<s>
In	O
quantum	B-Architecture
computing	I-Architecture
,	O
quantum	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
QFA	O
)	O
or	O
quantum	O
state	B-Architecture
machines	I-Architecture
are	O
a	O
quantum	O
analog	O
of	O
probabilistic	B-General_Concept
automata	I-General_Concept
or	O
a	O
Markov	O
decision	O
process	O
.	O
</s>
<s>
They	O
provide	O
a	O
mathematical	O
abstraction	O
of	O
real-world	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
Quantum	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
can	O
also	O
be	O
understood	O
as	O
the	O
quantization	O
of	O
subshifts	O
of	O
finite	O
type	O
,	O
or	O
as	O
a	O
quantization	O
of	O
Markov	O
chains	O
.	O
</s>
<s>
QFAs	O
are	O
,	O
in	O
turn	O
,	O
special	O
cases	O
of	O
geometric	O
finite	B-Architecture
automata	I-Architecture
or	O
topological	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
The	O
automata	O
work	O
by	O
receiving	O
a	O
finite-length	O
string	O
of	O
letters	O
from	O
a	O
finite	O
alphabet	O
,	O
and	O
assigning	O
to	O
each	O
such	O
string	O
a	O
probability	O
indicating	O
the	O
probability	O
of	O
the	O
automaton	O
being	O
in	O
an	O
accept	B-Architecture
state	I-Architecture
;	O
that	O
is	O
,	O
indicating	O
whether	O
the	O
automaton	O
accepted	O
or	O
rejected	O
the	O
string	O
.	O
</s>
<s>
The	O
languages	O
accepted	O
by	O
QFAs	O
are	O
not	O
the	O
regular	B-General_Concept
languages	I-General_Concept
of	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
,	O
nor	O
are	O
they	O
the	O
stochastic	B-General_Concept
languages	I-General_Concept
of	O
probabilistic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
There	O
is	O
a	O
simple	O
,	O
intuitive	O
way	O
of	O
understanding	O
quantum	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
One	O
begins	O
with	O
a	O
graph-theoretic	O
interpretation	O
of	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
DFA	B-General_Concept
)	O
.	O
</s>
<s>
A	O
DFA	B-General_Concept
can	O
be	O
represented	O
as	O
a	O
directed	O
graph	O
,	O
with	O
states	O
as	O
nodes	O
in	O
the	O
graph	O
,	O
and	O
arrows	O
representing	O
state	B-Application
transitions	I-Application
.	O
</s>
<s>
One	O
way	O
of	O
representing	O
such	O
a	O
graph	O
is	O
by	O
means	O
of	O
a	O
set	O
of	O
adjacency	B-Algorithm
matrices	I-Algorithm
,	O
with	O
one	O
matrix	O
for	O
each	O
input	O
symbol	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
list	O
of	O
possible	O
DFA	B-General_Concept
states	O
is	O
written	O
as	O
a	O
column	O
vector	O
.	O
</s>
<s>
For	O
a	O
given	O
input	O
symbol	O
,	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
indicates	O
how	O
any	O
given	O
state	O
(	O
row	O
in	O
the	O
state	O
vector	O
)	O
will	O
transition	O
to	O
the	O
next	O
state	O
;	O
a	O
state	B-Application
transition	I-Application
is	O
given	O
by	O
matrix	O
multiplication	O
.	O
</s>
<s>
One	O
needs	O
a	O
distinct	O
adjacency	B-Algorithm
matrix	I-Algorithm
for	O
each	O
possible	O
input	O
symbol	O
,	O
since	O
each	O
input	O
symbol	O
can	O
result	O
in	O
a	O
different	O
transition	O
.	O
</s>
<s>
The	O
entries	O
in	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
must	O
be	O
zero	O
's	O
and	O
one	O
's	O
.	O
</s>
<s>
For	O
any	O
given	O
column	O
in	O
the	O
matrix	O
,	O
only	O
one	O
entry	O
can	O
be	O
non-zero	O
:	O
this	O
is	O
the	O
entry	O
that	O
indicates	O
the	O
next	O
(	O
unique	O
)	O
state	B-Application
