Table Of Contenti
QUANTUM
COMPUTING
Jozef
Gruska
quantum measurement has the effect of ‘‘magnifying’’
one of the outcomes of quantum superposition
probabilistic, sequential
Only at this point do indeterminacy and probabilities
E.
T.
QUANTUM WORLD
CLASSICAL WORLD
Quantum computation is
deterministic
highly (exponentially) parallel
unitary
described by Schrodinger equation
using entanglement as a computational
resource
quantum
computation
(evolution)
Classical computation is
working with real probabilities
of computation are irreversibly lost
M
E
A
S
U
R
E
M
E
N
T
come in
quantum events from quantum to classical level
..
working with complex
is randomly picked up - all other results
amplitudes
ii
Chapter
�
FUND
AMENT
ALS
INTR
ODUCTION
The
p
o
w
er
of
quan
tum
computing
is
based
on
sev
eral
phenomena
and
la
ws
of
the
quan
tum
w
orld
that
are
fundamen
tally
di�eren
t
from
those
one
encoun
ters
in
classical
computing�
complex
probabilit
y
amplitudes�
quan
tum
in
terference�
quan
tum
parallelism�
quan
tum
en�
tanglemen
t
and
the
unitarit
y
of
quan
tum
ev
olution�
In
order
to
understand
these
features�
and
to
mak
e
a
use
of
them
for
the
design
of
quan
tum
algorithms�
net
w
orks
and
pro
cessors�
one
has
to
understand
sev
eral
basic
principles
whic
h
quan
tum
mec
hanics
is
based
on�
as
w
ell
as
the
basics
of
Hilb
ert
space
formalism
that
represen
ts
the
mathematical
framew
ork
used
in
quan
tum
mec
hanics�
The
c
hapter
starts
with
an
analysis
of
the
curren
t
in
terest
in
quan
tum
computing�
It
then
discusses
the
main
in
tellectual
barriers
that
had
to
b
e
o
v
ercome
to
mak
e
a
vision
of
the
quan
tum
computer
an
imp
ortan
t
c
hallenge
to
curren
t
science
and
tec
hnology
�
The
basic
and
sp
eci�c
features
of
quan
tum
computing
are
�rst
in
tro
duced
b
y
a
comparison
of
randomized
computing
and
quan
tum
computing�
An
in
tro
duction
to
quan
tum
phenomena
is
done
in
three
stages�
First�
sev
eral
classical
and
similar
quan
tum
exp
erimen
ts
are
analysed�
This
is
follo
w
ed
b
y
Hilb
ert
space
basics
and
b
y
a
presen
tation
of
the
elemen
tary
principles
of
quan
tum
mec
hanics
and
the
elemen
ts
of
classical
rev
ersible
computing�
LEARNING
OBJECTIVES
The
aim
of
the
c
hapter
is
to
learn
��
the
main
reasons
wh
y
to
b
e
in
terested
in
quan
tum
computing�
��
the
prehistory
of
quan
tum
computing�
��
the
sp
eci�c
prop
erties
of
quan
tum
computing
in
comparison
with
randomized
comput�
ing�
��
the
basic
exp
erimen
ts
and
principles
of
quan
tum
ph
ysics�
��
the
basics
of
Hilb
ert
space
theory�
��
the
elemen
ts
of
classical
rev
ersible
computing�
�
�
Chapter
��
Elemen
ts
Y
ou
ha
v
e
nothing
to
do
but
men
tion
the
quan
tum
theory�
and
p
eople
will
tak
e
y
our
v
oice
for
the
v
oice
of
science�
and
b
eliev
e
an
ything�
Bernard
Sha
w
������
Quan
tum
computing
is
a
big
and
gro
wing
c
hallenge�
for
b
oth
science
and
tec
hnol�
ogy
�
Computations
based
on
quan
tum
w
orld
phenomena�
pro
cesses
and
la
ws
o�er
radically
new
and
v
ery
p
o
w
erful
p
ossibilities
and
lead
to
di�eren
t
constrain
ts
than
computations
based
on
the
la
ws
of
classical
ph
ysics�
Moreo
v
er�
quan
tum
computing
seems
to
ha
v
e
the
p
oten
tial
to
deep
en
our
understanding
of
Nature
as
w
ell
as
to
pro
vide
more
p
o
w
erful
information
pro
cessing
and
comm
unication
to
ols�
A
t
the
same
time
the
main
theoretical
concepts
and
principles
of
quan
tum
mec
hanics
that
are
needed
to
grasp
the
basic
ideas�
mo
dels
and
the�
oretical
metho
ds
of
quan
tum
computing�
are
simple�
elegan
t
and
p
o
w
erful�
This
c
hapter
is
dev
oted
to
them�
In
tro
duction
of
the
basic
concepts
in
this
c
hapter
will
b
e
detailed
and
orien
ted
mainly
to
those
ha
ving
no�
or
close
to
no�
kno
wledge
of
quan
tum
ph
ysics
and
quan
tum
information
pro
cessing�
���
Wh
y
Quan
tum
Computing
Do
not
b
ecome
attac
hed
to
things
y
ou
lik
e�
do
not
main
tain
a
v
ersion
to
things
y
ou
dis�
lik
e�
Sorro
w�
fear
and
b
ondage
come
from
one�s
lik
es
and
dislik
es�
Buddha
Quan
tum
computing
is
without
doubt
one
of
the
hottest
topics
at
the
curren
t
fron
tiers
of
computing�
or
ev
en
of
the
whole
science�
It
sounds
v
ery
attractiv
e
and
lo
oks
v
ery
promising�
There
are
sev
eral
natural
basic
questions
to
ask
b
efore
w
e
start
to
explore
the
concepts
and
principles
as
w
ell
as
the
m
ystery
and
p
oten
tials
of
quan
tum
computing�
��
Wh
y
to
consider
quan
tum
computing
at
all�
The
dev
elopmen
t
of
classical
computers
is
still
making
enormous
progress
and
no
end
of
that
seems
to
b
e
in
sigh
t�
More�
o
v
er�
the
design
of
quan
tum
computers
seems
to
b
e
v
ery
questionable
and
almost
surely
enormously
exp
ensiv
e�
All
this
is
true�
Ho
w
ev
er�
there
are
at
least
four
v
ery
go
o
d
reasons
for
exploring
quan
tum
computing
as
m
uc
h
as
p
ossible�
�
Quan
tum
computing
is
a
c
hallenge�
A
v
ery
fundamen
tal
and
v
ery
natural
c
hallenge�
Indeed�
according
to
our
curren
t
kno
wledge�
our
ph
ysical
w
orld
is
fundamen
tally
quan�
tum
mec
hanical�
All
computers
are
ph
ysical
devices
and
all
real
computations
are
ph
ysical
pro
cesses�
It
is
therefore
a
fundamen
tal
c
hallenge�
and
actually
our
dut
y
�
to
explore
the
p
oten
tials�
la
ws
and
limitations
of
quan
tum
mec
hanics
to
p
erform
infor�
mation
pro
cessing
and
comm
unication�
Jozef
Grusk
a�
QUANTUM
COMPUTING
�
All
classical
computers
and
mo
dels
of
computers�
see
Grusk
a
�������
are
based
on
classical
ph
ysics
�ev
en
if
this
is
rarely
men
tioned
explicitly��
and
therefore
they
are
not
fully
adequate�
There
is
nothing
wrong
with
them�
but
they
do
not
seem
to
explore
fully
the
p
oten
tial
of
the
ph
ysical
w
orld
for
information
pro
cessing�
They
are
go
o
d
and
p
o
w
erful�
but
they
should
not
b
e
seen
as
re�ecting
our
full
view
of
information
pro
cessing
systems�
�
Moreo
v
er�
theoretical
results
obtained
so
far
pro
vide
evidence
that
quan
tum
compu�
tation
represen
ts
the
�rst
real
c
hallenge
to
the
mo
dern�
e�ciency
orien
ted�
v
ersion
of
the
Ch
urc
h�T
uring
thesis�
A
ny
r
e
asonable
mo
del
of
c
