Physical realization of Quantum computers (QC) have
been demonstrated using spins on a molecules in a NMR
experiment, polarization of light in a laser or the
energy of atoms in a trap. But the main problem with
these techniques lies either in scaling the number of
qubits to a useful value without losing the purity of
quantum state (decoherence time), or, complexity involved
in the setup to address and manipulate many qubits,
optical implantation for example [1].
In solid state systems, the spin states of either electron
or nuclei, form two-level systems which can be represented
as qubits. Quantum computer realizations rely on interaction
between spins. Also, localised spins are available via
confinement to quantum dots or impurity atoms. Fabrication
techniques for Quantum dots (QDs) are available down
to the single-electron spin but not has been implemented
with impurity atoms [1].
The basic idea was given by Loss and DiVincenzo (QDs),
which were later adapted to impurity spins by Kane,
and now have been extended to optically driven spin
based system by Rossi and Zoller and Sham et al [2].
All three approaches have been briefly described here.
Solid state QC: The Vincenzo criteria
According to DiVincenzo criteria for a system to be
a candidate for implementation of quantum computer [2],
it should:
1. Be a scalable system with well-characterized qubits.
2. Be initializable to a simple fiducial state: Includes
how well the qubits can be initialized, how quickly
they can be reset and how long the initialization takes
place.
3. Have much longer decoherence times: Longer than
gate operation time.
4. Have a universal set of quantum gates: Ability to
perform rotations of single qubits together with two
qubit coupling to perform all universal gate operations.
5. Permit high qubit-specific measurements capability:
Read out the state of a specific qubit with high accuracy.
6. Have the ability to interconvert stationary and
flying qubits: This can allow connection of different
parts of quantum computer and act as a bus.
7. Have the ability to faithfully transmit the flying
qubits between specified location.
I) Realization using nuclear spin
of 31P donors in Si
Kane proposed of building a solid-state quantum computer
by manipulating array of nuclear spin located on the
donor 31phosphorous atoms in silicon [3]. In his proposal
the nuclear spins of the 31phosphorus nuclei constitutes
the qubits; which is manipulated using a combination
of static magnetic, static electric and oscillating
radio-frequency magnetic fields. As shown in figure
single spin operation is performed by changing the voltage
on the metallic gate electrode (‘A’ gate)
located each above the 31phosphorous nuclei. Spin-flips
or electron-mediated interaction between the two nuclei
is carried out by pulsed rf field i.e. by applying voltage
to metallic electrode placed between them (‘J’
gate).
Fig 1: Silicon based quantum
computer (from Ref 4)
Measurements can be done by transferring the nuclear
spin polarization to the electrons and determining the
electron spin state by its effect on the orbital wave
function, which can be done by capacitance measurements
between adjacent gates. Requirement of donor nuclear
spin state I=0 and shallow donor with nuclear spin state
I=1/2 made Si:
31P system as the best candidate for its implementation.
Also it has been found out that for this system at T=1.5K,
the electron spin relaxation time of Si is thousand
of seconds (1 ms) and the nuclear spin relaxation time
for the donor 31P atom is 10 hr [5-7].
II) Electron spins in GaAs QDs:
Gated Qubits
DiVincenzo et al, proposed spin of a excess single
electron confined to a quantum dot as a qubit that can
be manipulated for quantum computations [8].
Coherent interaction between pair of qubits represents
the basic quantum computation. This can be manipulated
by the gating of the tunneling barrier between neighbouring
dots i.e. pair of qubits. If the barrier potential is
high the tunneling between the two dot is forbidden
and no change in qubit state is performed. However if
the potential is lowered down, spin gets subjected coupling
given by
Hs(t) =J(t) S1. S2 (1)
Where J(t) is the exchange coupling constant and Si
is the spin-1/2 operator for the dot i.
In the proposed method spin rotation is achieved by
exciting a pulsed magnetic field onto spin Si by a scanning
probe tip or alternatively fabricating an auxillary
dot (among the two dot pairs) made of insulating ferromagnetic
material. Lowering of potential causes electronic wave-function
overlaps for a fixed time thereby causing the rotation
of spins.