transition	I-Application
.	O
</s>
<s>
For	O
a	O
given	O
input	O
symbol	O
,	O
write	O
as	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
that	O
describes	O
the	O
evolution	O
of	O
the	O
DFA	B-General_Concept
to	O
its	O
next	O
state	O
.	O
</s>
<s>
The	O
set	O
then	O
completely	O
describes	O
the	O
state	B-Architecture
transition	I-Architecture
function	I-Architecture
of	O
the	O
DFA	B-General_Concept
.	O
</s>
<s>
Let	O
Q	O
represent	O
the	O
set	O
of	O
possible	O
states	O
of	O
the	O
DFA	B-General_Concept
.	O
</s>
<s>
Then	O
,	O
after	O
reading	O
input	O
symbols	O
from	O
the	O
input	O
tape	O
,	O
the	O
state	O
of	O
the	O
DFA	B-General_Concept
will	O
be	O
given	O
by	O
The	O
state	B-Application
transitions	I-Application
are	O
given	O
by	O
ordinary	O
matrix	O
multiplication	O
(	O
that	O
is	O
,	O
multiply	O
q0	O
by	O
,	O
etc	O
.	O
</s>
<s>
The	O
above	O
description	O
of	O
a	O
DFA	B-General_Concept
,	O
in	O
terms	O
of	O
linear	B-Architecture
operators	I-Architecture
and	O
vectors	O
,	O
almost	O
begs	O
for	O
generalization	O
,	O
by	O
replacing	O
the	O
state-vector	O
q	O
by	O
some	O
general	O
vector	O
,	O
and	O
the	O
matrices	O
by	O
some	O
general	O
operators	O
.	O
</s>
<s>
This	O
is	O
essentially	O
what	O
a	O
QFA	O
does	O
:	O
it	O
replaces	O
q	O
by	O
a	O
probability	O
amplitude	O
,	O
and	O
the	O
by	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Other	O
,	O
similar	O
generalizations	O
also	O
become	O
obvious	O
:	O
the	O
vector	O
q	O
can	O
be	O
some	O
distribution	O
on	O
a	O
manifold	B-Architecture
;	O
the	O
set	O
of	O
transition	B-Algorithm
matrices	I-Algorithm
become	O
automorphisms	O
of	O
the	O
manifold	B-Architecture
;	O
this	O
defines	O
a	O
topological	O
finite	B-Architecture
automaton	I-Architecture
.	O
</s>
<s>
Similarly	O
,	O
the	O
matrices	O
could	O
be	O
taken	O
as	O
automorphisms	O
of	O
a	O
homogeneous	O
space	O
;	O
this	O
defines	O
a	O
geometric	O
finite	B-Architecture
automaton	I-Architecture
.	O
</s>
<s>
The	O
first	O
is	O
the	O
non-deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	O
)	O
.	O
</s>
<s>
Likewise	O
,	O
the	O
state	B-Application
transition	I-Application
matrices	O
are	O
defined	O
in	O
such	O
a	O
way	O
that	O
a	O
given	O
column	O
can	O
have	O
several	O
non-zero	O
entries	O
in	O
it	O
.	O
</s>
<s>
A	O
well-known	O
theorem	O
states	O
that	O
,	O
for	O
each	O
DFA	B-General_Concept
,	O
there	O
is	O
an	O
equivalent	O
NFA	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
This	O
implies	O
that	O
the	O
set	O
of	O
languages	O
that	O
can	O
be	O
recognized	O
by	O
DFA	B-General_Concept
's	O
and	O
NFA	O
's	O
are	O
the	O
same	O
;	O
these	O
are	O
the	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Another	O
generalization	O
that	O
should	O
be	O
immediately	O
apparent	O
is	O
to	O
use	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
for	O
the	O
transition	B-Algorithm
matrices	I-Algorithm
,	O
and	O
a	O
probability	O
vector	O
for	O
the	O
state	O
;	O
this	O
gives	O
a	O
probabilistic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
The	O
transition	B-Algorithm
matrices	I-Algorithm
must	O
preserve	O
this	O
property	O
:	O
this	O
is	O
why	O
they	O
must	O
be	O
stochastic	O
.	O
</s>
<s>
Each	O
state	O
vector	O
should	O
be	O
imagined	O
as	O
specifying	O
a	O
point	O
in	O
a	O
simplex	O
;	O
thus	O
,	O
this	O
is	O
a	O
topological	B-General_Concept
automaton	I-General_Concept