omputation
c
an
b
e
e�ciently
simulate
d
by
pr
ob
a�
bilistic
T
uring
machines�
�
Quan
tum
computing
seems
to
b
e
a
m
ust
and
actually
our
destin
y�
As
miniaturization
of
computing
devices
con
tin
ues�
w
e
are
rapidly
approac
hing
the
microscopic
lev
el�
where
the
la
ws
of
the
quan
tum
w
orld
dominate�
By
Key
es
�������
an
extrap
olation
of
the
progress
in
miniaturization
sho
ws
that
around
����
computing
should
b
e
p
erformed
at
the
atomic
lev
el�
A
t
that
time�
if
the
dev
elopmen
t
k
eeps
con
tin
uing
as
hitherto�
one
electron
should
b
e
enough
to
store
one
bit�
and
the
energy
dissipation
of
�k
T
ln
�
should
b
e
su�cien
t
to
pro
cess
one
bit�
�
�
�
Th
us�
not
only
scien
ti�c
curiosit
y
and
c
hallenges�
but
also
tec
hnological
progress
requires
that
the
resources
and
p
oten
tials
of
quan
tum
computing
b
e
fully
explored�
�
�
Quan
tum
computing
is
a
p
oten
tial�
There
are
already
results
con
vincingly
demon�
strating
that
for
some
imp
ortan
t
practical
problems
quan
tum
computers
are
theoret�
ically
exp
onen
tially
more
p
o
w
erful
than
classical
computers�
Suc
h
results�
as
Shor�s
factorization
algorithm�
can
b
e
seen
as
apt
kil
lers
for
quan
tum
computing
and
ha
v
e
enormously
increased
activit
y
in
this
area�
In
addition�
the
la
ws
of
quan
tum
w
orld�
harv
ested
through
quan
tum
cryptograph
y�
can
o�er�
in
view
of
our
curren
t
kno
wledge�
unconditional
securit
y
of
comm
unication�
unac
hiev
able
b
y
classical
means�
�
Finally
�
the
dev
elopmen
t
of
quan
tum
computing
is
a
driv
e
and
giv
es
new
imp
etus
to
explore
in
more
detail
and
from
new
p
oin
ts
of
view
concepts�
p
oten
tials�
la
ws
and
limitations
of
the
quan
tum
w
orld
and
to
impro
v
e
our
kno
wledge
of
the
natural
w
orld�
The
study
of
information
pro
cessing
la
ws�
limitations
and
p
oten
tials
is
no
w
ada
ys
in
general
a
p
o
w
erful
metho
dology
to
extend
our
kno
wledge�
and
this
seems
to
b
e
partic�
ularly
true
for
quan
tum
mec
hanics�
Information
is
b
eing
iden
ti�ed
as
one
of
the
basic
�
A
t
this
p
oin
t
it
should
b
e
made
clear
that
quan
tum
computers
do
not
represen
t
a
c
hallenge
to
the
basic
Ch
urc
h�T
uring
thesis
concerning
computabilit
y�
They
cannot
compute
what
could
not
b
e
computed
b
y
classical
computers�
Their
main
adv
an
tage
is
that
they
can
solv
e
some
imp
ortan
t
computational
tasks
m
uc
h
more
e�cien
tly
than
classical
computers�
�
In
suc
h
a
case
it
will
b
e
necessary
to
include
in
the
design
and
description
of
computers
quan
tum
theory
and
suc
h
quan
tum
phenomena
as
sup
erp
osition
and
en
tanglemen
t�
to
obtain
correct
predictions
ab
out
computer
b
eha
viour�
Ho
w
ev
er�
the
clear
necessit
y
to
go
deep
er
in
to
the
quan
tum
lev
el
for
impro
ving
p
erformance
of
computers
do
es
not
immediately
imply
that
the
w
a
y
pursued
under
the
curren
t
in
terpretation
of
the
term
�quan
tum
computing�
is
the
only
one�
or
ev
en
the
b
est
one�
�
The
single
electron
transistor
is
already
under
dev
elopmen
t�
see
page
����
�
A
t
the
same
time
one
should
note
that
while
quan
tum
ph
ysics
has
b
een
already
for
a
long
time
essen
tial
to
the
understanding
of
the
op
erations
of
transistors
and
other
k
ey
elemen
ts
of
mo
dern
computers�
computation
remained
to
b
e
a
classical
pro
cess�
In
addition�
at
the
�rst
sigh
t
there
are
go
o
d
reasons
for
computing
and
quan
tum
ph
ysics
to
b
e
v
ery
far
apart
b
ecause
determinism
and
certain
t
y
required
from
computations
seem
to
b
e
in
strong
con
trast
with
uncertain
t
y
principle
and
probabilistic
nature
of
quan
tum
mec
hanics�
�
Chapter
��
Elemen
ts
and
p
o
w
erful
concepts
of
ph
ysics
and
quan
tum
en
tanglemen
t
is
an
imp
ortan
t
comm
u�
nication
resource�
Sev
eral
profound
insigh
ts
in
to
the
natural
w
orld
ha
v
e
already
b
een
obtained
on
this
basis�
�
Remark
�����
The
ab
ove
ide
as
ar
e
so
new
and
imp
ortant�
that
they
deserve
an
additional
analysis�
Historic
al
ly�
the
fundamental
principles
of
physics
�rst
c
onc
erne
d
the
pr
oblems
of
matter�what
things
ar
e
made
of
and
how
they
move�
L
ater�
the
pr
oblems
of
ener
gy
starte
d
to
b
e
r
e�e
cte
d
in
the
le
ading
principles
of
physics�how
ener
gy
is
cr
e
ate
d�
expr
esse
d
and
tr
ansforme
d�
As
the
next
stage
an
alternative
se
ems
to
b
e
to
lo
ok
to
information
pr
o
c
essing
for
a
new
sour
c
e
of
fundamental
principles
and
b
asic
laws�
F
or
example�
c
onc
erning
the
p
ar�
ticles�
the
questions
of
the
movement
of
p
articles
may
b
e
sup
erse
de
d
by
how
p
articles
c
an
b
e
utilize
d
for
information
pr
o
c
essing�
Final
ly�
let
us
observe
some
similarities
b
etwe
en
ener
gy
and
information�
Both
of
them
have
many
r
epr
esentations�
but
b
asic
principles�
and
also
e
quations�
hold
indep
endently
of
the
form
in
which
ener
gy
or
information
is
pr
esente
d�
The
incr
e
asing
imp
ortanc
e
of
information
pr
o
c
essing
principles
for
curr
ent
scienc
e
has
b
e
en
�rst�
c
orr
e
ctly�
r
e�e
cte
d
in
the
views
and
understanding
�due
to
L
andauer�
������
that
�information
is
physic
al�
and
in
the
c
orr
esp
onding
changes
of
emphases
on
the
essenc
e
and
ways
to
de
al
with
information
pr
o
c
essing
pr
oblems�
However�
it
c
ould
b
e
the
c
ase
that
this
is
only
the
�rst
step
and
p
erhaps
even
mor
e
fundamental
changes
in
the
principles
of
physics
c
ould
b
e
obtaine
d
fr
om
the
view
that
�physics
is
informational��
�
These
new
views
of
the
r
ole
of
information
in
quantum
physics
also
bring
new
p
otentials�
chal
lenges
and
questions
for
quantum
physics�
Is
the
wel
l
known
�weir
dness�
of
the
quantum
world
due
to
the
fact
that
physic
al
r
e
ality
is
governe
d
by
even
mor
e
b
asic
laws
of
the
infor�
mation
pr
o
c
essing
world�
Is
quantum
the
ory
a
the
ory
of
the
physic
al
or
of
the
information
world�
Can
the
study
of
quantum
information
help
to
de
al
with
the
most
b
asic
pr
oblems
quantum
the
ory
has�
As
an
example
of
a
change
of
r
ese
ar
ch
aims
in
physics
under
the
in�uenc
e
of
c
omputer
scienc
e
r
ese
ar
ch
p
ar
adigms�
c
onsider
quantum
evolution�
T
r
aditional
ly�
quantum
physics
has
