Quantum Control

Classical Input

Classical Output

QUANTUM WORLD

ψ in

ψ out

QUANTUM INFORMATION INSIDE

Preparation

Readout

Dynamics

Q.C. Paradigms

Paradigm Unitary

Gates

Measurement Prior

Entang.

Standard

CircuitYes No No

N

0108020No Yes No

R&B0010033

No Yes Yes

KLM

0006088Yes Yes No

Hilbert Space

Yes

Yes

Yes

Yes

Hilbert spaces are fungibleADJECTIVE: 1. Law. Returnable or negotiable in kind or by substitution, as a quantity of

grain for an equal amount of the same kind of grain. 2. Interchangeable.ETYMOLOGY: Medieval Latin fungibilis, from Latin fung (vice), to perform (in place of).

Unary systemD = 4

Subsystem division2 qubits; D = 4

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

We don’t live in Hilbert spaceA Hilbert space is endowed with structure by the physical system described by it, not vice versa.

The structure comes from preferred observables associated with spacetime symmetries that anchor Hilbert space to the external world.

Hilbert-space dimension is determined by physics. The dimension available for a quantum computation is a physical quantity that costs physical resources.

What physical resources are requiredto achieve a Hilbert-space dimension

sufficient to carry out a given calculation?

quant-ph/0204157

Hilbert space and physical resources

Hilbert-space dimension is a physical quantity that costs physical resources.

Single degree of freedom

Action quantifies thephysical resources.

Planck’s constant sets the scale.

Hilbert space and physical resourcesPrimary resource is

Hilbert-space dimension.Hilbert-space dimensioncosts physical resources.

Many degrees of freedom

Hilbert-space dimensionmeasured in qubit units.

Identical degreesof freedom

Number of degreesof freedom

quditsStrictly scalable resource requirement

Scalable resource requirement

Hilbert space and physical resourcesPrimary resource is

Hilbert-space dimension.Hilbert-space dimensioncosts physical resources.

Many degrees of freedom

x3, p3

x2, p2

x1, p1

x, p

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0

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0

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0

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1

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1

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1

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0 = 000

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1 = 001

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2 = 010

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3 = 011

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4 = 100

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5 = 101

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6 = 110

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7 = 111

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0 0 1

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0 1 1€

1 0 1

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1 1 1€

1 0 0

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1 1 0

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0

1

0

Quantum computing in a single atom

Characteristic scales are set by “atomic units”

Length Action EnergyMomentum

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 Action EnergyMomentum

Bohr

5 times the diameter of the Sun

Poor scaling in this physically unary quantum computer

Other requirements for a scalable quantum computerAvoiding an exponential demand for physical resources requires a

quantum computer to have a scalable tensor-product structure. This is a necessary, but not sufficient requirement for a scalable quantum computer. Are there other requirements?

1. Scalability: A scalable physical system with well characterized parts, usually qubits.2. Initialization: The ability to initialize the system in a simple fiducial state.3. Control: The ability to control the state of the computer using sequences of elementary universal gates.

4. Stability: Long decoherence times, together with the ability to suppressdecoherence through error correction and fault-tolerant computation.

5. Measurement: The ability to read out the state of the computer in a convenient product basis.

DiVincenzo’s criteria DiVincenzo, Fortschr. Phys. 48, 771 (2000)

Physical resources: classical vs. quantum

A few electrons on a capacitor

A pit on a compact disk

A 0 or 1 on the printed page

A smoke signal rising from a distant mesa

Classical bit

Quantum bit

A classical bit involves many degrees of freedom. Our scaling analysis applies, but with a basic phase-space scale of arbitrarily small. Limit set by noise, not fundamental physics.

The scale of irreducible resource requirements is always set by Planck’s constant.

10 βαψ +=

An electron spin in a semiconductor

A flux quantum in a superconductor

A photon of coupled ions

Energy levels in an atom

Why Atomic Qubits?

State Preparation

• Initialization• Entropy Dump

State Manipulation• Potentials/Traps • Control Fields• Particle Interactions

State Readout• Quantum Jumps• State Tomography• Process Tomography

Fluorescence

Laser cooling Quantum OpticsNMR

Optical Lattices

h ′ Γ =

43

k3 d 2

Scattering

Designing Optical Lattices

αij 13α0 2ij i ijkk

23

13

123

1

3/21/21/23/ 2

1/21/2S1/2

P3/2

Tensor Polarizability

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U(x) = −1

4α ij E i

*(x)E j (x)

Beff(x)~i E* ×E( )U0(x)~E(x)2U(x)=U0(x)−μ ⋅Beff(x)

Effective scalar + Zeeman interaction

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θ

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k

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−k

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r 1

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r 2

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U0 ~ E(x)2

~r ε 1 ⋅

r ε 2 cos(2kz) = cosθcos(2kz)

