Transcript nanoquant 1716

Quantum Control
Classical Input
Preparation
 in
QUANTUM WORLD
Dynamics
 out
Readout
Classical Outp
QUANTUM INFORMATION INSIDE
Q.C. Paradigms
Paradigm Unitary
Gates
Measurement
Prior Hilbert
Entang. Space
Standard
Circuit
Yes
No
No
Yes
N
0108020
No
Yes
No
Yes
R&B
0010033
No
Yes
Yes
Yes
KLM
Yes
Yes
No
Yes
0006088
Hilbert spaces are fungible
ADJECTIVE:
ETYMOLOGY:
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.
Medieval Latin fungibilis, from Latin fung (vice), to perform (in place of).
Subsystem division
2 qubits; D = 4
Unary system
D=4
Example: Rydberg atom
http://gomez.physics.lsa.umich.edu/~phil/qcomp.html
We don’t live in Hilbert space
A 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 required
to 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 the
physical resources.
Planck’s constant
sets the scale.
Hilbert space and physical resources
Primary resource is
Hilbert-space dimension.
Hilbert-space dimension
costs physical resources.
Many degrees of freedom
Number of degrees
of freedom
Hilbert-space dimension
measured in qubit units.
Identical degrees
of freedom
Scalable resource requirement
Strictly scalable
resource requirement
qudits
Hilbert space and physical resources
Primary resource is
Hilbert-space dimension.
Hilbert-space dimension
costs physical resources.
Many degrees of freedom
x3, p3
1 0 0


101
11 0
111
0


00 1
011
0
1
x 1 , p1


0
1
0
x 2 , p2
0


1

1

0  000 1  001 2  010 3  011 4  100 5  101 6  110 7  111
x, p
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 physically unary quantum computer
5 times the
diameter
of the Sun
Other requirements for a scalable quantum computer
Avoiding 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?
DiVincenzo’s criteria
DiVincenzo, Fortschr. Phys. 48, 771 (2000)
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 suppress
decoherence through error correction and fault-tolerant computation.
5. Measurement: The ability to read out the state of the computer in a
convenient product basis.
Physical resources: classical vs. quantum
Classical bit
A few electrons on a capacitor
A pit on a compact disk
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.
A 0 or 1 on the printed page
A smoke signal rising from a distant mesa
The scale of irreducible resource
Quantum bit
requirements is always set by
An electron spin in a semiconductor Planck’s constant.
A flux quantum in a superconductor
A photon of coupled ions
Energy levels in an atom
  0   1
Why Atomic Qubits?
State Preparation
• Initialization
• Entropy Dump
Laser cooling
State Manipulation
• Potentials/Traps
• Control Fields
• Particle Interactions
Quantum Optics
NMR
State Readout
• Quantum Jumps
• State Tomography
• Process Tomography
Fluorescence
Optical Lattices
Designing Optical Lattices
Tensor Polarizability
P3/2
-3/ 2
-1/2
1
S1/2
1/2
1
3
2
3
1
U(x)  -  ij E i* (x)E j (x)
4

-1/2
2
3
3/2
1
1 (
 ij  -  0 2 d ij  i e ijk s k )
3
1/2
Effective scalar + Zeeman interaction
U(x)  U0 (x) -   Beff (x)
U0 (x) ~ E(x)
2
Beff (x) ~ i(E*  E )
Lin--Lin Lattice
e1
e2
-k
k


U 0 ~ E(x) ~ e1  e2 cos(2kz)  cos cos(2kz)
2




QuickTime™ and a Animation dec ompres sor are needed to s ee this pic ture.
Beff ~ E (x)  E(x) ~ e1  e2 sin( 2kz)  sin  sin( 2kz)e z
*


   /3
   /2
Multiparticle Control
Controlled
Collisions
Dipole-Dipole Interactions
• Resonant dipole-dipole interaction
+
-
d
V dd ~ 3
r
2
(Quasistatic potential)
+
-
G tot  G G dd  2G
2
d
hG~ 3
D
(Dicke Superradiant State)
Figure of Merit
3
Vdd  

~  
G r 
Cooperative level shift
Bare
Coupled
g1e2
g1g2
Eg g
1 2
e1e2 
e1e2
e1e2
e1g2
Dressed
Vdd
D
g1g2

 
g1g2 
2 / 2

 s (D - iG)  sVd
(D - Vdd (r ) / )  i(G  Gdd (r )) / 2
Two Gaussian-Localized Atoms
r12
Three-Level Atoms
“Molecular” Spectrum
Atomic Spectrum
E
1 e
0 e
d2 d1
D
d3 d4
L
11
01 , 10
00
r
Molecular Hyperfine
“Molecular” Spectrum
Atomic Spectrum
0.8 GHz
F=2
F=1
F=2
F=1
5P1/2
1
1-
6.8 GHz
0
0
-

5S1/2
87Rb
Brennen et al.
PRA 65 022313 (2002)
Controlled-Phase Gate Fidelity
Figure of Merit:
E11  E00 - 2E01 DEc


hGij
hGij
Resolvability = Fidelity
Controlled-Phase Gate Fidelity
D/G (103 )
zo /
F
 0.05
D L  104 G
I L 3.2 kW/cm 2
Dz/z0
DC  D L
F  0.99
I C  10I L
Dz / z0  0.3

1 /   0.1( osc / 2  )  144 kHz
Leakage: Spin-Dipolar Interaction
d1  d2 - 3(d1  er )(d2  er )
V
r3
Noncentral force
D(m f 1  m f 2 )  0
azimuthally
symmetric trap


f  2,m f 1 f  2,m f  -1  f  2,m f  0 f  2,m f  0
m f  -1
mf  0
m f 1
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
Separated-Atom Cold-Collision
H  H1  H 2  Vint (r)  H cm  H rel
H rel
prel 1
2
2

  r - Dr  Vint (r)
2 2
4 2
(3)
Vint (r) 
aeff dreg
(r)
m
Short range interaction
 potential, well characterized
by a hard-sphere scattering with an “effective scattering
length”.

Energy Spectrum
aeff  0.5z0  0
5
4
E

Energy
3

2
1
TextEnd
0
-1
-2
0
1
2
Separation
Dz
z0
3
4
5
Shape Resonance
Quic kTime™ and a Animation dec ompres sor are needed to see this pic ture.
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?
http://info.phys.unm.edu/~deutschgroup
I.H. Deutsch, Dept. Of Physics and Astronomy
University of New Mexico
Collaborators:
• Physical Resource Requirements for Scalable Q.C.
Carl Caves (UNM), Robin Blume-Kohout (LANL)
• Quantum Logic via Dipole-Dipole Interactions
Gavin Brennen (UNM/NIST), Poul Jessen (UA),
Carl Williams (NIST)
• Quantum Logic via Ground-State Collisions
René Stock (UNM), Eric Bolda (NIST)