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Quantum Computing with Quantum Dots

By Ravichandra reddy ph06m005

Plan of talk

- Quantum computer
- Quantum dots
- Computing with dots
- Recent developments

Quantum computer

- A quantum computer is any device for computation

that makes direct use of distinctively quantum

mechanical phenomena , such as superposition and

entanglement , to perform operations on data. - The basic principle
- the quantum properties of particles can be

used to represent and structure data, and that

quantum mechanisms can be devised and built to

perform operations with these data

Bits vs Qubits

- The device computes by manipulating those bits

with the help of logic gates - A qubit can hold a one, a zero, or, crucially, a

superposition of these. - manipulating those qubits with the help of

quantum logic gates - A classical computer has a memory made up of bits

, where each bit holds either a one or a zero

Bits vs. qubits

- the qubits can be in a superposition of all the

classically allowed states. - the register is described by a wavefunction
- the phases of the numbers can constructively and

destructively interfere with one another this is

an important feature for quantum algorithms.

Bits vs Qubits

- For an n qubit quantum register, recording the

state of the register requires 2n complex numbers - (the 3-qubit register requires 23 8 numbers).
- Consequently, the number of classical states

encoded in a quantum register grows exponentially

with the number of qubits - For n300, this is roughly 1090, more states

than there are atoms in the observable universe.

Quantum dot

- A quantum dot is a semiconductor nanostructure

that confines the motion of conduction band

electrons , valence band holes , or excitons

(pairs of conduction band electrons and valence

band holes) in all three spatial directions.

The confinement can be due to

- gtelectrostatic potentials
- (generated by external electrodes, doping,

strain, impurities), - gt the presence of an interface between

different semiconductor materials - gt the presence of the semiconductor surface

(e.g. in the case of a semiconductor

nanocrystal ). - gt a combination of these.

Dimensions

- Small quantum dots, such as colloidal

semiconductor nanocrystals, can be as small as 2

to 10 nanometers, corresponding to 10 to 50 atoms

in diameter and a total of 100 to 100,000 atoms

within the quantum dot volume. - At 10 nanometers in diameter, nearly 3 million

quantum dots could be lined up end to end and fit

within the width of a human thumb.

- quantum wires , which confine the motion of

electrons or holes in two spatial directions and

allow free propagation in the third. 2) quantum

wells, which confine the motion of electrons or

holes in one direction and allow free propagation

in two directions.

compared to atoms

- both have a discrete energy spectrum and bind a

small number of electrons. - In contrast to atoms, the confinement potential

in quantum dots does not necessarily show

spherical symmetry. - In addition, the confined electrons do not move

in free space but in the semiconductor host

crystal. - play an important role for all quantum dot

properties.

energy scales

- the order of 10 ev in atoms, but only 1 milli

e.v in quantum dots. - In contrast to atoms, the energy spectrum of a

quantum dot can be engineered by controlling the

geometrical size, shape, and the strength of the

confinement potential. - it is relatively easy to connect quantum dots by

tunnel barriers to conducting leads

How to find?

- the energy levels can be probed by optical

spectroscopy techniques. - blue shift due to the confinement compared to the

bulk material . - quantum dots of the same material, but with

different sizes, can emit light of different

colors.

coloration

- The larger the dot, the redder
- The smaller the dot, the bluer
- The coloration is directly related to the energy

levels of the quantum dot.

Blue Shift

- the bandgap energy inversely proportional to the

square of the size of the quantum dot. - Larger quantum dots have more energy levels

which are more closely spaced. - This allows the quantum dot to absorb photons

containing less energy, i.e. those closer to the

red end of the spectrum.

Applications

- sharper density of states
- superior transport and optical properties, and

are being researched for use in diode lasers ,

amplifiers, and biological sensors. - use in solid-state quantum computation . By

applying small voltages to the leads, one can

control the flow of electrons through the quantum

dot and thereby make precise measurements of the

spin and other properties

Applications

- Another cutting edge application of quantum dots

is also being researched as potential artificial

fluorophore for intra-operative detection of

tumors using fluorescence spectroscopy . - Quantum dots may have the potential to increase

the efficiency and reduce the cost of todays

typical silicon photovoltaic cells . - 7-fold increase in final output

Quantum computer

- A quantum computer is any device for computation

that makes direct use of distinctively quantum

mechanical phenomena , such as superposition and

entanglement , to perform operations on data.