b
e
en
c
onc
erne
d
with
the
study
or
design
of
p
articular
quantum
systems
and
the
study
of
various
r
elate
d
fundamental
pr
oblems�
In
addition
to
these
pr
oblems
quantum
c
omputing
br
ought
up
new
gener
al
and
fundamental
questions�
Namely�
what
ar
e
the
b
est�
fr
om
wel
l
de�ne
d
quantitative
p
oint
of
views�
quantum
evolutions
to
solve
p
articular
algorithmic
or
c
ommunic
ation
tasks�
Or
a
pr
oblem
of
the
maximum
quantum
c
omputation
p
ower
achievable
in
a
quantum
system
of
a
c
ertain
dimension
and
disturb
anc
e
level
�Ste
ane�
����b��
and
of
the
way
to
r
e
ach
such
a
maximum�
New
fundamental
questions
in
quantum
me
chanics
ar
e
r
aise
d
also
in
c
onne
ction
with
the
fol
lowing
pr
oblem�
how
se
cur
e
ar
e�
or
c
an
b
e�
quantum
crypto
gr
aphic
pr
oto
c
ols�
F
or
�
F
or
example�
manifestations
of
quan
tum
nonlo
calit
y
that
go
b
ey
ond
en
tanglemen
t
�see
Bennett
et
al�
������
the
use
of
quan
tum
principles
for
secure
transmission
of
classical
information
�quan
tum
cryptograph
y��
the
use
of
quan
tum
en
tanglemen
t
for
reliable
transmission
of
quan
tum
states
o
v
er
a
distance
�quan
tum
telep
ortation��
the
p
ossibilit
y
of
preserving
quan
tum
coherence
in
the
presence
of
irrev
ersible
noise
pro
cesses
�quan
tum
error
correction
and
fault
toleran
t
computation��
In
addition�
b
y
Steane
�������
one
has
to
realize
that
historically
m
uc
h
of
fundamen
tal
ph
ysics
has
b
een
concerned
with
disco
v
ering
fundamen
tal
particles
of
Nature
and
the
equations
whic
h
describ
e
their
motions
and
in
teractions�
It
no
w
app
ears
that
a
di�eren
t
program
ma
y
b
e
equally
imp
ortan
t�
Namely
�
to
disco
v
er
the
w
a
ys
Nature
allo
ws�
and
prev
en
ts�
information
to
b
e
expressed
and
manipulated�
rather
than
particles
to
mo
v
e�
�
A
lot
of
researc
h
is
still
needed
to
determine
the
p
osition
and
real
role
information
pla
ys
in
ph
ysics�
The
extreme
views
go
ev
en
so
far
that
information
is
a
ph
ysical
quan
tit
y
�
similar
as
energy
in
thermo
dynamics
�Horo
dec
ki�
�����
and
Landauer�
�����
������
or
ev
en
that
information
is
deep
er
than
realit
y�a
substance
that
is
more
fundamen
tal
than
matter
and
energy�
Jozef
Grusk
a�
QUANTUM
COMPUTING
�
example�
the
question
how
much
information
c
an
b
e
extr
acte
d
fr
om
a
quantum
system
for
a
given
amount
of
exp
e
cte
d
disturb
anc
es�
These
questions
ar
e
of
fundamental
imp
ortanc
e
far
b
eyond
quantum
crypto
gr
aphy�
T
o
answer
these
questions�
new
the
or
etic
al
insights
and
also
new
exp
eriments
se
em
to
b
e
ne
e
de
d�
In
addition�
an
a
w
areness
has
b
een
emerging
also
in
the
foundations
of
computing
that
fundamen
tal
questions
regarding
computabilit
y
and
computational
complexit
y
are
in
a
deep
sense
questions
ab
out
ph
ysical
pro
cesses�
�
If
they
are
studied
on
a
mathematical
lev
el
then
the
underlying
mo
dels
ha
v
e
to
re�ect
fully
the
prop
erties
of
our
ph
ysical
w
orld�
This
in
particular
implies
that
computational
complexit
y
theory
has
to
b
e�
in
its
most
fundamen
tal
form�
based
on
mo
dels
of
quan
tum
computers�
�
��
Can
quan
tum
computers
do
what
classical
ones
cannot�
The
answ
er
de�
p
ends
on
the
p
oin
t
of
view�
It
can
b
e
YES�
Indeed�
the
simplest
example
is
generation
of
random
n
um
b
ers�
Quan
tum
algorithms
can
generate
truly
random
n
um
b
ers�
Deterministic
algorithms
can
generate
only
pseudo�random
n
um
b
ers�
Other
examples
come
from
the
sim�
ulation
of
quan
tum
phenomena�
On
the
other
hand�
the
answ
er
can
b
e
also
NO�
A
classical
computer
can
pro
duce
truly
random
n
um
b
ers
when
attac
hed
to
a
prop
er
ph
ysical
source�
��
Where
lie
the
di�erences
b
et
w
een
the
classical
and
quan
tum
information
pro
cessing�
Some
of
the
di�erences
ha
v
e
already
b
een
men
tioned�
Let
us
no
w
discuss
some
others�
Classical
information
can
b
e
read�
transcrib
ed
�in
to
an
y
medium��
duplicated
at
will�
transmitted
and
broadcasted�
Quan
tum
information�
on
the
other
hand�
cannot
b
e
in
general
read
or
duplicated
without
b
eing
disturb
ed�
but
it
can
b
e
�telep
orted�
�as
discussed
in
Section
�����
In
classical
randomized
computing�
a
computer
alw
a
ys
selects
one
of
the
p
ossible
com�
putation
paths�
according
to
a
source
of
randomness�
and
�what�could�happ
en�but�did�not�
has
no
in�uence
whatso
ev
er
on
the
outcome
of
the
computation�
On
the
other
hand�
in
quan
tum
computing�
exp
onen
tially
man
y
computational
paths
can
b
e
tak
en
sim
ultaneously
in
a
single
piece
of
hardw
are
and
in
a
sp
ecial
quan
tum
w
a
y
and
�what�could�happ
en�but�
did�not�
can
really
matter�
Acquiring
information
ab
out
a
quan
tum
system
can
inevitably
disturbs
the
state
of
the
system�
The
tradeo�
b
et
w
een
acquiring
quan
tum
information
and
creating
a
disturbance
of
the
system
is
due
to
quan
tum
randomness�
The
outcome
of
a
quan
tum
measuremen
t
has
a
random
elemen
t
and
b
ecause
of
that
w
e
are
unable
alw
a
ys
faithfully
infer
the
�initial�
state
of
the
system
from
the
measuremen
t
outcome�
�
An
understanding
has
emerged
that
eac
h
sp
eci�c
computation
is
p
erformed
b
y
a
ph
ysical
system
ev
olv�
ing
in
time
and�
consequen
tly
�
that
one
of
the
basic
problems
of
computing�
namely
�what
is
e�cien
tly
computable��
is
deeply
related
to
one
of
the
basic
problems
of
ph
ysics�
namely
�whic
h
dynamical
systems
are
ph
ysically
realizable��
�
The
follo
wing
citations
re�ect
a
dissatisfaction
with
the
fact
that
the
dev
elopmen
t
of
complexit
y
theory
ignored
one
of
its
most
fundamen
tal
tasks�
The
fact
that
this
had
b
een
so
is
in
one
w
a
y
explainable
but�
in
another
w
a
y
�
hardly
forgiv
able�
A�
Ek
ert
�������
Computers
ar
e
physic
al
obje
cts
and
c
omputations
ar
e
physic
al
pr
o
c
esses�
The
the
ory
of
c
omputation
is
not
a
br
anch
of
pur
e
mathematics�
F
undamental
questions
r
e
gar
ding
c
omputability
and
c
omputational
c
omplexity
ar
e
questions
ab
out
physic
al
pr
o
c
esses
that
r
eve
al
to
us
pr
op
erties
of
abstr
act
entities
such
as
numb
ers
or
ide
as�
Those
questions
b
elong
to
physics
r
ather
than
