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where and v and v are the velocities of a superluminal particle before and after the
measurement, and
Δ is a time interval.
v =
c
β
=
v /
c
When we let
and
, Eq.(10) can be rewritten as
h
Δ
E
Δ
t
(11)
β
( −
β
1
from Eq.(9), which gives the uncertainty relation between energy and time for superluminal
particles.
E NERGY C OST FOR Q UANTUM C OMPUTATION U TILIZING
T UNNELING P HOTONS
Hereafter we assume that tunneling photons which travel in an evanescent mode can
move with a superluminal group speed as confirmed by some experimenters.
We consider the computer system which consists of quantum gates utilizing quantum
tunneling photons to perform logical operations. Benioff showed that the computation speed
was close to the limit by the time-energy uncertainty principle [Benioff,1982].
Margolus and Levitin extended this result to a system with an averaged energy
< E
>
,
<
E
>=
π
/(
2
Δ
t
)
which takes time at least
to perform logical operations [Margolus and
Levitin, 1998; Lloyd, 2000].
Summing over all logical gates of operations, the total number of logic operations per
second is no more than
π
Δ
t
N
(12)
2
<
E
>
where Δ is a operational time of an elementary logical operation and N is the number of
consisting gates of the computer. From Eq.(12), energy spread for the quantum tunneling
photon (abbreviated QTP hereafter) gate becomes
ββ times the energy spread for the
logical gate using particles moving at sub-luminal speed including photons, then the total
number of logic operations per second for QTP gates can be given by
1
/
(
1
π
N
Δ
t
(13)
β
(
β
1
2
<
E
>
*
where
< E is an averaged energy for QTP gates.
As an uncertainty in the momentum of tunneling photons moving at the superluminal
speed can be given by
>
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