = 0 U (t)1 = % ( 2 2 $ 2 ' " + )

! On resonance and $=0: ! ! Uˆ (t) = cos Iˆ " isin Xˆ "=0 2 2 P (t) !/2 0 "=# 0 1 ˆ $ ' "=2# U (t = ! /") = #i& ) %1 0( 2! ˆ $1 0' U (t = 2! /") = #& ) #t/! %0 1( I. H. Deutsch, University of New Mexico Short Course in Quantum Information Two-Qubit Entangling Gates

Canonical Example: CNOT Example: CPhase

!1 0 0 0$ 00 "1 0 0 0 % 00 # & 0 1 0 0 01 $0 1 0 0 ' 01 # & $ ' #0 0 0 1& 10 $0 0 1 0 ' 10 # & $ ' "0 0 1 0% 11 #0 0 0 !1& 11

Example: SWAP SQRT of SWAP 1 0 0 0 00 !1 0 0 0$ 00 " % x $ 1+i 1!i ' #0 0 1 0& 01 0 0 01 # & $ 2 2 ' #0 1 0 0& 10 $0 1!i 1+i 0' 10 x # & 2 2 "0 0 0 1% 11 $ ' #0 0 0 1& 11

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Designing CPHASE Gates

ˆ ˆ ˆ ˆ ˆ ˆ Two-Qubit Separable Hamiltonian: H AB = H A + H B ! U AB = U A "U B

Suppose we have an interaction between qubits such that Hamiltonian is diagonal in the logical basis. !i" 00 ! E 00 $ #e & % !i" ( # E & e 10 ˆ # 01 & ˆ % ( H AB = U = ! = E t / AB !i"10 ij ij h # E10 & % e ( # & E % !i"11 ( " 11 % $ e '

Equivalent to CPhase by single-qubit rotations fo r when: # ! + ! " (! " ! ) = # $ t = h 11 00 10 01 E + E " (E " E ) 11 00 10 01

Requires interaction between qubits (inseparable) !ij " !i + ! j I. H. Deutsch, University of New Mexico Short Course in Quantum Information

The Tao of Quantum Computing

• Coupling qubits. • Control fields. Coherence

• Coupling to environment. • Coupling to neglected degrees of freedom. Decoherence

I. H. Deutsch, University of New Mexico Short Course in Quantum Information http://qist.lanl.gov/

I. H. Deutsch, University of New Mexico Short Course in Quantum Information A Variety of Platforms

Atomic-Molecular-Optical Solid-State • Ion traps. • Semiconducting quantum dots: • Excitons-electronics. • Neutral atom traps. • Spintronics. • Superconducting: • Linear optics. • Cooper-pair boxes. • Mesoscopic circuits.

Cross-Cutting Systems • NMR - Canonical example: Molecules in solvents. • Cavity QED - Canonical example: Single atom in a Fabry-Perot.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Carriers of Quantum Information

Electron / Nuclear Spins - Nuclear spins in molecular NMR. - Spintronics. - Atoms 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.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information I. H. Deutsch, University of New Mexico Short Course in Quantum Information Ion traps

Ernshaw’s Theorem !"E = 0 # !2V = 0

Secular motion in oscillating trap

V(x,z)

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Paul Trap

Typical numbers

Dimensions: r0 ~ 10 µm !1cm Voltage: U˜ ~ 100 ! 500V, U ~ 0 ! 50V

rf-frequency: !rf ~ 100 kHz "100MHz Secular frequencies: ! ~ 10 kHz "10MHz rf I. H. Deutsch, University of New Mexico Short Course in Quantum Information Normal modes of motion

“center of mass” “stretch mode”

!cm = !0 !st = 3!0

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Courtesy of R. Blatt Encoding quantum information

Internal states

P3 / 2 Ca+ 1 D5 / 2 P1/ 2 + D3 / 2 Be Two-photon One-photon optical Raman 0 1 S1/ 2 S1/ 2 0 Hyperfine sublevels

Motional states 2

1

0

I. H. Deutsch , University of New Mexico Short Course in Quantum Information Readout

P3 / 2 P3 / 2 P 1/ 2 1 D5 / 2 ! D3 / 2 " 1 0 S1/ 2 S1/ 2 0

“Cycling transition”: Many photons scattered.