mathematics�
J�
Bec
kman
et
al
�������
The
the
ory
of
c
omputation
would
b
e
b
o
otless
if
the
c
omputations
that
it
describ
es
c
ould
not
b
e
c
arrie
d
out
using
physic
al
ly
r
e
alizable
devic
es�
Henc
e
it
is
r
e
al
ly
a
task
of
physics
to
char
ac�
terize
what
is
c
omputable�
and
to
classify
the
e�ciency
of
c
omputations�
The
physic
al
world
is
quantum
me
chanic
al�
Ther
efor
e�
the
foundations
of
the
the
ory
of
c
omputation
must
b
e
quantum
me
chanic
al
as
wel
l�
The
classic
al
the
ory
of
c
omputation
should
b
e
viewe
d
as
an
imp
ortant
sp
e
cial
c
ase
of
a
mor
e
gener
al
the
ory�
�
Chapter
��
Elemen
ts
P
erhaps
the
main
di�erence
b
et
w
een
classical
and
quan
tum
information
pro
cessing
lies
in
the
fact
that
quan
tum
information
can
b
e
enco
ded
in
m
utual
correlations
b
et
w
een
remote
parts
of
ph
ysical
systems
and
quan
tum
information
pro
cessing
can
mak
e
essen
tial
use
of
this
phenomena�called
en
tanglemen
t�not
a
v
ailable
for
classical
information
pro
cessing�
Another
big
di�erence
b
et
w
een
the
classical
and
quan
tum
w
orlds
that
strongly
in�uences
quan
tum
information
pro
cessing
stems
from
the
fact
that
the
relationship
b
et
w
een
a
system
and
its
subsystems
is
di�eren
t
in
the
quan
tum
w
orld
than
in
the
classical
w
orld�
F
or
example�
the
states
of
a
quan
tum
system
comp
osed
of
quan
tum
subsystems
cannot
b
e
in
general
decomp
osed
in
to
states
of
these
subsystems�
��
Can
quan
tum
computers
solv
e
some
practically
imp
ortan
t
problems
m
uc
h
more
e�cien
tly�
Y
es�
F
or
example�
in
teger
factorization
can
b
e
done
in
p
olynomial
time
on
quan
tum
computers
what
seems
to
b
e
imp
ossible
on
classical
computers�
Searc
hing
in
unordered
database
can
b
e
done
pro
v
ably
with
less
queries
on
quan
tum
computer�
��
Where
do
es
the
p
o
w
er
of
quan
tum
computing
come
from�
On
one
side�
quan
tum
computation
o�ers
enormous
parallelism�
The
size
of
the
computational
state
space
is
exp
onen
tial
in
the
ph
ysical
size
of
the
system
and
the
energy
a
v
ailable�
A
quan
tum
bit
can
b
e
in
an
y
of
a
p
oten
tially
in�nite
n
um
b
er
of
states
and
quan
tum
systems
can
b
e
sim
ultaneously
in
sup
erp
osition
of
exp
onen
tially
man
y
of
the
basis
states�
A
linear
n
um
b
er
of
op
erations
can
create
an
exp
onen
tially
large
sup
erp
osition
of
states
and�
in
parallel�
an
exp
onen
tially
large
n
um
b
er
of
op
erations
can
b
e
p
erformed
in
one
step�
Secondly
�
it
is
the
branc
hing
and
quan
tum
in
terference
that
create
parallel
computation
and
constructiv
e�destructiv
e
sup
erp
ositions
of
states
and
can
amplify
or
destro
y
the
impacts
of
some
computations�
Due
to
this
fact�
w
e
can�
in
spite
of
the
p
eculiarities
of
quan
tum
measuremen
ts�
utilize
quan
tum
parallelism�
Thirdly
�
it
is
mainly
the
existence
of
so�called
�en
tangled
states�
that
mak
es
quan
tum
computing
more
p
o
w
erful
than
classical
and
allo
ws
ev
en
v
ery
distan
t
parts
of
systems
to
b
e
strongly
tied�
This
creates
a
base
for
dev
eloping
and
exploring
quan
tum
telep
ortation
and
other
phenomena
that
are
outside
of
the
realm
of
the
classical
w
orld�
�����
After
all
this
excitemen
t
let
us
start
to
deal
with
more
prosaic
and
�harder
�questions�
��
Where
are
the
dra
wbac
ks
and
b
ottlenec
ks
of
quan
tum
computing�
There
are�
unfortunately
�
quite
a
few�
Let
us
men
tion
here
only
t
w
o
of
them�
�
Quan
tum
computing
can
pro
vide
enormous
parallelism�
Ho
w
ev
er�
there
are
also
enor�
mous
problems
with
harnessing
the
p
o
w
er
of
its
parallelism�
According
to
the
basic
principles
of
quan
tum
mec
hanics�
a
�pro
jection�
measuremen
t
pro
cess
can
get
out
of
�large�
quan
tum
sup
erp
osition
only
one
classical
result�
randomly
c
hosen�
and
the
remaining
quan
tum
information
can
b
e
irrev
ersibly
destro
y
ed�
�
An
in
teraction
of
a
quan
tum
system
with
its
en
vironmen
t
can
lead
to
the
the
so�
called
decoherence
e�ects
and
can
greatly
in�uence�
or
ev
en
completely
destro
y
�
subtle
quan
tum
in
terference
mec
hanisms�
This
app
ears
to
mak
e
long
reliable
quan
tum
com�
putations
practically
imp
ossible�
��
Ho
w
feasible
are
�p
o
w
erful�
quan
tum
computers
and
really
imp
ortan
t
quan
tum
information
pro
cessing
applications�
It
is
to
o
early
to
giv
e
a
de�nite
answ
er�
On
one
side�
there
is
a
strong
scien
ti�c
b
elief�
based
on
long
term
exp
eriences
of
science�
that
something
v
ery
imp
ortan
t
will
come
out
of
the
researc
h
in
quan
tum
computing�
Jozef
Grusk
a�
QUANTUM
COMPUTING
�
On
the
other
hand�
one
has
to
admit
that
man
y
of
the
curren
t
exciting
results
concerning
quan
tum
computing
should
b
e
seen
as
Gedank
en
exp
erimen
ts�
Namely
�
one
w
orks
with
systems
�exp
erimen
ts�
that
p
erhaps
do
not
exist�
or
cannot
b
e
p
erformed
in
the
real
w
orld�
or
only
with
enormous
di�cult
y
�
but
do
not
con
tradict
an
y
kno
wn
la
w
within
a
�certain�
consisten
t
theory
of
quan
tum
mec
hanics�
�
Suc
h
considerations�
systems
and
results
are
usually
tak
en
as
b
eing
in
principle
acceptable�
In
addition�
in
the
recen
t
y
ears
quite
impressiv
e
progress
has
b
een
made
on
the
exp
eri�
men
tal
lev
el
and
w
a
ys
ha
v
e
b
een
found
to
deal
with
man
y
problems
that
seemed
to
prev
en
t
the
utilization
of
the
p
o
w
er
of
quan
tum
computing�
Esp
ecially
exp
erimen
tal
quan
tum
cryp�
tograph
y
has
made
formidable
progress
to
sho
w
that
long
distance
optical
�b
er�
op
en�air
and
ev
en
earth�satellites
quan
tum
k
ey
generation
seems
to
b
e
feasible�
Finally
�
it
seems
quite
safe
to
assume
that
either
quan
tum
computing
will
meet
its
exp
ectations
or
something
new
and
imp
ortan
t
will
b
e
learned
and
our
kno
wledge
of
Nature
will
b
e
enhanced�
��
Are
not
curren
t
computers
quan
tum�
No�
in
spite
of
the
fact
that
curren
t
com�
puters
use
elemen
ts�
for
example
semiconductors�
whose
functioning
cannot
b
e
explained
without
quan
tum
mec
hanics�
Curren
t
computers
are
in
some
v
ery
restricted
sense
quan
tum
mec
hanical
b
ecause
ev
erything
can
b
e
seen
as
b
eing
quan
tum
mec
hanical�
In
spite
of
that�
curren
t
computers
are
not
considered
as
fully
quan
tum
mec
hanical�
The
main
di�erence
b
et
w
een
a
classical
and
a
quan
tum
computer
is
on
the
information
storage
and
pro
cessing
lev
el�
In
classical
computers
information
is
recorded
in
macroscopic
t
w
o�lev
el
systems�
called
bits�
represen
ting
t
w
o
bit
v
alues�
In
quan
tum
computers
information
is
recorded
and
pro�
cessed
at
microscopic
lev
el
using
t
w
o�lev
el
quan
tum
systems�
called
quan
tum
bits�
that
can
b
e
in
an
y
quan
tum
sup
erp
osition
of
quan
tum
states
corresp
onding
to
t
w
o
classical
bits�
��
Can
quan
tum
computers
ev
en
tually
replace
classical
ones�
Nob
o
dy
kno
ws�
but
this
do
es
not
seem
to
b
e
so�
at
least
not
in
the
near
future�
Both
classical
and
quan
tum
computers
ha
v
e
their
strong
and
w
eak
p
oin
ts�
and
it
seems
curren
tly
that
they
can
supp
ort�
but
not
replace�
eac
h
other�
���
Prehistory
of
Quan
tum
Computing
The
past
is
but
the
b
eginning
of
a
b
e�
ginning�
and
all
that
is
and
has
b
een
is
but
the
t
wiligh
t
of
the
da
wn�
Herb
ert
Georg
W
ells
�����������
Since
����
w
e
ha
v
e
b
een
witnessing
a
rapid
gro
wth
of
the
ra
w
p
erformance
of
computers
with
resp
ect
to
their
sp
eed
and
memory
size�
An
imp
ortan
t
step
in
this
dev
elopmen
t
w
as
the
in
v
en
tion
of
transistors�
whic
h
already
use
some
quan
tum
e�ects
in
their
op
eration�
Ho
w
ev
er�
it
is
clear
that
if
suc
h
an
increase
in
p
erformance
of
computers
con
tin
ues�
then
after
��
y
ears�
our
c
hips
will
ha
v
e
to
con
tain
��
��
gates
and
op
erate
at
a
��
��
Hz
clo
c
k
rate
�
The
term
�Gedank
en
exp
erimen
t�
is
used
in
sev
eral
meanings�
Sometimes
it
is
required
that
the
corre�
sp
onding
systems
or
exp
erimen
ts
are
in
principle
p
ossible�
Sometimes
it
is
su�cien
t
that
no
ph
ysical
la
w
is
kno
wn
that
w
ould
not
allo
w
suc
h
an
exp
erimen
t�
�
Chapter
��
Elemen
ts
�th
us
deliv
ering
��
��
logic
op
erations
p
er
second�
��
�
It
seems
that
the
only
w
a
y
to
ac
hiev
e
that
is
to
learn
to
build
computers
directly
out
of
the
la
ws
of
quan
tum
ph
ysics�
In
order
to
come
up
seriously
with
the
idea
of
quan
tum
information
pro
cessing�
and
to
dev
elop
it
so
far
and
so
fast�
it
has
b
een
necessary
to
o
v
ercome
sev
eral
in
tellectual
barriers�
The
most
basic
one
concerned
an
imp
ortan
t
feature
of
quan
tum
ph
ysics�rev
ersibilit
y
�see
Section
�����
��
None
of
the
kno
wn
mo
dels
of
univ
ersal
computers
w
as
rev
ersible�
This
barrier
w
as
o
v
ercome
�rst
b
y
Bennett
��
�������
who
sho
w
ed
the
existence
of
univ
ersal
re�
v
ersible
T
uring
mac
hines�
and
then
b
y
T
o�oli
������
�����
and
F
redkin
and
T
o�oli
�������
who
sho
w
ed
the
existence
of
univ
ersal
classical
rev
ersible
gates�
��
The
second
in
tellectual
barrier
w
as
o
v
ercome
b
y
Benio�
������
�����
����a�
who
sho
w
ed
that
quan
tum
mec
hanical
computational
pro
cesses
can
b
e
at
least
as
p
o
w
erful
as
classical
computational
pro
cesses�
He
did
that
b
y
sho
wing
ho
w
a
quan
tum
system
can
sim
ulate
actions
of
the
classical
rev
ersible
T
uring
mac
hines�
Ho
w
ev
er�
his
�quan
tum
computer�
w
as
not
fully
quan
tum
y
et
and
could
not
outp
erform
classical
ones�
The
o
v
ercoming
of
these
basic
in
tellectual
barriers
had
signi�can
t
and
broad
conse�
quences�
Relations
b
et
w
een
ph
ysics
and
computation
started
to
b
e
in
v
estigated
on
a
more
general
and
deep
er
lev
el�
This
has
also
b
een
due
to
the
fact
that
rev
ersibilit
y
results
im�
plied
the
theoretical
p
ossibilit
y
of
zero�energy
computations�
��
A
W
orkshop
on
Ph
ysics
and
Computation
started
to
b
e
organized
and
in
his
k
eynote
sp
eec
h
at
the
�rst
of
these
w
ork�
shops�
in
�����
R�
F
eynman
������
��
ask
ed
an
imp
ortan
t
question�
Can
�quantum�
physics
b
e
�e�ciently�
simulate
d
by
�classic
al�
c
omputers�
A
t
the
same
time
he
sho
w
ed
go
o
d
rea�
sons
to
b
eliev
e
that
the
answ
er
is
negativ
e�
Namely
�
that
it
app
ears
to
b
e
imp
ossible
to
sim
ulate
a
general
quan
tum
ph
ysical
system
on
a
probabilistic
T
uring
mac
hine
without
an
exp
onen
tial
slo
wdo
wn
��
�
Moreo
v
er�
he
sp
eculated
that
one
could
deal
with
the
problem
b
y
allo
wing
computers
to
run
according
to
the
la
ws
of
quan
tum
mec
hanics�
In
other
w
ords�
that
quan
tum
computers
could
b
e
exp
onen
tially
more
p
o
w
erful
than
classical
ones
and
could
��
Due
to
these
facts�
the
concern
w
as
v
oiced
quite
a
while
ago
on
the
p
ossible
negativ
e
e�ects
that
quan
tum
phenomena
could
induce
in
the
�classical�
op
erations
of
computers�
F
or
example�
what
fundamen
tal
limits
could
Heisen
b
erg�s
uncertain
t
y
principle
imp
ose
on
memory
c
hips
whose
bits
are
stored
in
single
electron
states�
This
approac
h
w
as
later
sup
erseded�
as
w
e
shall
see�
b
y
more
optimistic�
more
constructiv
e
and
more
am
bitious
aims
to
harness
the
p
o
w
er
of
quan
tum
mec
hanics
to
p
erform
computations�
��
Rev
ersibilit
y
is
actually
not
an
exclusiv
e
phenomenon
of
the
quan
tum
w
orld�
Rev
ersibilit
y
also
o
ccurs
in
the
classical
ph
ysics�
It
is
only
the
ph
ysics
of
large
systems
�classical
but
also
quan
tum�
that
is
not
rev
ersible�
The
fact
is
that
classical
computationally
rev
ersible
systems
suggested
b
y
Bennett
and
others�
as
discussed
later�
w
ere
not
practically
realizable�
This
brough
t
up
the
idea
of
considering
quan
tum
rev
ersible
information
pro
cessing
systems�
��
F
or
earlier
references
see
Section
���
in
App
endix�
��
Bennett
������
traces
the
need
to
think
seriously
ab
out
the
thermo
dynamics
of
men
tal
pro
cesses
�and
computation
w
as
though
t
of
this
w
a
y
in
the
nineteen
th
cen
tury��
bac
k
to
the
famous
parado
x
of
�Maxw
ell�s
demon�
from
�����
whic
h
seemed
to
violate
the
second
la
w
of
thermo
dynamics�
see
App
endix�
Section
������
��
Actually
�
the
original
motiv
ation
for
studying
the
rev
ersibilit
y
of
computation
came
from
the
in
terest
in
determining
the
ultimate
thermo
dynamic
costs
of
elemen
tary
information
pro
cessing
op
erations�
esp
ecially
b
ecause
heat
remo
v
al
has
alw
a
ys
b
een
a
ma
jor
