Download Practical realization of Quantum Computation

Practical realization of
Quantum Computation
Superconducting qubits
Electrons on liquid Helium
Cavity QED
Lecture 19
QC implementation proposals
Bulk spin
Resonance (NMR)
Optical
Atoms
Solid state
Linear optics Cavity QED
Trapped ions Optical lattices
Electrons on He
Semiconductors
Nuclear spin Electron spin Orbital state
qubits
qubits
qubits
http://courses.washington.edu/bbbteach/576/
Superconductors
Flux
qubits
Charge
qubits
Superconductivity
Superconductivity is a phenomenon
occurring in certain materials at
extremely low temperatures,
characterized by exactly zero electrical
resistance and the exclusion of the
interior magnetic field (the Meissner
effect).
A magnet levitating above a high-temperature superconductor, cooled with liquid
nitrogen. Persistent electric current flows on the surface of the superconductor,
acting to exclude the magnetic field of the magnet (the Meissner effect). This
current effectively forms an electromagnet that repels the magnet.
1911
Walter Meissner
“Meissner effect”
Heike Kamerlingh Onnes
Superconductivity in He
1933
Martinis (NIST)
phase qubit
Martinis (UCSB)
two-qubit gate (87% fidelity)
1957 1962
Schnirman et al. – theoretical
proposal for JJ qubits
Devoret group (Saclay)
first Cooper Pair Box qubit
Nakamura, Tsai (NEC)
Rabi oscillations in CPB
Lukens, Han (SUNY SB)
Flux qubit
Supercurrent
through a nonsuperconducting
gap
Bardeen, Cooper, Schrieffer
Theory of Superconductivity
Superconducting qubits – a timeline
1997 1998 1999 2000
2002
2006
Superconductivity
The electrical resistivity of a metallic conductor decreases gradually
as the temperature is lowered. However, in ordinary conductors such
as copper and silver, impurities and other defects impose a lower
limit. Even near absolute zero a real sample of copper shows a nonzero resistance.
The resistance of a superconductor, on the other hand, drops
abruptly to zero when the material is cooled below its "critical
temperature", typically 20 kelvin or less. An electrical current flowing
in a loop of superconducting wire can persist indefinitely with no
power source. Like ferromagnetism and atomic spectral lines,
superconductivity is a quantum mechanical phenomenon. It cannot
be understood simply as the idealization of "perfect conductivity" in
classical physics.
Superconductivity
Superconductors are also able to maintain a current with no applied
voltage whatsoever. Experimental evidence points to a current lifetime of
at least 100,000 years, and theoretical estimates for the lifetime of
persistent current exceed the lifetime of the universe.
In a normal conductor, an electrical current may be visualized as a fluid
of electrons moving across a heavy ionic lattice. The electrons are
constantly colliding with the ions in the lattice, and during each collision
some of the energy carried by the current is absorbed by the lattice and
converted into heat (which is essentially the vibrational kinetic energy of
the lattice ions.) As a result, the energy carried by the current is
constantly being dissipated. This is the phenomenon of electrical
resistance.
Superconductivity
The situation is different in a superconductor. In a conventional
superconductor, the electronic fluid cannot be resolved into individual
electrons. Instead, it consists of bound pairs of electrons known as
Cooper pairs. This pairing is caused by an attractive force between
electrons from the exchange of phonons.
Due to quantum mechanics, the energy spectrum of this Cooper pair
fluid possesses an energy gap, meaning there is a minimum amount of
energy ∆E that must be supplied in order to excite the fluid. Therefore, if
∆E is larger than the thermal energy of the lattice (given by kT, where k
is Boltzmann's constant and T is the temperature), the fluid will not be
scattered by the lattice. The Cooper pair fluid is thus a superfluid,
meaning it can flow without energy dissipation.
Superconductivity
Superconductivity- the persistence of the resistantless electric currents.
Certain metals lose their resistance when the temperature is lowered
below a certain critical temperature ( which is different for different metals).