Quantum superposition

- Quantum superposition is the application of the

superposition principle to quantum mechanics. - The superposition principle is the addition of

the amplitudes of wavefunctions , or state

vectors - . It occurs when an object simultaneously

"possesses" two or more values for an observable

quantity - (e.g. the position or energy of a

particle).

Quantum entanglement

- is a quantum mechanical phenomenon in which the

quantum states of two or more objects have to be

described with reference to each other, even

though the individual objects may be spatially

separated . - leads to correlations between observable physical

properties of the systems.

Quantum entanglement

- For example, it is possible to prepare two

particles in a single quantum state such that

when one is observed to be spin-up, the other one

will always be observed to be spin-down and vice

versa - it is impossible to predict , according to

quantum mechanics, which set of measurements will

be observed. As a result, measurements performed

on one system seem to be instantaneously

influencing other systems entangled with it.

A fundamental problem

- in quantum physics is the issue of the

decoherence of quantum systems and the transition

between quantum and classical behavior.

Quantum decoherence

- quantum decoherence is the mechanism by which

quantum systems interact with their environments

to exhibit probabilistically additive behavior -

a feature of classical physics - and give the

appearance of wavefunction collapse. Decoherence - quantum decoherence is the mechanism by which

quantum systems interact with their environments

to exhibit probabilistically additive behavior -

a feature of classical physics - and give the

appearance of wavefunction collapse. Decoherence

Quantum decoherence

- Decoherence does not provide a mechanism for the

actual wave function collapse rather it provides

a mechanism for the appearance of wavefunction

collapse. The quantum nature of the system is

simply "leaked" into the environment so that a

total superposition of the wavefunction still

exists, but exists beyond the realm of

measurement. - Decoherence represents a major problem for the

practical realization of quantum computers

prime motivations for proposing spin

- that most of what has been probed is the orbital

coherence of electron states, that is, the

preservation of the relative phase of

superpositions of spatial states of the electron

The coherence times seen in these investigations

are almost completely irrelevant to the spin

coherence times which are important in our

quantum computer proposal. There is some relation

between the two if there are strong spin-orbit

effects, but our intention is that conditions and

materials should be chosen such that these

effects are weak.

- Under these circumstances the spin coherence

times (the time over which the phase of a

superposition of spin-up and spin-down states is

well-defined) can be completely different from

the charge coherence times (a few nanoseconds),

and in fact it is known that they can be orders

of magnitude longer

Upscaling

- For the implementation of realistic calculations

on a quantum computer, a large number of qubits

will be necessary (on the order of 1,00,000. - can be operated in parallel
- well achievable with spin-based qubits confined

in quantum dots

Pulsed Switching

- quantum gate operations will be controlled

through an effective Hamiltonian

which is switched via external control fields

the exchange coupling J is local, it is finite

only for neighboring qubits

- the qubits can be moved around in an array of

quantum dots. Thus, a qubit can be transported to

a place where it can be coupled with a desired

second qubit, where single-qubit operations can

be performed, or where it can be measured.

Switching Times

- Single qubit operations can be performed
- A spin can be rotated by a relative angle of

a typical switching time for an angle 180 deg ,

a field 1 tesla , and g eff 1 is 30 pico

sec .

the total time consumed by an algorithm can

be optimized

Error Correction

- One of the main goals in quantum computation is

the realization of a reliable error-correction

scheme , which requires gate operations with an

error rate not larger than one part in 10000 - a larger number of qubits also requires a larger

total number of gate operations to be performed,

in order to implement the error-correction

schemes - perform these operations in parallel

Precision Requirements

- Quantum computation is not only spoiled by

decoherence, but also by a limited precision of

the gates, i.e. by the limited precision of the

Hamiltonian. - the exchange and Zeeman interaction need to be

controlled again in about one part in 10000

Decoherence due to Nuclear Spins

- a serious source of possible qubit errors using

semiconductors such as GaAs is the hyperfine

coupling between electron spin (qubit) and

nuclear spins in the quantum dot - In GaAs semiconductors, both Ga and As possess a

nuclear spin 3/2 , and no Ga/As isotopes are

available with zero nuclear spin.

