practical realization of quantum computationmsafrono/650/lecture19old.pdf · optical atoms solid...

Practical realization of Practical realization of

Quantum ComputationQuantum Computation

Superconducting Superconducting qubitsqubits

Electrons on liquid HeliumElectrons on liquid Helium

Cavity QEDCavity QED

Lecture 19Lecture 19

QC implementation proposals

Bulk spin Resonance (NMR)

Optical Atoms Solid state

Linear optics Cavity QED

Trapped ions Optical lattices

Electrons on He Semiconductors Superconductors

Nuclear spin qubits

Electron spinqubits

Orbital statequbits

Flux qubits

Charge qubits

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SuperconductivitySuperconductivity

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 Meissnereffect).

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.

Superconducting qubits Superconducting qubits –– a timelinea timeline

1911

Heike

Kam

erlin

ghOnn

esSu

percon

duct

ivity

in H

e

1933

Walte

r Meissne

r“M

eissne

reffe

ct”

1957

Schn

irman

et al. –

theo

retic

alpr

opos

al fo

r JJ qu

bits

1962Su

percur

rent

thro

ugh

a no

n-su

percon

ductin

gga

p

1997

Bard

een,

Coo

per,

Schr

ieffe

rTh

eory

of S

uper

cond

uctiv

ity

1998

Dev

oret

grou

p (S

aclay)

first C

oope

r Pair B

ox q

ubit

2000

Luke

ns, H

an (SU

NY

SB)

Flux

qub

it

2002

Martin

is (N

IST)

phase qu

bit

1999

Nak

amur

a, T

sai (

NEC

)Ra

bi o

scillat

ions

in C

PB

2006

Martin

is (U

CSB)

two-

qubi

t gat

e (8

7% fi

delit

y)


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 non-zero resistance.

The resistance of a superconductor, on the other hand, drops abruptly to zero when the material is cooled below its "criticaltemperature", 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.


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.


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.


SuperconductivitySuperconductivity-- the persistence of the the persistence of the resistantlessresistantless electric currents.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) theoryis 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


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 canattract another electron.

The binding energy of the two electrons is very small, 1meV, and thepairs 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.


- - 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 occupiedstate 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 resistanceas there is an energy gap between their energy and the energies of the states to which they can scatter.

Flux quantization in superconductorsFlux 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.

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.∫ dr�

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

(0)

r

( / ) ( ') '

r

2

0'( ) ( )

ei

eψ ψ=

∫ A r dr

r r

h

Flux quantization in superconductorsFlux quantization in superconductors

∫ dr�

(0)

r

( / ) ( ') '

r

2

0'( ) ( )

ei

eψ ψ=

∫ A r dr

r r

h

When we consider a closed path S around the cylinder which starts at point r0 we get

S

r0

(0) (0)( / ) ( ') ' / 2 2 '( ) ( ) ( )i

eie

e eψ ψ ψ Φ= =∫ A r dr

r r rh

h�

As the electron wave function should not be multivaluedas we go around the cylinder we get the condition

2 2 = , 0, 1, 2,...

e nn n

e

ππ

Φ= → Φ =

h

h

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?How this effect can be used?

The main attraction of the Aharonov-Bohm effect is the possibility to useit 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 switchfrom the constructive to destructive electron interference?

( ) ( )

346

19 6 2

=

1.05 10 J s= 5.1 10

1.6 10 20 10

e

B TeA C m

π

π π −−

− −

∆Φ

× × ⋅∆ ≈ ≈ ×

× ×

h

h

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 junctionJosephson junction

Josephson junction: Josephson junction:

a thin insulator sandwiched a thin insulator sandwiched

between two superconductorsbetween two superconductors

insulator

superconductors

0 sinJ J δ=

phase difference 2 1δ θ θ= −

There is a current flow across the junction in the absence of an applied voltage!

Depends on the tunneling probability of the electron pairs

Superconducting devicesSuperconducting devices

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 .

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 Φis an integral product of . 2 / e Φ h

The current will vary with Φ and has maxima at . e

nπΦ

=h

SQUID magnetometer (Superconductind QUantum Interference Device)

This promising This promising

implementation of implementation of

quantum information quantum information

involves involves

nanofabricated nanofabricated

superconducting superconducting

electrodes coupled electrodes coupled

through Josephson through Josephson

junctions. Possible junctions. Possible

qubits are charge qubits are charge

qubits, flux qubits, qubits, flux qubits,

and hybrid qubits.and hybrid qubits.

Superconducting quantum computing

Josephson Junction Charge (NEC)Josephson Junction Charge (NEC)

Two-qubit device

Pashkin et al., Nature, 421(823), 2003

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

JJ Flux (Delft)JJ Flux (Delft)

The qubit representation is

a quantum of current (flux)

moving either clockwise or

counter-clockwise around

the loop.

Charge qubitCharge qubit

Circuit diagram of a

cooper pair box circuit.

The island (dotted line)

is formed by the

superconducting

electrode between the

gate capacitor and the

junction capacitance.

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

state of the qubit is determined by the number of Cooper

pairs which have tunneled across the junction. In contrast

with the charge state of an atomic or molecular ion, the

charge states of such an "island" involve a macroscopic

number of conduction electrons of the island. The quantum

superposition of charge states can be achieved by tuning

the gate voltage U that controls the chemical potential of the

island. The charge qubit is typically read-out by

electrostatically coupling the island to an extremely sensitive

electrometer such as the radio-frequency single-electron

transistor.

Flux qubitsFlux qubits

In quantum computing, flux qubits (also known as persistent current qubits) are micro-meter 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 counter-clockwise. 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 qubitsFlux 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://http://qist.lanl.gov/qcomp_map.shtmlqist.lanl.gov/qcomp_map.shtml

““Scalable physical system Scalable physical system with wellwith well--characterized qubitscharacterized qubits””

The system is physical – it is amicrofabricated device withwires, capacitors and such

The system is in principlequite scalable. Multiplecopies of a qubit can beeasily fabricated using thesame lithography, etc.