Main point of the theory, known as Bardeen-Cooper-Schrieffer (BCS) theory
is that in normal metals the electrons behave as fermions,
while in superconducting state they form “Cooper pairs” and behave
like bosons.
-
-
-
-
-
-
Singe electrons- the wave function is
antisymmetric under exchange
Cooper pairs - the wave function is
symmetric under exchange
Superconductivity
Normally electrons do not form pairs as they repel each other. However,
inside the material the electrons interact with ions of the crystal lattice.
Very simplify, the electron can interact with the positively charged
background ions and create a local potential disturbance which can
attract another electron.
The binding energy of the two electrons is very small, 1meV, and the
pairs dissociate at higher temperatures.
At low temperatures, the electrons can exists in the bound
states (from Cooper pairs).
From BCS theory we learn that the lowest state of the system is the
one in which Cooper pairs are formed.
Superconductivity
-
-
-
-
-
-
Singe electrons- only one electron can
occupy a particular state
Cooper pairs – the above restriction no longer applies
as electron pairs are bosons and very large number
of pairs can occupy the same state
1. Therefore, the electron pairs do not have to move from an occupied
state to unoccupied one to carry current.
2. The normal state is an excited state which is separated from the
ground state (in which electrons form Cooper pairs) by an energy gap.
Therefore, electrons do not suffer scattering which a source of resistance
as there is an energy gap between their energy and the energies
of the states to which they can scatter.
Flux quantization in superconductors
We consider a superconductor in form of a hollow
cylinder which is placed in an external magnetic
field, which is parallel to the axis of the cylinder.
∫ dr
The magnetic field is expelled from the
superconductor (Meissner effect) and vanishes
within it. Therefore, Cooper pairs move in the
region of B=0, and we can apply the results
which we previously developed.
If the wave function of the Cooper pair in the absence of the field is ψ(0),
then in the presence of the field we have
r
ψ '(r ) = ψ (0) (r )e
(i 2e / h ) ∫ A (r ') dr '
r0
Flux quantization in superconductors
r
ψ '(r ) = ψ (0) (r )e
S
r0
∫ dr
(i 2e / h ) ∫ A (r ') dr '
r0
When we consider a closed path S around the
cylinder which starts at point r0 we get
ψ '(r ) = ψ
(0)
(r )e
(i 2e / h ) ∫ A (r ')dr '
= ψ (0) (r )ei 2e Φ / h
As the electron wave function should not be multivalued
as we go around the cylinder we get the condition
2e Φ
nπ h
= 2nπ → Φ =
,
e
h
n = 0, 1, 2,...
And the flux enclosed by the superconducting cylinder (or ring) is quantized!
This effect has been experimentally verified which confirmed that the current in
superconductors is carried by the pair of the electrons and not the individual electrons.
How this effect can be used?
The main attraction of the Aharonov-Bohm effect is the possibility to use
it in switching devices, i.e. to use the change in magnetic filed to change
the state of the device from 0 to 1.
How much do we have to change the magnetic field to switch
from the constructive to destructive electron interference?
∆Φ =
πh
e
πh
π ×1.05 ×10−34 J ⋅ s
−6
∆B =
≈
≈
5.1
×
10
T
−19
−6
2
eA (1.6 ×10 C )( 20 × 10 m )
for 20µm x 20µm device
This is a very small field! The Earth’s magnetic field is about 40µT.
It is very difficult to practically use.
Josephson junction
superconductors
J = J 0 sin δ
insulator
Josephson junction:
a thin insulator sandwiched
between two superconductors
phase difference δ = θ 2 − θ1
Depends on the tunneling probability of the electron pairs
There is a current flow across the junction in the
absence of an applied voltage!
Superconducting devices
Extremely interesting devices may be designed with a superconducting
loop with two arms being formed by Josephson junctions.
The operation of such devices is based on the fact that the phase difference
around the closed superconducting loop which encloses the magnetic flux Φ
is an integral product of 2e Φ / h .
eΦ
= nπ .
The current will vary with Φ and has maxima at
h
The control of the current through the superconducting loop is the
basis for many important devices. Such loops may be used
in production of low power digital logic devices, detectors, signal
processing devices, and extremely sensitive magnetic field
measurement instruments .