engineering
concern
in
the
design
of
classical
computers�
limiting
the
densit
y
with
whic
h
activ
e
comp
onen
ts
could
b
e
pac
k
ed�
In
the
b
eginnings
of
the
mo
dern
computer
era
there
w
as
a
folklore
b
elief�
going
bac
k
to
a
v
on
Neumann�s
lecture
in
����
�see
Burks�
������
that
at
least
k
T
ln
�
of
energy
is
needed
p
er
bit
op
eration�
A
ttempts
to
pro
v
e
this
misleading
folklore
b
elief
led
Landauer
to
the
disco
v
ery
of
rev
ersible
computing�
��
Ric
hard
P
�
F
eynman
������������
an
American
ph
ysicist�
His
main
scien
ti�c
con
tributions
w
ere
in
quan
tum
electro
dynamics
and
in
the
study
of
in
teractions
of
elemen
tary
particles�
He
ga
v
e
a
mathematical
description
of
helium�
F
eynman
receiv
ed
the
����
Nob
el
prize
for
ph
ysics
for
his
con
tributions
to
quan
tum
electro
dynamics�
He
has
also
b
een
kno
wn
for
his
extraordinary
capabilities
to
explain
ph
ysical
phenomena
and
his
lectures
and
textb
o
oks
represen
t
an
additional
imp
ortan
t
con
tributions
to
mo
dern
ph
ysics�
��
Actually
�
this
is
no
w
ada
ys
in
tuitiv
ely
prett
y
ob
vious
b
ecause
n
in
teracting
��state
quan
tum
systems
ma
y
ha
v
e
up
to
�
n
basis
states�
Jozef
Grusk
a�
QUANTUM
COMPUTING
�
b
e
a
�rst
reasonable
mo
del
of
computation
that
do
es
not
ob
ey
the
mo
dern
Ch
urc
h�T
uring
thesis�
��
The
third
in
tellectual
barrier
that
had
to
b
e
o
v
ercome
w
as
a
lac
k
of
a
prop
er
mo
del
for
a
univ
ersal
quan
tum
computing
device
capable
of
sim
ulating
e�ectiv
ely
an
y
other
quan�
tum
computer�
The
�rst
step
to
o
v
ercome
this
barrier
w
as
done
b
y
Deutsc
h
������
who
elab
orated
F
eynman�s
ideas
and
dev
elop
ed
a
�theoretically�
ph
ysically
realisable
mo
del
of
quan
tum
computers�
a
quan
tum
ph
ysical
analogue
of
a
probabilistic
T
uring
mac
hine�
whic
h
mak
es
full
use
of
the
quan
tum
sup
erp
osition
principle�
and
on
an
y
giv
en
input
pro
duces
a
random
sample
from
a
probabilit
y
distribution�
Deutsc
h
conjectured
that
it
migh
t
b
e
more
e�cien
t
than
a
classical
T
uring
mac
hine
for
certain
computations�
He
also
sho
w
ed
the
existence
of
a
univ
ersal
quan
tum
T
uring
mac
hine
�that
could
consequen
tly
sim
ulate
an
y
ph
ysical
pro
cess
and
exp
erimen
t�
and
also
a
mo
del
of
quan
tum
net
w
orks�a
quan
tum
analog
of
classical
sequen
tial
logical
circuits�
Ho
w
ev
er�
his
mo
del
of
the
univ
ersal
T
uring
mac
hine
had
the
dra
wbac
k
that
the
sim
ulation
of
other
quan
tum
T
uring
mac
hines
�QTM��
could
b
e
exp
onen
tial�
��
This
problem
w
as
then
o
v
ercome
b
y
Bernstein
and
V
azirani
������
and
Y
ao
�������
They
sho
w
ed
the
existence
of
univ
ersal
quan
tum
T
uring
mac
hines
capable
of
sim
ulating
other
quan
tum
T
uring
mac
hines
in
p
olynomial
time�
�F
or
a
full
pro
of
see
Bern�
stein
and
V
azirani
��������
The
pap
er
of
Bernstein
and
V
azirani
������
laid
the
foundations
of
quan
tum
complexit
y
theory
�
In
addition�
Y
ao
������
sho
w
ed
that
QTM
and
quan
tum
circuits
compute
in
p
olynomial
time
the
same
class
of
functions�
This
result
implies
that
the
concept
of
quan
tum
computation
in
p
olynomial
time
is
robust
enough
and
indep
enden
t
of
the
mac
hine
mo
dels�
In
parallel
with
the
dev
elopmen
t
of
the
basic
mo
dels
of
quan
tum
computing
an
e�ort
w
as
put
in
to
o
v
ercoming
the
fourth
in
tellectual
barrier�
Can
quan
tum
computing
b
e
really
more
p
o
w
erful
than
classical
computing�
Are
there
some
go
o
d
reasons
to
assume
that
quan
tum
computing
could
bring
an
essen
tial
�exp
onen
tial�
sp
eed�up
of
computations
for
at
least
some
imp
ortan
t
information
pro
cessing
problems�
This
w
as
a
v
ery
imp
ortan
t
issue
b
ecause
it
w
as
clear
that
an
y
design
of
a
quan
tum
computer
w
ould
require
o
v
ercoming
a
n
um
b
er
of
large
scien
ti�c
and
engineering
barriers
and
therefore
it
w
as
needed
to
kno
w
whether
the
prop
osed
mo
del
of
quan
tum
computer
o�ers�
at
least
theoretically
�
an
y
substan
tial
b
ene�t
o
v
er
the
classical
computers�
In
spite
of
the
fact
that
this
problem
has
not
y
et
b
een
completely
resolv
ed
there
is
already
strong
evidence
that
this
is
so�
It
w
as
�rst
sho
wn
b
y
Deutsc
h
and
Jozsa
�������
that
there
are
problems
unkno
wn
to
b
e
in
P
that
could
b
e
solv
ed
in
p
olynomial
time
on
quan
tum
computers�
and
therefore
b
elong
to
the
class
QEP
of
problems
solv
able
with
certain
t
y
in
p
olynomial
time
on
quan
tum
computers�
By
recasting
the
original
Deutsc
h�Jozsa
problem�
in
the
framew
ork
of
so�called
�promise
problems��
Berthiaume
and
Brassard
������
����a�
����b�
�����
pro
v
ed
the
�rst
separation
results
in
the
relativized
quan
tum
complexit
y
theory
�
F
or
example�
they
sho
w
ed
that
there
is
an
oracle
A
suc
h
that
QEP
A
��
ZPP
A
�they
pro
v
ed
the
existence
of
an
oracle
for
whic
h
there
are
computational
problems
that
QTM
can
solv
e
in
p
olynomial
time
with
certain
t
y
�
but
eac
h
probabilistic
T
uring
mac
hine
to
solv
e
these
problems
with
certain
t
y
needs
exp
onen
tial
time
for
some
inputs�
These
results
w
ere
�rst
impro
v
ed
b
y
Bernstein
and
��
R�
F
reiv
alds
called
m
y
atten
tion
to
the
fact
that
Y
u�
Manin
already
in
����
in
his
b
o
ok
�Computable
and
uncomputable�
p
oin
ted
out
explicitly
the
p
oten
tial
adv
an
tages
of
quan
tum
computing
�exp
onen
tial
n
um
b
er
of
basis
states
to
w
ork
with
sim
ultaneously�
and
emphasized
a
need
to
design
a
theory
of
quan
tum
automata
that
w
ould
b
e
abstract
enough
and
w
ould
ha
v
e
a
prop
er
balance
b
et
w
een
mathematical
principles
and
fundamen
tal
principles
of
quan
tum
mec
hanics
without
sp
eci�cation
of
some
ph
ysical
realizations�
��
Deutsc
h
cen
tered
his
atten
tion
on
the
computabilit
y
and
not
on
complexit
y
issues�
��
Chapter
��
Elemen
ts
V
azirani
������
and
later
b
y
Simon
�������
He
pro
v
ed
the
follo
wing
result
that
w
as
at
that
time
the
strongest
argumen
t
in
fa
v
or
of
the
sup
eriorit
y
of
quan
tum
computers
o
v
er
classical
ones�
Theorem
�����
Ther
e
exists
an
or
acle
r
elative
to
which
ther
e
is
a
pr
oblem
solvable
in
p
oly�
nomial
time
�with
b
ounde
d
err
or
pr
ob