But: the qubits can never be madeperfectly identical (unlike atoms). Each qubit will have slightly differentenergy levels; qubits must be characterized individually.

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““ability to initialize qubit stateability 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 off = 1 - 10 GHz (which corresponds to T = hf/kB = 50 - 500 mK)

If the system is cooled down to T0 = 10 mK, the ground stateoccupancy is, according to Boltzmann distribution:

P|0> = exp(-hf/kBT0) = 0.82 – 0.98

Lower temperature dilution refrigerators mean better qubitinitialization!

““(relative) long coherence times(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 gatesuniversal 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 313313313313, 1432 (2006) –entanglement of two phase qubits (Martinis’ group – UCSB)

““qubitqubit--specific measurementspecific 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 Superconducting qubits -- pros and conspros 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 beimpossible

Superconducting qubits Superconducting qubits –– what can wewhat can weexpect in near term?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 thesame chip as the qubits themselves).

Electrons onElectrons onLiquid HeliumLiquid Helium

http://wwwhttp://www--drecam.cea.fr/Images/astImg/375_1.gifdrecam.cea.fr/Images/astImg/375_1.gif

Electrons are weakly attracted by the image charge (ε = 1.057for 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 in1-D look like hydrogen:En = −R/n2 (n = 1, 2, . . .), R = ∑2m/2ħ2

Rydberg energy is about 8K, andthe effective Bohr radius is about8 nm.

Electrons on HeliumElectrons on Helium

Electrons on Helium Electrons on Helium -- 22Liquid 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 mKit 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

“Quantum Computing with Electrons Floating on Liquid Helium”P. M. Platzman, M. I. Dykman, Science 284 284 284 284 pp. 1967 – 1969 (1999).

The original proposalThe original proposal

The qubit is formed by the two lowest energy states of thetrapped 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 introducesStark shift of the levels.

Single qubit operations are performed by applying microwavesat the Stark-shifted frequency. Expected Rabi frequencies of theorder of hundreds of MHz

Patterned bottom electrodesPatterned bottom electrodesElectrons on surface of LHe of thickness dddd(typically about 1 micron) will form a 2-D solid with lattice constant approximately equal to dddd. (This is because the Coulomb energy e2/dddd 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.

TwoTwo--qubit gatesqubit gatesTwo-qubit gates via dipole-dipole interaction (similar to the liquid state NMR QC).

For a dipole moment (er), the interaction energy betweenqubits separated by distance d is (er)2/d3. At 1 micron separationthe 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 quantumlogic gates like the CNOT gate.

However, it is strongly distance-dependent. Thus, interactions arelimited to nearest neighbors.

The readoutThe 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.wmi.badw.de/SFB631/tps/dipoletrap_and_cavity.jpghttp://www.wmi.badw.de/SFB631/tps/dipoletrap_and_cavity.jpg

http://www2.nict.go.jp/http://www2.nict.go.jp/

http://http://www.quantumoptics.ethz.chwww.quantumoptics.ethz.ch//

http://courses.washington.edu/bbbteach/576/

Cavity Quantum Cavity Quantum ElectroDynamicsElectroDynamics• 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, high-finesse resonator.

F = 2√R/(1 – R), where R is mirror reflectivity

power incirculating power

loss

Microwave resonatorsMicrowave resonators

S. Haroche, “Normal Superior School”

• 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.

The optical cavityThe 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.

G. G. RempeRempe -- MPQMPQ

• 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...

M. Chapman - GATech

• These cavities need to be phenomenally good to get into a regime where single photons trapped inside interact strongly with the atoms.

The technology: mirrorsThe technology: mirrors

• To make g >> κ we need:

• a small-volume cavity to increase g

• a very high-finesse cavity to reduce κ

• “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.

M. Chapman - GATech

Qubits: single atoms or ionsQubits: single atoms or ions(also, artificial atoms)(also, artificial atoms)

• A cavity QED system is usually

combined with and atom or ion trap

• 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

D5/2

D3/2

Qubit preparation and detectionQubit preparation and detection

S1/214.5 GHz

P3/2

P1/2

Cycling transition

(cooling/detection)

σ+

|1,1⟩

π

111Cd+

|1,-1⟩ |1,0⟩

|0,0⟩

|2,2⟩|1,1⟩

• 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 upscattering laser light appears as“bright”, while the other stateappears as “dark”.

• Both the preparation and thedetection steps have beendemonstrated to work with over99% efficiency with trapped ions.

Other qubits: photonsOther 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 polarizationPhoton polarizationPhoton polarizationPhoton polarization is the quantum mechanicaldescription of the classical polarized sinusoidal plane electromagnetic wave.

Note on polarizationNote on polarization

Photon polarization is the quantum mechanical

description of the classical polarized sinusoidal plane

electromagnetic wave. In electrodynamics, polarizationis 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 zindicating 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. WikipediaWikipediaWikipediaWikipedia

Note on polarizationNote on polarizationThe 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).

WikipediaWikipediaWikipediaWikipedia

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

~100 µm

Thin ion trap inside a cavity (Monroe/Chapman, Blatt)

Optical lattice confining atoms inside a cavity (M. Chapman)

Cavity field used to trap atoms (G. Rempe)

Other cavities: whispering gallery resonatorsOther cavities: whispering gallery resonators

Whispering cavity resonator laser(http://physics.okstate.edu/shopova/research.html)

• Quality factors of 108 and greater

• 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...

J. Kimble (Caltech)

Challenges of cavity QED QCChallenges 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.

Weakness1. 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.

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