SQUID magnetometer (Superconductind QUantum Interference Device)
Superconducting quantum computing
This promising
implementation of
quantum information
involves
nanofabricated
superconducting
electrodes coupled
through Josephson
junctions. Possible
qubits are charge
qubits, flux qubits,
and hybrid qubits.
Josephson Junction Charge (NEC)
One-qubit device can control the
number of Cooper pairs of
electrons in the box, create
superposition of states.
Superconducting device, operates
at low temperatures (30 mK).
Nakamura et al., Nature, 398(786), 1999
Two-qubit device
Pashkin et al., Nature, 421(823), 2003
JJ Flux (Delft)
The qubit representation is
a quantum of current (flux)
moving either clockwise or
counter-clockwise around
the loop.
Charge qubit
In quantum computing, a charge qubit is a superconducting
qubit whose basis states are charge states (ie. states which
represent the presence or absence of excess Cooper pairs
in the island).
A charge qubit is formed by a tiny superconducting island
(also known as a Cooper-pair box) coupled by a Josephson
junction to a superconducting reservoir (see figure). The
Circuit diagram of a
cooper pair box circuit. state of the qubit is determined by the number of Cooper
pairs which have tunneled across the junction. In contrast
The island (dotted line) with the charge state of an atomic or molecular ion, the
is formed by the
charge states of such an "island" involve a macroscopic
superconducting
number of conduction electrons of the island. The quantum
electrode between the superposition of charge states can be achieved by tuning
gate capacitor and the the gate voltage U that controls the chemical potential of the
island. The charge qubit is typically read-out by
junction capacitance.
electrostatically coupling the island to an extremely sensitive
electrometer such as the radio-frequency single-electron
transistor.
Flux qubits
In quantum computing, flux qubits (also
known as persistent current qubits) are micrometer sized loops of superconducting metal
interrupted by a number of Josephson
junctions. The junction parameters are
engineered during fabrication so that a
persistent current will flow continuously when
an external flux is applied.
The computational basis states of the qubit are defined by the
circulating currents which can flow either clockwise or counterclockwise. These currents screen the applied flux limiting it to multiples
of the flux quanta and give the qubit its name. When the applied flux
through the loop area is close to a half integer number of flux quanta
the two energy levels corresponding to the two directions of circulating
current are brought close together and the loop may be operated as a
qubit.
Flux qubits
Computational operations are performed by pulsing the
qubit with microwave frequency radiation which has an
energy comparable to that of the gap between the
energy of the two basis states. Properly selected
frequencies can put the qubit into a quantum
superposition of the two basis states, subsequent pulses
can manipulate the probability weighting that qubit will
be measured in either of the two basis states, thus
performing a computational operation.
http://qist.lanl.gov/qcomp_map.shtml
“Scalable physical system
with well-characterized qubits”
The system is physical – it is a
microfabricated device with
wires, capacitors and such
The system is in principle
quite scalable. Multiple
copies of a qubit can be
easily fabricated using the
same lithography, etc.
But: the qubits can never be made
perfectly identical (unlike atoms).
Each qubit will have slightly different
energy levels; qubits must be
characterized individually.
http://courses.washington.edu/bbbteach/576/
http://courses.washington.edu/
bbbteach/576/
“ability to initialize qubit state”
Qubits are initialized by cooling to low temperatures (mK)
in a dilution refrigerator. This is how:
Energy splittings between qubit states are of the order of
f = 1 - 10 GHz (which corresponds to T = hf/kB = 50 - 500 mK)
If the system is cooled down to T0 = 10 mK, the ground state
occupancy is, according to Boltzmann distribution:
P|0> = exp(-hf/kBT0) = 0.82 – 0.98
Lower temperature dilution refrigerators mean better qubit
initialization!
“(relative) long coherence times”
Coherence times from a fraction of a nanosecond (charge qubits)
to tens of nanoseconds (flux) to microseconds (“quantronium”).
