Quantum Computing with Quantum Dots

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 Quantum Computing with Quantum Dots
By Ravichandra reddy ph06m005
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Plan of talk

• Quantum computer
• Quantum dots
• Computing with dots
• Recent developments

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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

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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

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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.

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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.

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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.

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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.

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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.

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• 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.

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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.

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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

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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.

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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.

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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.

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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

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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

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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.

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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).

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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.

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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.

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A fundamental problem

• in quantum physics is the issue of the
  decoherence of quantum systems and the transition
  between quantum and classical behavior.

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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

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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

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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.

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• 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

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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

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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
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• 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.

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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
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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

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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

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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.

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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

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• . 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.

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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 )
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• 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 

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• 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.

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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

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• 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 

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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.

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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

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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.

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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.

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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.

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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.

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• 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

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• Thank u !

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