ability�
on
a
quantum
c
omputer�
but
any
pr
ob
abilistic
T
uring
machine
with
b
ounde
d
err
or
pr
ob
ability
solving
this
pr
oblem
�using
the
or
acle�
wil
l
r
e
quir
e
exp
onential
time
�at
le
ast
�
n��
steps�
on
in�nitely
many
inputs
�of
length
n��
Results
of
Bernstein
and
V
azirani
������
and
Simon
������
pro
vide
formal
evidence
that�
in
the
relativized
setting�
QTM
are
more
p
o
w
erful
than
PTM�
��
Ho
w
ev
er�
all
these
problems
w
ere
quite
arti�cial�
V
ery
imp
ortan
t
and
m
uc
h
needed
steps
along
these
lines
ha
v
e
b
een
the
results
of
Shor
������
�����
who�
building
on
the
w
orks
of
the
ab
o
v
e
men
tioned
authors�
esp
ecially
on
Simon�s
metho
d�
sho
w
ed
ho
w
to
factor
in
tegers�
and
ho
w
to
compute
discrete
logarithms
in
p
olynomial
time
on
p
oten
tial
quan
tum
computers�t
w
o
problems
of
crucial
imp
ortance
for
public�k
ey
cryptograph
y�
Due
to
these
results
quan
tum
computing�
that
till
then
used
to
b
e
considered
as
a
curios�
it
y
for
few
visionaries�
started
to
b
e
of
broader
scien
ti�c�
and
not
only
scien
ti�c�
in
terest�
An
in
tensiv
e
searc
h
started
to
disco
v
er
ph
ysical
principles
and
pro
cesses
that
could
ev
en
tually
mak
e
quan
tum
computation
practical�
Moreo
v
er�
sev
eral
groups
of
exp
erimen
tal
ph
ysicists
around
the
w
orld
ha
v
e
b
egun
pro
jects
to
explore
exp
erimen
tally
the
basic
principles
of
quan
tum
computing�
The
next
question
to
address
w
as
whether
one
can
build
a
practically
successful
quan
tum
computer�
Could
quan
tum
computing
b
e
brough
t
from
a
visionary
stage
to
an
exp
erimen
tal
stage
�and
later
to
an
engineering
stage��
This
question
is
still
to
b
e
answ
ered�
An
in
tensiv
e
e�ort
to
deal
with
quan
tum
computer
design
problems
has
brough
t
some
remark
able
success�
but
also
rev
ealed
new
problems�
On
one
hand
success
came
in
an
unexp
ected
area�
Quan
tum
cryptograph
y�in
whic
h
one
tries
to
exploit
quan
tum
phenomena
��
to
transmit
quan
tum
information
in
suc
h
a
w
a
y
that
undetectable
ea
v
esdropping
is
imp
ossible�
has
already
reac
hed
an
exp
erimen
tal
stage�
There
has
also
b
een
success
in
the
e�ort
to
�nd
su�cien
tly
simple
rev
ersible
quan
tum
gates
that
could
b
e
used
to
build
p
oten
tial
quan
tum
computers�
The
classical
univ
ersal
rev
ersible
gates
ha
v
e
three
inputs
and
outputs�
Sleator
and
W
einfurter
�������
Barenco
������
and
DiVincenzo
������
ha
v
e
sho
wn
univ
ersal
t
w
o
bit
quan
tum
gates�
This
has
b
een
an
imp
ortan
t
result
b
ecause
the
problem
to
con
trol
in
teraction
of
three
particles
seems
to
b
e
m
uc
h
more
complex
than
for
the
case
of
t
w
o
particles�
In
addition�
Barenco
������
and
Llo
yd
������
ha
v
e
sho
wn
that
almost
an
y
quan
tum
t
w
o�bit
gate
is
univ
ersal�
These
results
greatly
simpli�ed
the
searc
h
for
ph
ysical
implemen
tations
of
quan
tum
computational
net
w
orks�
On
the
other
hand�
it
has
also
turned
out
that
the
�rst
mo
dels
of
quan
tum
computers
w
ere
o
v
ersimpli�ed
and
that
for
quan
tum
computing
to
come
to
an
exp
erimen
tal
or
ev
en
engineering
stage
man
y
fundamen
tal
problems
still
need
to
b
e
solv
ed�
The
necessit
y
of
examining
impacts
of
inaccuracies�
emissions
and
coupling
with
the
en
vironmen
t
of
an
y
realistic
device
on
the
capabilit
y
of
quan
tum
computing
to
meet
their
promises
has
long
b
een
emphasized
b
y
Landauer
�������
Esp
ecially
problems
decoherence
causes
made
man
y
to
b
eliev
e
that
it
is
in
principle
imp
ossible
to
design
reliably
enough
functioning
quan
tum
��
Ho
w
ev
er�
it
is
necessary
to
mak
e
clear
that
the
question
whether
quan
tum
computers
allo
w
one
to
obtain
essen
tially
more
computational
p
o
w
er
has
not
y
et
b
een
completely
satisfactorily
answ
ered�
��
Heisen
b
erg�s
uncertain
t
y
principle�see
Section
������
Jozef
Grusk
a�
QUANTUM
COMPUTING
��
computer�
��
The
situation
started
to
lo
ok
almost
hop
eless�
A
breakthrough
came
after
o
v
ercoming
another
in
tellectual
barrier�
it
w
as
realised
that
the
situation
is
not
as
bad
as
it
lo
oks
and
that
ph
ysics
do
es
not
need
to
rely
on
itself
only
in
the
searc
h
for
ho
w
to
o
v
ercome
problems
of
the
imp
erfections
of
op
erations�
emission
and
of
the
decoherence�
Mathematics
and
informatics
seem
to
b
e
able
to
help
signi�can
tly
�
The
�rst
imp
ortan
t
and
encouraging
result
w
as
due
to
Bernstein
and
V
azirani
�������
They
sho
w
ed
that
quite
w
eak
precision
requiremen
ts
are
su�cien
t
for
quan
tum
computing�only
logarithmic
precision
for
inputs
and
gates
is
needed�
Disco
v
ery
of
error�correcting
co
des
b
y
Shor
�������
and
so
on
b
y
man
y
others�
allo
w
ed
one
to
cop
e
with
decoherence
and
op
erational
imp
erfections
during
transmission
and
storage
of
quan
tum
information�
�In
b
ehind
there
w
as
a
k
ey
disco
v
ery
that
quan
tum
noise�errors�
in
principle
con
tin
uous�
can
b
e
view
ed
and
dealt
with
as
b
eing
discrete��
The
disco
v
ery
of
quan
tum
fault�toleran
t
computations
b
y
Shor
������
allo
w
ed
one
to
cop
e
with
decoherence
and
imprecisions
during
pro
cessing
of
quan
tum
information�
��
The
disco
v
ery
of
�concatenated
co
des�
�Knill
and
La�amme�
�����
and
�quan
tum
rep
eaters�
�Briegel�
������
allo
ws
one
to
cop
e
with
the
problem
of
storage
and
transmission
of
quan
tum
information
for
a
long
time
and
long
distance
with
desirable
reliabilit
y
�
Quan
tum
cryptograph
y
has
also
con
tributed
to
an
a
w
areness
that
quan
tum
computing
is
full
of
pitfalls�
not
fully
understo
o
d
y
et�
In
�����
Brassard�
Cr�p
eau�
Jozsa
and
Langlois
surprised
the
comm
unit
y
b
y
the
claim
�pro
of
�
that
a
quan
tum
bit
commitmen
t
proto
col
pro
v
ably
un
break
able
b
y
b
oth
parties
is
p
ossible�
It
to
ok
three
y
ears
to
�nd
out�
b
y
Lo
and
Chau
������
����a�
and
Ma
y
ers
�������
that
prop
osed
proto
cols
are�
in
principle�
insecure�
Another
in
tellectual
barrier
w
as
o
v
ercome
b
y
con
tributions
of
Cirac
and
Zoller
�������
They
sho
w
ed�
at
least
on
the
lab
oratory
lev
el�
that
in
the
searc
h
for
tec
hnology
to
build
quan
tum
pro
cessors
and
computers
one
do
es
not
need