Correspond to about 10 – 1000 operations before decoherence.
Many sources of noise (it’s solid state!)
“universal set of quantum gates”
Single qubit gates: applying microwaves
(1 – 10 GHz) for a prescribed period of
time.
Two-qubit gates: via capacitive or
inductive coupling of qubits.
Science 313
313, 1432 (2006) –
entanglement of two phase qubits
(Martinis’ group – UCSB)
“qubit-specific measurement”
Measurement depends on the type of qubit.
Charge qubit readout: amplifier with bimodal
response corresponding to the state of the qubit.
Flux and phase qubits readout: built-in DC-SQUID that detects
the change of flux.
Superconducting qubits - pros and cons
• Cleanest of all solid state qubits.
• Fabrication fairly straightforward,
uses standard microfab techniques
• Gate times of the order of ns
(doable!)
• Scaling seems straightforward
• Need dilution refrigerators
(and not just for noise reduction)
•No simple way to couple to
flying qubits (RF photons not good)
• Longer coherence needed, may be
impossible
Superconducting qubits – what can we
expect in near term?
• More research aimed at identifying and quantifying the
major source(s) of decoherence.
• Improved control of the electromagnetic environment –
sources, wires, capacitors, amplifiers.
• Entanglement demonstrations in other types of SC qubits.
• Integration of the qubit manipulation electronics (on the
same chip as the qubits themselves).
http://qist.lanl.gov/qcomp_map.shtml
Electrons on
Liquid Helium
http://wwwhttp://www-drecam.cea.fr/Images/astImg/375_1.gif
Electrons on Helium
Electrons are weakly attracted by the image charge (ε = 1.057
for LHe); the 1-D image potential along z is:
-∑/z , where ∑ = (ε-1)e2/4(ε+1)
They are prevented from penetrating helium surface by a high
(~ 1eV) barrier.
Bound states in this potential in
1-D look like hydrogen:
En = −R/n2 (n = 1, 2, . . .), R = ∑2m/2ħ2
Rydberg energy is about 8K, and
the effective Bohr radius is about
8 nm.
Electrons on Helium - 2
Liquid helium film must be cooled down to mK temperatures in order to
reduce the vapor pressure (which would otherwise wreak havoc with
among the electrons)
It is well known that below about 2.2 K He-4 turns superfluid. At few mK
it is pure He II.
These features are crucial for the QC
proposal with electrons on LHe. The main
source of noise (heating) for the electrons
trapped on the surface is the ripplons.
http://silvera.physics.harvard.edu/bubbles.htm
The original proposal
“Quantum Computing with Electrons Floating on Liquid Helium”
P. M. Platzman, M. I. Dykman, Science 284 pp. 1967 – 1969 (1999).
The qubit is formed by the two lowest energy states of the
trapped electron. Given R = 8K = 170 GHz, the n = 1 and the
n = 2 levels are split by about 125 GHz.
Presence of electric fields from bias electrodes introduces
Stark shift of the levels.
Single qubit operations are performed by applying microwaves
at the Stark-shifted frequency. Expected Rabi frequencies of the
order of hundreds of MHz
Patterned bottom electrodes
Electrons on surface of LHe of thickness d
(typically about 1 micron) will form a 2-D
solid with lattice constant approximately
equal to d. (This is because the Coulomb
energy e2/d
d is of the order 20 K >> kbT at
10 mK).
In order to control the locations of the
electrons, as well as to be able to
individually address each qubits, the
bottom electrode of the capacitor is
patterned. This also provides confinement
in the plane of the LHe film.
Electrons can be physically raised and
lowered by controlling the voltages on the
patterned electrodes.
Two-qubit gates
Two-qubit gates via dipole-dipole interaction (similar to the
liquid state NMR QC).
For a dipole moment (er), the interaction energy between
qubits separated by distance d is (er)2/d3. At 1 micron separation
the interaction energy is estimated to be about 10 MHz.
The frequency of the coupling is qubit state-dependent (because
(er) is state-dependent). This forms the basis of the quantum
logic gates like the CNOT gate.
