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Optimal Gate Control of
Quantum-dot Cellular Automata
Yousof Mardoukhi, Esa Räsänen
[email protected], [email protected]
Introduction
System and Methodology
Quantum dots positioned on a certain lattice
structure, usually square lattices, have a physical realization of cellular automata. This system is called Quantum-dot Cellular Automata
(QCA).
QCAs can be used for next generation of computer circutes as their time switching is fast and
their energy consumption is considerabily low.
The aim of this project[1] was to achieve controlability over QCAs by applying a local voltage
gate and manipulate the charge distribution of
a single electron QCA as a very first step for
further complicated studies.
System The Hamiltonian of the system is defined as follow,
p2
H(x, y, t) =
+ Vc (x, y) + U (x, y, t)
2
(2)
where Vc (x, y) is the model confining potential for the QDs. It is given by a square lattice of
cavities (M × N ) having a shape of a Fermi function given by,
(
)
1
h√
i
:
1
≤
i
≤
N,
1
≤
j
≤
M
− max
(x−i a)2 +(y−j a)2 −R /ξ
+1
e
where ξ = 0.25 is the softness of the potential and a = 6 a.u. is the lattice constant and R = 2
a.u. is the radius of the quantum dot.
Voltage Gate U (x, y, t) in the Hamiltonian is the pontential of the voltage gate which can be splited
up into spatial and time varying parts
Control Scheme
The optimisation is done by optimal control thery (OCT). the time varying part of the field
is optimised iteratively by solving Schrödinger
equation. By introducing the Lagrange multiplier |χ(t)i the control equations are derived.
i∂t |Ψ(t)i = H(r, t)|Ψ(t)i, |Ψ(0)i = |ΦI i (1a)
i∂t |χ(t)i = H(r, t)|χ(t)i
(1b)
|χ(T )i = Θ(|r − r0 |)|rihr|Ψ(T )i
1
f (t) = − hχ(t)|g(r)|Ψ(t)i.
α
(1c)
(1d)
The target operatior is a heaviside distribution
function wich implies the localisation of the particle within a certain region of space. α is the
fluence factor which is kept constant during the
opitmisation.
Conclusion
We have elaborated a scheme within quantum
optimal control theory to manipulate electric
charge in quantum-dot lattices that could be applied as quantum.
We have demonstrated that high-delity local
control of single-electron charge in quantum dot
lattices is possible up to N' 5 for both (1×N)
and (2×N) lattices.
Our study sheds light for the future controllability of a quantum-dot cellular automata in the
case of elementary majority gates like AND and
OR.
We were also successful in identification of two
bit digits 0 and 1 thanks to the symmetry properties of the lattice sturcture.
U (x, y, t) = g(x, y) f (t),
where f (t) = A sin(ωt) and g(x, y) is modeled as a Gaussian pulse shape:
β
−(x−x0 )2 −(y−y0 )2 ]/(2σ 2 )
[
√
e
(σ 2π)2
Results
1×N: We start from the delocalised ground state. The voltage gate acts on the very left
most dot and the target is aimed to localize the charge density on the very right most QD.
The optimisation is done on the time-depentent part
f (t), while keeping the spatial part g(x, y) and the
fluence of the initial field constant.The yield at the
U
T
end of pulse duration is found to be 91.4%. The
initial pulse without optimisation has a yield of
only 10%. Even after applying a frequency filter at
ωth = 5 a.u. the yield becomes 84.7%. The threshold
frequency as an instance for GaAs is 8.6THz which
can be expected to be reached [2].
2×N: This case study is more realistic since the
lattice size of 2×2 is the premitive cell of QCA. As
there are two symmetry axes present, two voltage
gates are considered. In the special case of 2×2 since
the cell has C and σ symmetries, it is controlable
even with one voltage gate for realisation of digit 0
and 1.
For a lattice size of 2×5 with the same parameters for
the voltage gates as previous case of 1×6, the initial
pulse yield is only 2.5% which after the optimisation
the yield is increased to 94.2%. By imposing a fre- Figure 1: (a) GS charge density distribution.
The
target
dot
is
marked
by
T.
(b)
Final
state
quency filter ωth = 5 a.u. the yield is reduced to
charge distribution after optimisation. (c) Pulse
87.6%.
References
U
T
[2] Wirth et al. Synthesized light transients. Science,
334(6053):195–200, 2011.
U
T
U
T
Acknowledgements
The work was supported by the European Communitys FP7 through the CRONOS project, grant agreement
no. 280879, the Academy of Finland, and COST Action CM1204 XLIC. CSC Scientic Computing Ltd. is
acknowledged for computational resources.
(4)
where β = 16.04 a.u. and σ = 1.6 a.u. and the tuple (x0 , y0 ) is the position of the center of
the voltage gate.
shapes and (d) corresponding Fourier spectra
[1] Y. Mardoukhi E. Räsänen. Towards a quantum dot
cellular automata with optimal gate control. submitted, 2014.
(3)
2×2: This cell is premitive cell for the purpose of
computation. The diagonal charge distribution corresponds
to bit 0 and anti-diagonal one to bit 1. The essence of
this representation of bits lies within the symmetry of the
configuration. In Fig. 2(b) and (c) since the target is
not distinguishable the charge will be distributed equally
between the two QDs positioned on anti-diagonal lattice
points. The yield of transition from bit 0 to 1 and vice versa
is > 98%. The field parameters is the same as above.
Figure 2: Middle row: Field-driven (U) localization of the
electron charge from the ground state (b1) to the diagonal
configuration defined as bit 1 (b2). Lower row: Transformation
of bit 0, corresponding to a diagonal configuration (c1) to an
off-diagonal configuration defined as bit 1 (c2).