Quantum Control Classical Input Classical Output QUANTUM WORLD QUANTUM INFORMATION INSIDE...
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Transcript of Quantum Control Classical Input Classical Output QUANTUM WORLD QUANTUM INFORMATION INSIDE...
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
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
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
QuickTime™ and aAnimation decompressorare needed to see this picture.
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
€
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
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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
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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
QuickTime™ and aAnimation decompressorare needed to see this picture.
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