However, it is strongly distance-dependent. Thus, interactions are
limited to nearest neighbors.
The readout
“In order to read out the wave function at some time tf , when the
computation is completed, we apply a reverse field E+ to the capacitor...”
Qubit readout relies on state-dependent
electron tunneling when a reversed bias
field is applied to the capacitor.
Problems: reading out the whole system
at once; need to detect single electrons
reliably
Conclusions....
• A “neat” and certainly very unique approach
• Builds on ideas from the superconducting qubits, trapped ions,
quantum dots
• The experiment is harder than theory. Some theoretical
predictions unrealistic.
http://www.quantumoptics.ethz.ch
http://www.quantumoptics.ethz.ch//
http://courses.washington.edu/
bbbteach/576/
http://www2.nict.go.jp/
http://www.wmi.badw.de/SFB631/tps/dipoletrap_and_cavity.jpg
http://qist.lanl.gov/qcomp_map.shtml
Cavity Quantum ElectroDynamics
• In cavity QED we want to achieve conditions where single photon interacts
so strongly with an atom that it causes the atom to change its quantum state.
• This requires concentrating the electric field of the photon to a very small
volume and being able to hold on to that photon for an extended period of
time.
• Both requirements are achieved by confining photons into a small, highfinesse resonator.
F = 2√R/(1 – R), where R is mirror reflectivity
power in
circulating power
loss
Microwave resonators
• Microwave photons can be
confined in a cavity made of good
metal. Main source of photon loss
(other than dirt) is electrical
resistance.
• Better yet, use superconductors!
Cavity quality factors (~ the finesse)
reach ~ few × 108 for microwave
photons at several to several tens of
GHz.
• Microwave cavities can be used to
couple to highly-excited atoms in
Rydberg states. There are proposals to
do quantum computation with
Rydberg state atoms and cavities.
S. Haroche, “Normal Superior School”
The optical cavity
• The optical cavity is usually a standard Fabry-Perot optical resonator that
consists of two very good concave mirrors separated by a small distance.
• The length of the cavity is stabilized to
have a standing wave of light resonant
or hear-resonant with the atomic
transition of interest.
• Making a good cavity is part black
magic, part sweat and blood...
G. Rempe - MPQ
• These cavities need to be phenomenally good to get
into a regime where single photons trapped inside
interact strongly with the atoms.
M. Chapman - GATech
The technology: mirrors
• To make g >> κ we need:
• a small-volume cavity to increase g
• a very high-finesse cavity to reduce κ
M. Chapman - GATech
• “clean” cavity to reduce other losses
• Strong-coupling cavities use super-polished mirrors (surface roughness
less order of 1 Å, flatness λ/100) to reduce losses due to scattering at the
surface.
• Mirrors have highly-reflective multi-layer dielectric coatings (reflectivity
at central wavelength better than 0.999995, meaning finesse higher than
500000).
• Mirrors have radius of curvature of 1 – 5 cm, and small diameter. Mirror
spacing is 100 micron down to 30 micron. These features of the cavities
make for stronger confinement of photons for higher g.
Qubits: single atoms or ions
(also, artificial atoms)
• A cavity QED system is usually
combined with and atom or ion trap
D5/2
D3/2
• Two-level system formed by either
the hyperfine splitting of the ground
state (“hyperfine” qubit) or by the
ground state and a metastable
excited state (“optical” qubit)
• The atom can interact with the
laser field (“classical” field) and the
cavity field (“quantum” field)
• Qubit state preparation and
detection techniques are well
established and robust
Qubit preparation and detection
• Initialization of the qubits state is via optical pumping: applying a laser
light that is decoupled from a single quantum state
• Detection by selectively exciting one of the qubit states into a fast
cycling transition and measuring photon rate. May also start by
“shelving” one of the qubit states to a metastable excited state, then
applying resonant laser light. The qubit state that ends up
scattering laser light appears as
“bright”, while the other state
|1,1〉
|2,2〉
P3/2
appears as “dark”.
• Both the preparation and the
detection steps have been
demonstrated to work with over
99% efficiency with trapped ions.