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Beff ~ E*(x) × E(x) ~r ε 1 ×

r ε 2 sin(2kz) = sinθ sin(2kz)ez

Lin-θ-Lin Lattice

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θ =π /2

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θ =π /3

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Multiparticle Control

θyxz

Controlled Collisions

Dipole-Dipole Interactions

• Resonant dipole-dipole interaction

+-

+-

tot dd 2

(Quasistatic potential)(Dicke Superradiant State)

Vdd ~d

2

r3

h ~

d2

D3

Figure of Merit

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κ ≡Vdd

h ′ Γ ~

Dr

⎛ ⎝ ⎜

⎞ ⎠ ⎟3

Cooperative level shift

Heff =HA1 + HA2 +Vdd(r) + HAL −i

h2ΓA1 +ΓA2 +Γdd(r)( )

e1e2

g1g2

e1g2 g1e2

Bare

ψ

ψ

e1e2

g1g2

Vdd

Coupled

ψ

g1g2′

e1e2′

Dressed

Eg1g2

=hΩ2 / 2

Δ−Vdd r( ) / h( )+i Γ +Γdd r( )( ) / 2≈sh(Δ−iΓ)+sVdd

r12

Two Gaussian-Localized Atoms

ΕS+PΔΣRcΣg

+

r12r = 6x0

Σu

−ΠgΠu

r2 ψ rel (r)2

Spinless Molecular Potentials

Require r −Rc >>x0

Three-Level Atoms

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0α

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0β

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1β

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1α

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ω01

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e α

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e β

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ωc

Atomic Spectrum

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00

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01 , 10

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11

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ωL

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Δ

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δ1

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δ2

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δ3

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δ4

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0 +e

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1+ e

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r

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E

“Molecular” Spectrum

Molecular Hyperfine

Atomic Spectrum

0 +

1+ 1−

0 −

5P1/2

5S1/2

F=2

F=1

F=2F=1

87Rb

0.8 GHz

6.8 GHz

“Molecular” Spectrum

Brennen et al.PRA 65 022313 (2002)

Pij =1−e−Γijtint ≈Γijtint =

π hΓij

E11 + E00 −2E01

=πκ

Error Probability

Resolvability = Fidelity

κ

E11 E00 2E01

h ij

Ec

h ij

Figure of Merit:

Controlled-Phase Gate Fidelity

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z/z0

€

Δ/Γ ×(103 )F

Controlled-Phase Gate Fidelity

zo / D=0.05

ΔL =104 Γ

1/τ ≅0.1(ωosc / 2π)=144kHz€

IL=3.2 kW/cm2

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IC = 10I L

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C = ΔL

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z / z0 = 0.3

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F ≈0.99

Leakage: Spin-Dipolar Interaction

€

V =

r d 1 ⋅

r d 2 − 3(

r d 1 ⋅

r e r)(

r d 2 ⋅

r e r )

r3

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(m f 1 + m f 2) = 0

azimuthally symmetric trapNoncentral force

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f = 2,m f =1 f = 2,m f = −1 ⇒ f = 2,m f = 0 f = 2,m f = 0

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m f =1

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m f = −1

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m f = 0

Suppressing Leakage Through Trap

Energy and momentum conservation suppress spin flip for localized and separated atoms.

Dimer Control• Lattice probes dimer dynamics

• Localization fixes internuclear coordinate

2. Ground-bound

F,F+1( )

F,F( )Bext

1. - Excited groundFeshbach

0g−

r2π Rabi flop

++

11 −11ψ auxmolecule

Separated-Atom Cold-Collision

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H = H1 + H2 + Vint (r) = Hcm + Hrel

Hrel =prel

2μ+

1

2μω2 r − Δr

2+ Vint (r)

Vint (r) =4πh2

maeff δreg

(3) (r)

Short range interaction potential, well characterized by a hard-sphere scattering with an “effective scattering length”.

0 1 2 3 4 5-2

-1

0

1

2

3

4

5

Separation

EnergyTextEnd

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aeff = 0.5z0 > 0

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z

z0

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E

hω

Energy Spectrum

Shape Resonance

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Molecular bound state, near dissociation, plays the role of an auxiliary level for controlled phase-shift.

Dreams for the Future

• Qudit logic: Improved fault-tolerant thresholds?

• Topological lattice - Planar codes?

Carl Caves (UNM), Robin Blume-Kohout (LANL)

http://info.phys.unm.edu/~deutschgroup

Gavin Brennen (UNM/NIST), Poul Jessen (UA),Carl Williams (NIST)

I.H. Deutsch, Dept. Of Physics and AstronomyUniversity of New Mexico

Collaborators:• Physical Resource Requirements for Scalable Q.C.

• Quantum Logic via Dipole-Dipole Interactions

René Stock (UNM), Eric Bolda (NIST)

• Quantum Logic via Ground-State Collisions