Two scheme for qubit measurement was given: First involves
switchable tunneling into supercooled paramagnetic (PM)
dot. For performing measurements electron tunnels to
PM dot, nucleating from a metastable state to a ferromagnetic
domain whose magnetization can be measured by conventional
means. Other includes spin dependent switchable ‘spin-valve
(SV)’ tunnel barrier. When measurement needs to
be performed the spin valve is switched so that only
the up-spin electron passes into a third QD. The presence
of electron on third dot can be measured externally
and gives indication that spin had been up.
Fig 2: Two coupled quantum
dots 1 and 2, containing one single excess electron
(e) with spin 1/2 (from Ref 4).
Also, for the preparation of the initial state of the
system is, a simple initial state like all spins up
can be created if the system is cooled in a uniform
magnetic field; where acceptable spin polarization of
electrons are achievable at cryogenic temperatures.
Minimum decoherence time for systems are found to be
atleast 1ms for electrons in GaAs [4].
III) Optically Measured QD qubits
Zoller et al have proposed a spin based optical quantum
computation via Pauli blocking in QDs [9]. The device
is based on the spin of electrons confined on to a semiconductor
quantum dots. The qubit of the system is represented
by the excess electron spin in a quantum dot defining
|0ñ and |1ñ. Two qubit gate quantum operations
by the spin were mediated by exciton interactions. Quantum
registers in the proposed scheme consisted of arrays
of GaAs based QDs each containing an excess electron
for conduction.
Quantum operations are performed based on Pauli’s
blocking mechanism in quantum dots. Quantum dots can
be individually addressed and the Coulomb interactions
are obtained by shining a polarized laser pulse on the
quantum dot. Due to Pauli’s exclusion principle
a heavy electron pair is created in the s-shell only
if the excess electron present in the QD has a spin
projection of 1/2. Thus precise spin control of the
switching on and off of further Coulomb interactions
is made possible. Presence of photo generated electron
pair is required only during the gating, after which
latter is annihilated via a second laser pulse. Thus
switching on and off the exciton interactions. The dephasing
times
of ground state exciton phonon is approximately 1ns
limited by excitonic life-time. Figure 3 below shows
the proposed device.
Fig. 3. Quantum dots and energy level scheme. Left:
the excess electron is in state | - 1/2ñ ? |0ñ
and the transition induced by a polarized light is blocked.
Right: the excess
electron is in | + 1/2ñ ? |1ñ
and the exciton can be excited. (from Ref 9)
Conclusion
Although semiconductor QC system offer inherent scalability
apart from offering compatibility with present microelectronics
industry and a well defined two-level system, but the
approach however suffers from background impurity levels
problems, single spin read-write operations and extension
to 2-D arrays via gate techniques is challenging. Further
work is being done and is required to address these
issues in detail.
References:
[1] Nielsen, M. A. & Chuang, I. J. Quantum Computation
and Quantum Information; Cambridge Univ. Press, Cambridge,
2000.
[2] Solid State Approaches to Quantum Information Processing
and Quantum Computing: A Quantum Information Science
and Technology Roadmap April 2004. URL: http://qist.lanl.gov
[6] Wilson, D.K. et al; Phys. Rev. 124, 1068–1083
(1961).
[7] Waugh, J. S. et al; Phys. Rev. B 37, 4337–4339
(1988).
[8] Loss, D. and D.P.DiVincenzo, “Quantum computation
with quantum dots,” PhysicalReview A 57, 120–126
(1998).
[9] Pazy, E., E.Biolatti, T.Calarco, I.D'Amico, P.Zanardi
F.Rossi, and P.Zoller “Spin-based optical quantum
gates via Pauli blocking in semiconductor quantum dots,”
(19-Sep-2001) preprint cond-mat/0109337.
Contibutor: "Rajan Sharma
is doing a PhD in Nanobioelectronics at School of Electrical
and Electronics Engineering. He completed MS in Nanoelectronics
with distinction from University of Leeds. He graduated
from NIT Jalandhar, India with a first class degree
and distinction in Instrumentation and Control Engineering.
Currently he is working on his PhD project of nanowire
self assembly of nanowires with biomolecules at Nanobioelectronics
lab."