Ion “lights up” when in 0

I. H. Deutsch, University of New Mexico Short Course in Quantu m Information 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 1 osc e 0 ! = ! ! drive eg eg g 2 “carrier transition” g 1 g 0 I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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 1 osc e 0 ! = ! " ! ! drive eg osc eg g 2 “red sideband” g 1 g 0 I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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 1 osc e 0 ! = ! + ! ! drive eg osc eg g 2 “blue sideband” g 1 g 0 I. H. Deutsch, University of New Mexico Short Course in Quantum Information Rabi Oscillations

Experimental Results

Carrier Transition

Blue sideband

I. H. Deutsch, University of New Mexico Short Course in Quantum Information • Ions in a linear Paul trap. • Use quantized normal modes as a “quantum bus”. • Resolved sidebands couple internal-external qubits.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Two-qubit gate: Cirac-Zoller I

Atomic qubit { g , e } Motional qubit { 0 , 1 }

Employ “auxiliary” level aux . Apply a 2! rotation on the blue-sideband of g ! aux e 1 e 0 aux 1 aux 0

2! g 1 g 0 g 0 ! g 0 Achieves C-Phase between e 0 ! e 0 internal and external qubit g 1 ! " g 1 e 1 ! e 1 I. H. Deutsch, University of New Mexico Short Course in Quantum Information Two-qubit gate: Cirac-Zoller II

Motional qubit as “quantum bus”: Oscillation mode shared.

• Swap control-qubit with motional qubit

e c 1 e 0 c !-pulse on red-sideband ! U = !iX g 1 c g c 0

g c g t 0 ! g c g t 0 g c e t 0 ! g c e t 0

e c g t 0 ! "i g c g t 1

e c e c 0 ! "i g c e c 1

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Two-qubit gate: Cirac-Zoller II

Motional qubit as “quantum bus”: Oscillation mode shared.

• C-Phase between motional qubit and target

e 1 t aux 1 e t 0 t aux t 0 2! U = !I g 1 t g t 0 g c g t 0 ! g c g t 0 g e 0 ! g e 0 c t c t

"i e c g t 1 ! +i e c g t 1

"i e c e t 1 ! "i e c e t 1

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Two-qubit gate: Cirac-Zoller II

Motional qubit as “quantum bus”: Oscillation mode shared.

• Swap motional qubit back with control

e c 1 e 0 c !-pulse on red-sideband U = !i" x ! g 1 c g c 0 g g 0 ! g g 0 c t c t

g c e t 0 ! g c e t 0 +i e c g t 1 ! e c g t 0

"i e c e t 1 ! " e c e t 0

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Experiments on Cirac-Zoller Gate • 1995 Monroe et al.: Entangling motional and internal qubit.

!/2 %!/2

Q-circuit = Ramsey Interferometer

• 2003 Schmidt-Kaler et al.: Multiqubit entanglement

CNOT gate

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Early Issues with Ion Trap QC

• Anomalous Heating

- Fluctuating patch potentials. - Poor scaling with trap size. - Want large oscillation freq.

• Dense normal mode spacing

I. H. Deutsch, University of New Mexico Short Course in Quantum Information A more flexible trap architecture

I. H. Deutsch, University of New Mexico Short Course in Quantum Information I. H. Deutsch, University of New Mexico Short Course in Quantum Information Scaling Up: Multiplex Segmented Trap

D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002).

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Qubits in Quantum Dots

Science 309, 2184 (2005).

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Qubits in Josephson Junctions

Science 299, 189 (2003).

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Qubits in Neutral Atoms

$=0 Ramsey Fringes

$=! $=2!

Nature 425, 937 (2003).

I. H. Deutsch, University of New Mexico Short Course in Quantum Information Qubits in Photons

Nature 409, 46 (2001) Nature 426, 264 (2003)

I. H. Deutsch, University of New Mexico Short Course in Quantum Information 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.

I. H. Deutsch, University of New Mexico Short Course in Quantum Information