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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

>