,	O
with	O
the	O
simplex	O
being	O
the	O
manifold	B-Architecture
,	O
and	O
the	O
stochastic	B-Algorithm
matrices	I-Algorithm
being	O
linear	O
automorphisms	O
of	O
the	O
simplex	O
onto	O
itself	O
.	O
</s>
<s>
By	O
contrast	O
,	O
in	O
a	O
QFA	O
,	O
the	O
manifold	B-Architecture
is	O
complex	O
projective	O
space	O
,	O
and	O
the	O
transition	B-Algorithm
matrices	I-Algorithm
are	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Each	O
point	O
in	O
corresponds	O
to	O
a	O
quantum-mechanical	O
probability	O
amplitude	O
or	O
pure	O
state	O
;	O
the	O
unitary	B-Algorithm
matrices	I-Algorithm
can	O
be	O
thought	O
of	O
as	O
governing	O
the	O
time	O
evolution	O
of	O
the	O
system	O
(	O
viz	O
in	O
the	O
Schrödinger	O
picture	O
)	O
.	O
</s>
<s>
A	O
worthy	O
point	O
to	O
contemplate	O
is	O
the	O
distributions	O
that	O
result	O
on	O
the	O
manifold	B-Architecture
during	O
the	O
input	O
of	O
a	O
language	O
.	O
</s>
<s>
Put	O
in	O
other	O
words	O
,	O
the	O
machine	O
learning	O
methods	O
used	O
to	O
train	O
hidden	O
Markov	O
models	O
generalize	O
to	O
QFAs	O
as	O
well	O
:	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
and	O
the	O
forward-backward	B-Algorithm
algorithm	I-Algorithm
generalize	O
readily	O
to	O
the	O
QFA	O
.	O
</s>
<s>
As	O
with	O
an	O
ordinary	O
finite	B-Architecture
automaton	I-Architecture
,	O
the	O
quantum	O
automaton	O
is	O
considered	O
to	O
have	O
possible	O
internal	O
states	O
,	O
represented	O
in	O
this	O
case	O
by	O
an	O
-state	O
qubit	O
.	O
</s>
<s>
The	O
state	B-Application
transitions	I-Application
,	O
transition	B-Algorithm
matrices	I-Algorithm
or	O
de	O
Bruijn	O
graphs	O
are	O
represented	O
by	O
a	O
collection	O
of	O
unitary	B-Algorithm
matrices	I-Algorithm
,	O
with	O
one	O
unitary	B-Algorithm
matrix	I-Algorithm
for	O
each	O
letter	O
.	O
</s>
<s>
That	O
is	O
,	O
given	O
an	O
input	O
letter	O
,	O
the	O
unitary	B-Algorithm
matrix	I-Algorithm
describes	O
the	O
transition	O
of	O
the	O
automaton	O
from	O
its	O
current	O
state	O
to	O
its	O
next	O
state	O
:	O
</s>
<s>
Because	O
the	O
left-action	O
of	O
on	O
reverses	O
the	O
order	O
of	O
the	O
letters	O
in	O
the	O
string	O
,	O
it	O
is	O
not	O
uncommon	O
for	O
QFAs	O
to	O
be	O
defined	O
using	O
a	O
right	O
action	O
on	O
the	O
Hermitian	B-Algorithm
transpose	I-Algorithm
states	O
,	O
simply	O
in	O
order	O
to	O
keep	O
the	O
order	O
of	O
the	O
letters	O
the	O
same	O
.	O
</s>
<s>
A	O
regular	B-General_Concept
language	I-General_Concept
is	O
accepted	O
with	O
probability	O
by	O
a	O
quantum	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
,	O
if	O
,	O
for	O
all	O
sentences	O
in	O
the	O
language	O
,	O
(	O
and	O
a	O
given	O
,	O
fixed	O
initial	O
state	O
)	O
,	O
one	O
has	O
.	O
</s>
<s>
As	O
should	O
be	O
readily	O
apparent	O
,	O
if	O
the	O
initial	O
state	O
is	O
the	O
pure	O
state	O
or	O
,	O
then	O
the	O
result	O
of	O
running	O
the	O
machine	O
will	O
be	O
exactly	O
identical	O
to	O
the	O
classical	O
deterministic	B-General_Concept
finite	I-General_Concept
state	I-General_Concept
machine	I-General_Concept
.	O
</s>
<s>
In	O
particular	O
,	O
there	O
is	O
a	O
language	O
accepted	O
by	O
this	O
automaton	O
with	O
probability	O
one	O
,	O
for	O
these	O
initial	O
states	O
,	O
and	O
it	O
is	O
identical	O
to	O
the	O
regular	B-General_Concept
language	I-General_Concept
for	O
the	O
classical	O
DFA	B-General_Concept
,	O
and	O
is	O