to
w
ait
till
some
�unobtainium�
is
a
v
ailable�
but
that
one
can
start
with
the
existing
tec
hnologies
with
whic
h
there
are
already
ric
h
exp
erimen
tal
exp
eriences�
�Of
course�
this
is
not
the
whole
story
�
One
also
has
to
realize
that
ev
en
if
it
migh
t
b
e
p
ossible
to
build
small
quan
tum
computers�
scaling
up
to
mac
hines
large
enough
to
mak
e
really
imp
ortan
t
computations
could
presen
t
fundamen
tal
di�culties��
���
F
rom
Randomized
to
Quan
tum
Computation
A
comparison
of
probabilistic
T
uring
��
mac
hines
�PTM��
with
quan
tum
T
uring
mac
hines
�QTM�
will
allo
w
us
to
see�
in
an
easy
and
transparen
t
w
a
y
similarities
and
di�erences
��
P
essimism
that
tec
hnology
cannot
b
e
made
reliable
enough
to
realize
useful
computations
is
not
a
new
phenomenon
in
the
short
history
of
mo
dern
computers�
F
or
example�
in
the
autobiograph
y
of
K�
Zuse
�������
there
is
a
story
ab
out
sceptical
reactions
to
his
talk
in
����
in
whic
h
he
an
ticipated
�based
on
discussions
with
Sc
hrey
er�that
ab
out
����
tub
es
w
ould
b
e
needed
to
build
an
electronic
computer�
�A
t
that
time
the
biggest
electronic
devices
w
ere
broadcasting
stations
with
few
h
undreds
of
v
alv
es��
Similarly
�
the
idea
that
ENIA
C
with
its
�����
tub
es
could
w
ork
for
a
su�cien
tly
long
time
w
as
for
that
time
an
enginnering
phan
tasy
that
w
ould
hardly
get
through
a
gran
ting
agency
of
�p
eace
time��
��
Actually
Landauer�s
constan
t
c
hallenge
of
�visionaries�
to
sho
w
a
really
w
ork
able
path
to
the
future
has
b
een
of
immense
signi�cance
for
making
correct
researc
h
agenda
in
quan
tum
computing�
Quan
tum
computing
is
an
excellen
t
example
of
the
rapid
progress
in
science
and
tec
hnology
that
can
b
e
ac
hiev
ed
b
y
optimists
and
visionaries
if
they
closely
co
op
erate
with�
and
listen
to�
sceptics
and
p
essimists
directing
constructiv
ely
the
e�ort
of
visionaries
and
optimists
on
the
k
ey
problems
to
attac
k�
��
Alan
M�
T
uring
�����������
an
English
mathematician�
He
wrote
fundamen
tal
pap
ers
on
computabilit
y
and
arti�cial
in
telligence
and
in
v
en
ted
a
computation
mo
del
b
earing
his
name�
During
the
Second
W
orld
W
ar
T
uring
participated
in
the
cryptanalysis
pro
ject
UL
TRA
in
Bletc
hley
P
ark
and
in
the
design
of
the
�rst
p
o
w
erful
electronic
computer
Colossus�
After
the
w
ar
he
sup
ervised
the
design
and
building
of
A
CE�
a
large
electronic
digital
computer
at
the
National
Ph
ysical
Lab
oratory
�
��
Chapter
��
Elemen
ts
b
et
w
een
these
t
w
o
basic
mo
dels
of
classical
and
quan
tum
computing�
In
this
w
a
y
w
e
can
also
demonstrate
the
adv
an
tages
and
problems
quan
tum
computing
has�
There
are
go
o
d
reasons
to
start
our
in
tro
duction
to
quan
tum
computing
b
y
compar�
ing
probabilistic
and
quan
tum
T
uring
mac
hines�
Probabilistic
T
uring
mac
hines
represen
t
no
w
ada
ys
the
most
imp
ortan
t
mo
del
of
classical
computing�
P
olynomial
time
computation
on
probabilistic
T
uring
mac
hines
stands
for
a
formal
equiv
alen
t
of
�feasibilit
y�
in
classical
computing�
In
addition�
similarly
to
classical
T
uring
mac
hines�
quan
tum
T
uring
mac
hines
w
ere
historically
the
�rst
really
fully
quan
tum
and
p
o
w
erful
mo
del
of
quan
tum
computing�
�����
Probabilistic
T
uring
mac
hines
F
ormally
�
a
�one�tap
e�
probabilistic
T
uring
mac
hine�
on
a
�nite
set
Q
of
states
and
the
�nite
alphab
et
��
is
giv
en
b
y
a
transition
function
�
�
�
�
Q
�
�
�
Q
�
f��
��
�g
�
�
���
��
assigning
to
eac
h
p
ossible
transition
a
probabilit
y
in
suc
h
a
w
a
y
that
for
eac
h
con�guration
��
c
�
and
all
its
successor�con�gurations
c
�
�
�
�
�
�
c
k
�
the
follo
wing
lo
cal
probabilit
y
condition
is
satis�ed�
If
p
i
�
�
�
i
�
k
�
is
the
probabilit
y
�
assigned
b
y
�
�
of
the
transition
from
c
�
to
c
i
�
then
�see
Figure
���a��
k
X
i��
p
i
�
��
c
c
c
c
c
c
c
c
c
c
p
p
p
p
α
0
1
2
k-1
k
1
2
1
2
0
k-1
k
k-1
k
α
k-1
k
(a) PTM
(b) QTM
|α | + |α | = 1
p + p + .... + p + p = 1
2
1
1
k-1
k
2
k-1
k
α
α
2
2
2
.....+
2
|α | + |α |+
2
1
Figure
����
Lo
cal
probabilit
y
conditions
This
condition
is
often
written
in
the
follo
wing
form�
if
��
�
�
q
�
�
�
�
�
Q�
then
X
��
�q
�d����Q�f�����g
�
��
�
�
q
�
�
�
�
q
�
d�
�
��
��
A
con�guration
is
a
full
description
of
the
global
state
of
a
PTM�
It
can
b
e
seen
as
ha
ving
the
form
w
�
q
w
�
�
where
w
�
w
�
is
the
curren
t
con
ten
t
of
the
tap
e�
q
is
the
curren
t
state
and
the
curren
t
p
osition
of
the
head
of
the
PTM
is
on
the
cell
with
the
�rst
sym
b
ol
of
w
�
�
Jozef
Grusk
a�
QUANTUM
COMPUTING
��
On
the
base
of
the
transition
function
�
of
a
PTM
M
w
e
can
assign
probabilities
to
all
edges�
to
all
no
des
and
also
to
all
con�gurations
of
eac
h
lev
el
of
an
y
con�guration
tree
of
T
�
The
probabilit
y
assigned
to
an
edge
c
�
c
�
of
suc
h
a
tree
is
giv
en
directly
b
y
�
and
represen
ts
the
probabilit
y
that
computation
go
es�
in
one
step�
from
c
to
c
�
�
F
rom
that
w
e
can
assign
a
probabilit
y
to
eac
h
no
de
N
of
an
y
con�guration
tree�
see
Figure
���a�
as
the
pro
duct
of
all
probabilities
assigned
to
the
edges
on
the
path
from
the
ro
ot
to
N
�
�The
probabilit
y
assigned
to
the
ro
ot
is
de�ned
to
b
e
���
The
probabilit
y
assigned
to
an
arbitrary
no
de
N
is
therefore
the
probabilit
y
that
a
computation
starting
at
the
ro
ot
reac
hes
the
no
de
N
�
2
2
2
2
2
2
2
c
1
2
a
b
b
0.5
0.5
0.5
2
2
2
2
2
(c) invalid computation
2
2
0.5
2
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.25 0.25 0.25
0.25
(a) PTM
1
c
1
(b) QTM
c
d
0.5
0.5
-0.5
0.5
2
2
d
2
2
Figure
����
Con�guration
trees
with
probabilities
and
the
probabilit
y
amplitudes
It
ma
y
happ
en
that
at
a
certain
lev
el
of
a
con�guration
tree
there
are
sev
eral
o
ccurrences
c
���
�
�
�
�
�
c
�m�
of
the
same
con�guration
c�
see
Figure
���a�
In
suc
h
a
case�
if
p
i
is
the
c
c
c
p
p
α
α
α
p
1
2
c
c
c
2
(1)
(2)
1
(1)
(2)
m
m
(m)
(m)
(a) PTM
(b) QTM
Figure
����
Multiple
o
ccurrences
of
the
same
con�guration
probabilit
y
assigned
to
the
o
ccurrence
c
�i�
of
the
con�guration
c�
then
the
total
probabilit
y