111Cd+
π
P1/2
Cycling transition
(cooling/detection)
σ+
|0,0〉
14.5 GHz
S1/2
|1,-1〉
|1,0〉
|1,1〉
Other qubits: photons
• Cavity QED quantum computing makes use of photons to both mediate
the atomic qubit entanglement and to transfer quantum information over
long distances.
• Photon detection: PBS
(polarization beam splitter) and
single photon counters
Note on polarization. Photon polarization is the quantum mechanical
description of the classical polarized sinusoidal plane electromagnetic wave.
Note on polarization
Photon polarization is the quantum mechanical
description of the classical polarized sinusoidal plane
electromagnetic wave. In electrodynamics, polarization
is the property of electromagnetic waves, such as light,
that describes the direction of their transverse electric
field.
The electric field vector may be arbitrarily divided into
two perpendicular components labelled x and y (with z
indicating the direction of travel). For a simple harmonic
wave, where the amplitude of the electric vector
varies in a sinusoidal manner, the two components
have exactly the same frequency. However, these
components have two other defining characteristics that
can differ. First, the two components may not
have the same amplitude. Second, the two components
may not have the same phase, that is they may not
reach their maxima and minima at the same time.
Wikipedia
Note on polarization
The shape traced out in a fixed plane by the electric vector as such a plane wave
passes over it, is a description of the polarization state. The following figures show
some examples of the evolution of the electric field vector (blue) with time
(the vertical axes), along with its x and y components (red/left and green/right), and
the path traced by the tip of the vector in the plane (purple).
Wikipedia
Notes on polarization
The polarization of a classical sinusoidal plane wave traveling
in the z direction can be characterized by the Jones vector
where the angle θ describes the relation between the
amplitudes of the electric fields in the x and y directions.
Polarization applet
http://webphysics.davidson.edu/physlet_resources/dav_optics/Examples/polarization.html
Combining atom trapping and cavity
Optical lattice confining atoms
inside a cavity (M. Chapman)
~100 µm
Thin ion trap inside a cavity (Monroe/Chapman,
Blatt)
Cavity field used to trap atoms (G. Rempe)
Other cavities: whispering gallery resonators
• Quality factors of 108 and greater
Whispering cavity resonator laser
(http://physics.okstate.edu/shopova/research.html)
J. Kimble (Caltech)
• Simple (sort-of) technology – just
make a nice, smooth glass sphere
~50 micron in diameter...
• Evanescent field extends only a
fraction of the wavelength (i.e. ~100
nm) outside the sphere – need to
place atoms close to the surface.
• “Artificial atoms” such as quantum
dots can be used...
Challenges of cavity QED QC
• Cavity QED quantum computing attempts to combine two
very hard experimental techniques: the high-finesse optical
cavity and the single ion/atom trapping. This is not just
doubly-very-hard, but may well be (very-hard)2
• Assuming “hard” > 1, we have “very hard” >> 1,
and (“very hard”)2 >> “very hard”
• However, the benefits of cavity QED, namely, the
connection of static qubits to flying qubits, are very exciting
and are well worth working hard for.
Strengths
1. Ability to interconvert material and photonic
qubits.
2. Source of deterministic single photons and
entangled photons.
3. Cavity QED systems provide viable platforms for
distributed quantum computing implementations
for both neutral atom and trapped ions.
4. Well understood systems from a theoretical
standpoint. The cavity QED system has been an
important paradigm of quantum optics.
Weakness
1. Ultimate performance of systems is dependent on
advances in mirror coating and polishing technologies.
Current mirror reflectivities, while adequate to achieve the
strong coupling limit, are still ~!100 times lower than the
theoretical limit imposed by Rayleigh scattering in the
coating. Additionally, smaller mirror curvature would provide
for large coherent coupling rates.
2. The role of the atomic motional degree of freedom in the
cavity gate operation and subsequent evolution needs to be
better understood both experimentally and theoretically.
3. Need to combine two already very hard to implement
technologies.
http://qist.lanl.gov/qcomp_map.shtml