given	O
by	O
the	O
regular	B-Language
expression	I-Language
:	O
</s>
<s>
More	O
subtle	O
behaviour	O
occurs	O
when	O
the	O
matrices	O
and	O
are	O
not	O
so	O
simple	O
;	O
see	O
,	O
for	O
example	O
,	O
the	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
as	O
an	O
example	O
of	O
a	O
quantum	B-General_Concept
finite	I-General_Concept
state	I-General_Concept
machine	I-General_Concept
acting	O
on	O
the	O
set	O
of	O
all	O
possible	O
finite	O
binary	O
strings	O
.	O
</s>
<s>
As	O
of	O
2019	O
,	O
most	O
quantum	B-Architecture
computers	I-Architecture
are	O
implementations	O
of	O
measure-once	O
quantum	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
,	O
and	O
the	O
software	O
systems	O
for	O
programming	O
them	O
expose	O
the	O
state-preparation	O
of	O
,	O
measurement	O
and	O
a	O
choice	O
of	O
unitary	O
transformations	O
,	O
such	O
the	O
controlled	O
NOT	O
gate	O
,	O
the	O
Hadamard	B-Algorithm
transform	I-Algorithm
and	O
other	O
quantum	O
logic	O
gates	O
,	O
directly	O
to	O
the	O
programmer	O
.	O
</s>
<s>
The	O
primary	O
difference	O
between	O
real-world	O
quantum	B-Architecture
computers	I-Architecture
and	O
the	O
theoretical	O
framework	O
presented	O
above	O
is	O
that	O
the	O
initial	O
state	O
preparation	O
cannot	O
ever	O
result	O
in	O
a	O
point-like	O
pure	O
state	O
,	O
nor	O
can	O
the	O
unitary	O
operators	O
be	O
precisely	O
applied	O
.	O
</s>
<s>
There	O
is	O
no	O
quantum	O
analog	O
to	O
the	O
push-down	B-Application
automaton	I-Application
or	O
stack	B-Application
machine	I-Application
.	O
</s>
<s>
The	O
above	O
constructions	O
indicate	O
how	O
the	O
concept	O
of	O
a	O
quantum	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
can	O
be	O
generalized	O
to	O
arbitrary	O
topological	O
spaces	O
.	O
</s>
<s>
In	O
place	O
of	O
the	O
unitary	B-Algorithm
matrices	I-Algorithm
,	O
one	O
uses	O
the	O
isometries	O
of	O
the	O
Riemannian	O
manifold	B-Architecture
,	O
or	O
,	O
more	O
generally	O
,	O
some	O
set	O
of	O
open	O
functions	O
appropriate	O
for	O
the	O
given	O
topological	O
space	O
.	O
</s>
<s>
The	O
set	O
of	O
accept	B-Architecture
states	I-Architecture
can	O
be	O
taken	O
to	O
be	O
some	O
arbitrary	O
subset	O
of	O
the	O
topological	O
space	O
.	O
</s>
<s>
One	O
then	O
says	O
that	O
a	O
formal	O
language	O
is	O
accepted	O
by	O
this	O
topological	B-General_Concept
automaton	I-General_Concept
if	O
the	O
point	O
,	O
after	O
iteration	O
by	O
the	O
homeomorphisms	O
,	O
intersects	O
the	O
accept	O
set	O
.	O
</s>
<s>
The	O
behaviour	O
of	O
topological	B-General_Concept
automata	I-General_Concept
is	O
studied	O
in	O
the	O
field	O
of	O
topological	O
dynamics	O
.	O
</s>
<s>
The	O
quantum	O
automaton	O
differs	O
from	O
the	O
topological	B-General_Concept
automaton	I-General_Concept
in	O
that	O
,	O
instead	O
of	O
having	O
a	O
binary	O
result	O
(	O
is	O
the	O
iterated	O
point	O
in	O
,	O
or	O
not	O
in	O
,	O
the	O
final	O
set	O
?	O
</s>
<s>
Thus	O
,	O
it	O
follows	O
that	O
the	O
quantum	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
is	O
just	O
a	O
special	O
case	O
of	O
a	O
geometric	B-General_Concept
automaton	I-General_Concept
or	O
a	O
metric	B-General_Concept
automaton	I-General_Concept
 '	O
,	O
where	O
is	O
generalized	O
to	O
some	O
metric	O
space	O
,	O
and	O
the	O
probability	O
measure	O
is	O
replaced	O
by	O
a	O
simple	O
function	O
of	O
the	O
metric	O
on	O
that	O
space	O
.	O
</s>