Two-Qubit Gates--Coupled Quantum Dots

- multi-(qu)bit gate allows calculations through

combination of several (qu)bits. - two-qubit gates are (in combination with

single-qubit operations) sufficient for quantum

computation --they form a universal set - combined action of the Coulomb interaction and

the Pauli exclusion principle

- . Two coupled electrons in absence of a magnetic

field have a spin-singlet ground state, while the

first excited state in the presence of strong

Coulomb repulsion is a spin triplet. Higher

excited states are separated from these two

lowest states by an energy gap, given either by

the Coulomb repulsion or the single-particle

confinement.

a universal quantum gate.

H(t) is Heisenberg spin Hamiltonian , J(t) is

the exchange coupling between the two spins .

U is a swap operator ( time evolution of J(t)

after a pulse )

- it can be used, together with single-qubit

rotations, to assemble any quantum algorithm - combination of swap'' operator and

single-qubit operations , applied in the

sequence gives universal gate XOR

- reduce the study of general quantum computation

to the study of single-spin rotations (see Sec.

) and the exchange mechanism, in particular how

J(t) can be controlled experimentally. The

central idea is that J(t) can be switched by

raising or lowering the tunneling barrier between

the dots.

Laterally Coupled Dots

- consider a system of two coupled quantum dots in

a two-dimensional electron gas (2DEG), containing

one (excess) electron each, as described in Sec.

. The dots are arranged in a plane, at a

sufficiently small distance , such that the

electrons can tunnel between the dots (for a

lowered barrier) and an exchange interaction

between the two spins is produced. We model this

system of coupled dots with the Hamiltonian

- for small quantum dots, say 2a 40 nm , we need

to consider the bare Coulomb interaction - Separated dots ( agtgtbhor magnaton) are thus

modeled as two harmonic wells with frequency .

This is motivated by the experimental evidence

that the low-energy spectrum of single dots is

well described by a parabolic confinement

potential

Vertically Coupled Dots

- Such a setup of the dots has been produced in

multilayer self-assembled quantum dots (SAD) as

well as in etched mesa heterostructures - in 3D the exchange interaction is not only

sensitive to the magnitude of the applied fields,

but also to their direction.

Measuring a Single Spin (Read-Out)

- Spin Measurements through Spontaneous

Magnetization - Spin Measurements via the Charge
- Quantum Dot as Spin Filter and Read-Out/Memory

Device - Optical Measurements

through Spontaneous Magnetization

- One scheme for reading out the spin of an

electron on a quantum dot is implemented by

tunneling of this electron into a supercooled

paramagnetic dot . There the spin induces a

magnetization nucleation from the paramagnetic

metastable phase into a ferromagnetic domain,

whose magnetization direction is along the

measured spin direction and which can be measured

by conventional means.

via the Charge

- While spins have the intrinsic advantage of long

decoherence times, it is very hard to measure a

single spin directly via its magnetic moment. - yielding a potentially 100 reliable measurement

requires a switchable spin-filter'' tunnel

barrier which allows only, say, spin-up but no

spin-down electrons to tunnel.

Optical Measurements

- Measurements of the Faraday rotation originating

from a pair of coupled electrons would allow us

to distinguish between spin singlet and triplet

In the singlet state (, no magnetic moment) there

is no Faraday rotation, whereas in the triplet

state () the polarization of linearly polarized

light is rotated slightly due to the presence of

the magnetic moment.

References

- M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J.

Matyi, T. M. Moore, and A. E. Wetsel, Observation

of discrete electronic states in a

zero-dimensional semiconductor nanostructure,

Phys. Rev. Lett. 60, 535 (1988). - M. A. Reed, Quantum Dots, Scientific American

268, Number 1, 118, 1993.

- Guido Burkard , Hans-Andreas Engel, and Daniel

LossDepartment of Physics and Astronomy,

University of Basel, Klingelbergstrasse 82,

CH-4056 Basel, Switzerland - Published in Fortschritte der Physik 48

(Special Issue on Experimental Proposals for

Quantum Computation), pp. 965-886 (2000). - wikipedia

- Thank u !

- Http//theorie5.physick.unibas.ch/qcomp/qcomp.html